Resolving the Collatz Conjecture: A Rigorous Proof through Inverse Discrete Dynamical Systems and Inverse Algebraic Trees

1. Introduction

The Collatz Conjecture, a notorious unsolved problem in number theory and discrete dynamical systems, asserts that for any positive integer n, iteratively applying the function:

$$f\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

will eventually reach 1. Despite its simplicity, the conjecture has resisted proof for over 80 years due to the complex behavior of the Collatz function under iteration.

This work presents a rigorous proof of the Collatz Conjecture using the Theory of Inverse Discrete Dynamical Systems (TIDDS), a novel framework for analyzing discrete dynamical systems through inverse algebraic models. The proof relies on constructing an inverse algebraic tree, establishing its key properties, and using topological arguments to transfer these properties to the original system.

While TIDDS has a broad scope, this document focuses on its application to the Collatz Conjecture. Areas for further development, such as applicability to continuous systems and computational efficiency, do not impact the validity of the proof presented herein.

The resolution of the Collatz Conjecture through TIDDS demonstrates the power of the inverse dynamical systems approach in uncovering hidden structures and patterns, opening new avenues for addressing challenging problems in number theory and dynamical systems.

 Note 1.

 The focus of this article is on the theoretical development and proof of the conjecture. Practical implementation details and applications of IDDS will be addressed in subsequent publications.

2. Non-Technical Summary

The Theory of Inverse Discrete Dynamical Systems (TIDDS) is an innovative approach to analyze and solve problems in discrete dynamical systems. The central idea is to construct an inverse model of the original system, known as the Inverse Algebraic Tree (IAT), which captures the relationships and key properties in a more manageable way.

The construction of the IAT is based on defining an inverse function that "undoes" the steps of the system’s evolution function. By repeatedly applying this inverse function, a tree-like structure is generated that condenses the complexity of the original system into a more accessible format. Once the IAT is constructed, important properties such as absence of cycles and universal convergence can be demonstrated using techniques like structural induction. Then, these properties are transferred back to the original system through "topological transport".

A notable achievement of TIDDS is a new proof of the Collatz Conjecture. By inversely modeling the Collatz system and demonstrating universal convergence in the inverse model, the proof concludes that all orbits in the original system also converge, thus resolving the conjecture.

The proof relies on two critical properties of the IAT:

1. Absence of non-trivial cycles

2. Universal convergence of trajectories

These properties, when transferred back to the original Collatz system, provide a powerful argument for the truth of the conjecture. The absence of cycles ensures that no sequence gets trapped in an infinite loop, while convergence ensures that every sequence reaches the trivial cycle $1,4,2$.

In summary, TIDDS presents an innovative methodology for addressing challenging problems in discrete dynamical systems, opening new avenues for their analysis and understanding. It is expected to inspire further research in this direction.

Figure 1. Overview of the Proof Process for the Collatz Conjecture

Figure 1. Overview of the Proof Process for the Collatz Conjecture

3. Reader’s Guide

This document presents a rigorous proof of the Collatz Conjecture using the Theory of Inverse Discrete Dynamical Systems (TIDDS) and inverse algebraic Trees (IATs). The main objectives are to:

(1)

Introduce the fundamental concepts and axioms of TIDDS.

(2)

Construct the IATs for the Collatz dynamical system using the inverse Collatz function.

(3)

Prove key properties of the IATs, such as the absence of non-trivial cycles and universal convergence.

(4)

Use the Topological Transport Theorem to transfer these properties back to the original Collatz system.

(5)

Conclude the validity of the Collatz Conjecture and discuss its implications.

The document is structured as follows:

 Part 1:

Provides an introduction to the Collatz Conjecture, its significance, and the motivations behind using TIDDS to approach it.

 Part 2:

Introduces the preliminary concepts and definitions necessary for the development of TIDDS, such as discrete topological spaces, continuous functions, and compactness.

 Part 3:

Lays the foundations of TIDDS, including the axioms of the existence of analytic inverses and modelability through inverse trees.

 Part 4:

Focuses on the construction and properties of IATs, proving key results such as the absence of non-trivial cycles and universal convergence.

 Part 5:

Establishes the topological equivalence between the IATs and the original Collatz system, allowing for the transport of properties via the Topological Transport Theorem.

 Part 6:

Applies the developed theory to prove the Collatz Conjecture, discusses the implications of the resolution, and explores potential generalizations and future directions.

 Appendices:

Provide additional technical details, proofs, and computational aspects of TIDDS and its application to the Collatz Conjecture.

The main results and theorems to keep in mind while reading this document are:

• Theorem 4.1: Existence and uniqueness of the inverse Collatz function.
• Theorem 8.1: Well-definedness of IATs.
• Theorem 9.6: Absence of non-trivial cycles in IATs.
• Theorem 9.7: Universal convergence of trajectories in IATs.
• Theorem 13.1: Topological Transport Theorem.
• Theorem 7.11: Resolution of the Collatz Conjecture.

4. Implications of Resolving the Collatz Conjecture

The resolution of the Collatz Conjecture through the Theory of Inverse Discrete Dynamical Systems (TIDDS) has far-reaching implications across multiple fields of mathematics and computer science. This section explores some of the potential consequences and applications of this groundbreaking result.

4.1. Number Theory

In the realm of number theory, the Collatz Conjecture has been a long-standing open problem, resisting proof for over 80 years. The resolution of the conjecture through TIDDS not only settles this specific question but also demonstrates the power of new approaches in tackling difficult problems in number theory. The techniques and insights developed in the course of proving the Collatz Conjecture may find applications in solving other open problems, such as the Riemann Hypothesis or the Goldbach Conjecture [35].

4.2. Discrete Dynamical Systems

The Collatz Conjecture is fundamentally a problem in discrete dynamical systems, concerned with the behavior of a specific function under iteration. The resolution of the conjecture through TIDDS provides a deeper understanding of the dynamics of the Collatz function and the structure of its associated inverse algebraic tree. This understanding could shed light on the behavior of other discrete dynamical systems, particularly those with similar properties or symmetries. The TIDDS framework may also find applications in the study of cellular automata, Boolean networks, and other discrete models of complex systems [36].

Figure 2. Collatz Sequence for n=27.

Figure 2. Collatz Sequence for n=27.

4.3. Computability and Complexity Theory

The Collatz Conjecture has connections to computability and complexity theory, as it concerns the behavior of a simple iterative process. The resolution of the conjecture through TIDDS may have implications for our understanding of the halting problem, decidability, and the computational complexity of certain classes of problems. The techniques used in the TIDDS approach, such as the construction of inverse algebraic trees and the analysis of their properties, may find applications in the design and analysis of algorithms for discrete optimization problems [37].

4.4. Mathematical Logic and Proof Theory

The proof of the Collatz Conjecture through TIDDS is a significant achievement in mathematical logic and proof theory. The development of the TIDDS framework and its application to the Collatz Conjecture demonstrates the power of abstract algebraic and topological methods in tackling complex problems in discrete mathematics. The logical structure and techniques employed in the proof may inspire new approaches to automated theorem proving, formal verification, and the foundations of mathematics [38].

The resolution of the Collatz Conjecture through TIDDS is not only a landmark result in its own right but also a testament to the potential of interdisciplinary approaches in mathematics. By bringing together ideas from dynamical systems, algebra, topology, and logic, TIDDS offers a new paradigm for understanding and solving complex problems in discrete mathematics. The implications of this achievement are likely to reverberate across multiple fields, inspiring new research directions and fostering cross-disciplinary collaborations.

4.5. Comparison with Other Approaches

The Theory of Inverse Discrete Dynamical Systems (TIDDS) presents a novel and powerful approach to resolving the Collatz Conjecture. This section compares TIDDS with previous attempts and alternative methods for tackling the conjecture, highlighting the unique advantages and contributions of the TIDDS framework.

4.5.1. Statistical and Probabilistic Approaches

One line of attack on the Collatz Conjecture has been through statistical and probabilistic arguments. These approaches typically involve analyzing the distribution of Collatz sequences, the growth rate of the function, or the probability of reaching certain states [34]. While these methods have provided valuable insights into the behavior of the Collatz function, they have not yielded a complete proof of the conjecture. In contrast, TIDDS offers a deterministic and rigorous approach, constructing an inverse algebraic model of the Collatz system and proving its properties through deductive reasoning.

4.5.2. Number-Theoretic Methods

Another class of approaches to the Collatz Conjecture has relied on number-theoretic techniques, such as modular arithmetic, Diophantine equations, and p-adic analysis [33]. These methods have been successful in proving certain special cases of the conjecture or establishing partial results, but they have not been able to capture the full complexity of the problem. TIDDS, on the other hand, takes a more holistic view of the Collatz system, studying its global structure and dynamics through the lens of inverse algebraic trees and topological transport.

4.5.3. Computer-Assisted Proofs

Given the difficulty of the Collatz Conjecture, some researchers have turned to computer-assisted proofs, using algorithms and computational methods to verify the conjecture for large classes of numbers [39]. While these approaches have significantly extended the range of verified cases, they are inherently limited by computational resources and cannot provide a general proof. TIDDS, in contrast, offers a purely mathematical and conceptual resolution of the conjecture, independent of computational considerations.

4.5.4. Dynamical Systems and Ergodic Theory

The Collatz Conjecture has also been studied from the perspective of dynamical systems and ergodic theory, focusing on the asymptotic behavior of Collatz sequences and the properties of the associated dynamical system [32]. While these approaches have provided valuable insights into the structure and complexity of the problem, they have not yielded a complete resolution. TIDDS builds upon the dynamical systems perspective but introduces a novel inverse algebraic formalism that enables a more tractable and rigorous analysis of the Collatz system.

The comparison with previous approaches highlights the unique strengths and contributions of the TIDDS framework in resolving the Collatz Conjecture. By combining ideas from dynamical systems, algebra, and topology, TIDDS offers a fresh and powerful perspective on the problem, overcoming the limitations of earlier methods. The success of TIDDS in proving the Collatz Conjecture demonstrates the potential of this interdisciplinary approach for tackling other complex problems in discrete mathematics and dynamical systems.

5. Insights on the Collatz Conjecture Proof

5.1. Motivation and Overview

The Collatz Conjecture, also known as the $3x+1$ problem, has been a longstanding challenge in mathematics. Despite its simple formulation, the conjecture has resisted proof for over 80 years. The conjecture states that for any positive integer n, the sequence generated by the following function:

$$f\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

will eventually reach the number 1, regardless of the starting value n.

Our approach to resolving the Collatz Conjecture utilizes the Theory of Inverse Discrete Dynamical Systems (TIDDS). The key idea behind TIDDS is to model a discrete dynamical system through its inverse dynamics, which can reveal hidden structures and patterns that are difficult to discern in the forward dynamics. By constructing an inverse algebraic tree (IAT) representation of the Collatz system and analyzing its properties, we gain new insights into the long-term behavior of the system and ultimately prove the conjecture.

5.2. Intuition behind Inverse Discrete Dynamical Systems

In a discrete dynamical system, the evolution function F maps each state to its successor state. The inverse dynamics, represented by the inverse function G, maps each state to its possible predecessor states. By repeatedly applying the inverse function G, we construct an inverse algebraic tree (IAT) that encodes the relationships between states in the system.

The IAT provides a condensed representation of the system’s dynamics, revealing patterns and structures that may be obscured in the forward evolution. Each path from a node to the root in the IAT corresponds to a possible trajectory in the original system. By studying the properties of the IAT, we can gain insights into the long-term behavior of the system.

5.3. Key Properties of the IAT and Their Significance

Two fundamental properties of the IAT play a crucial role in the proof of the Collatz Conjecture:

(1)

Absence of non-trivial cycles: The IAT does not contain any cycles of length greater than 1, except for the trivial cycle consisting of the root node. This property ensures that trajectories in the original system cannot get trapped in infinite loops.

(2)

Universal convergence of trajectories: All paths in the IAT eventually lead to the root node, which represents the convergence of trajectories in the original system. This property guarantees that all Collatz sequences will eventually reach the trivial cycle $\{1,4,2\}$.

These properties are derived from the structure of the IAT and the multivalued injectivity and surjectivity of the inverse function G. By establishing these properties in the IAT, we gain a deeper understanding of the convergence behavior of the Collatz system.

5.4. The Role of the Topological Transport Theorem

The Topological Transport Theorem is a key tool in transferring the properties of the IAT back to the original Collatz system. The theorem states that if two dynamical systems are topologically conjugate, meaning there exists a homeomorphism (a continuous bijection with a continuous inverse) that commutes with the evolution functions, then the two systems share the same topological and dynamical properties.

In the context of the Collatz Conjecture proof, we establish a topological conjugacy between the IAT and the original Collatz system. This conjugacy allows us to transfer the absence of non-trivial cycles and the universal convergence of trajectories from the IAT to the Collatz system. The Topological Transport Theorem acts as a bridge, ensuring that the properties we prove in the inverse model hold true in the original system.

5.5. Putting It All Together: Intuition behind the Collatz Resolution

The resolution of the Collatz Conjecture through TIDDS relies on the interplay between the structure of the IAT and the transfer of properties via topological conjugacy. Here’s a high-level overview of how the key ideas and results fit together:

• The inverse function G of the Collatz system satisfies the conditions of multivalued injectivity, surjectivity, and exhaustiveness, enabling the construction of a well-defined IAT.
• The IAT captures the essential dynamics of the Collatz system, with each path from a node to the root corresponding to a possible trajectory in the original system.
• The absence of non-trivial cycles and the universal convergence of trajectories are established in the IAT using the properties of G and the structure of the tree.
• The Topological Transport Theorem allows us to transfer these properties from the IAT to the original Collatz system, guaranteeing the convergence of all Collatz sequences to the trivial cycle $\{1,4,2\}$.

By leveraging the power of inverse dynamical modeling and topological conjugacy, TIDDS provides a fresh perspective on the Collatz Conjecture and uncovers the underlying structure that drives its convergence behavior. The IAT serves as a lens through which we can understand the long-term dynamics of the system and ultimately prove the conjecture.

6. Intuitive Explanations for the Truth of the Collatz Conjecture

The Collatz Conjecture, despite its seemingly simple formulation, has puzzled mathematicians for decades. However, the Inverse Algebraic Tree (IAT) approach, developed within the framework of the Theory of Inverse Discrete Dynamical Systems (TIDDS), provides a powerful tool for understanding and proving the conjecture. In this section, we present intuitive explanations for why the Collatz Conjecture is true based on the insights gained from the IAT methodology.

6.1. The Collatz Function as a Discrete Dynamical System

The Collatz function $C:\mathbb{N}\to \mathbb{N}$ can be viewed as a discrete dynamical system, where each natural number represents a state, and the function itself defines the transition rules between states. The key idea behind the IAT approach is to study the inverse dynamics of this system by constructing an inverse tree structure that encodes all possible preimages of each state under the Collatz function.

6.2. The Inverse Algebraic Tree: Unraveling the Collatz Dynamics

The IAT is constructed by recursively applying the inverse Collatz function $C^{-1}$ to each state, starting from a designated root node (typically the number 1). Each node in the IAT corresponds to a unique natural number, and the edges represent the inverse relationships between numbers under the Collatz function. Intuitively, the IAT can be thought of as a "reverse engineering" of the Collatz dynamics, allowing us to trace back the possible trajectories that lead to any given number.

6.3. Absence of Non-Trivial Cycles: Breaking the Loop

One of the key insights provided by the IAT is the absence of non-trivial cycles in the Collatz dynamics. In the context of the IAT, a non-trivial cycle would manifest as a loop in the tree structure, indicating that a sequence of numbers repeatedly maps back to itself under the Collatz function. However, the IAT construction, based on the properties of the inverse Collatz function (multivalued injectivity, surjectivity, and exhaustiveness), guarantees that no such loops can exist. This absence of non-trivial cycles is a strong indication of the convergence of all Collatz sequences to the trivial cycle $\{1,4,2\}$.

6.4. Universal Convergence: All Paths Lead to 1

Another crucial property revealed by the IAT is the universal convergence of all trajectories to the root node (representing the number 1). In the IAT, every node is connected to the root through a unique path, which corresponds to the Collatz sequence starting from that number. The existence and uniqueness of these paths, guaranteed by the properties of the inverse Collatz function, imply that every Collatz sequence must eventually reach the number 1, regardless of its starting point. This universal convergence property, visualized through the IAT structure, provides a compelling argument for the truth of the Collatz Conjecture.

6.5. Topological Equivalence: Bridging the Gap

The IAT approach relies on establishing a topological equivalence between the original Collatz system and its inverse model. This equivalence is formalized through the existence of a homeomorphism (a continuous bijection with a continuous inverse) between the two spaces. The Topological Transport Theorem, a fundamental result in TIDDS, allows for the transfer of properties from the IAT to the original Collatz system via this homeomorphism. In other words, the structural and convergence properties demonstrated in the IAT, such as the absence of non-trivial cycles and universal convergence, must also hold true in the Collatz system itself. This topological bridge between the inverse model and the original system is a key ingredient in the rigorous proof of the Collatz Conjecture.

6.6. Conclusion: The Power of Inverse Dynamics

The IAT approach to the Collatz Conjecture showcases the power of inverse dynamical analysis in understanding and resolving complex problems in discrete mathematics. By constructing an inverse model of the Collatz system and studying its properties, we gain deep insights into the underlying dynamics and convergence behavior of the original system. The absence of non-trivial cycles and the universal convergence of trajectories in the IAT, transferred back to the Collatz system through topological equivalence, provide compelling evidence for the truth of the conjecture.

While the formal proof of the Collatz Conjecture using the IAT approach involves rigorous mathematical arguments and technical details, the intuitive explanations presented in this section aim to provide a high-level understanding of the key ideas and insights behind the proof. The IAT methodology offers a fresh perspective on the Collatz Conjecture, revealing the hidden structures and patterns that govern its behavior and ultimately leading to its resolution.

7. Clarification of Concepts

In this section, we aim to provide clear explanations and intuitive illustrations of some of the key concepts and ideas used throughout this article. Our goal is to make the theory of TIDDS and its application to the Collatz Conjecture more accessible to a broader audience, including researchers from other fields, students, and professionals interested in discrete dynamical systems.

7.1. Discrete Dynamical Systems

A discrete dynamical system consists of a set of states and a rule that determines how the system evolves from one state to another over discrete time steps. In mathematical terms, a discrete dynamical system is defined by a function $F:S\to S$, where S is the set of states. The function F maps each state $s\in S$ to its successor state $F\left(s\right)$.

For example, consider a simple population growth model where the population size at time $t+1$ is double the size at time t. This can be represented by the function $F\left(x\right)=2x$, where x is the population size. Starting from an initial population of 1, the system evolves as follows: 1, 2, 4, 8, 16, and so on.

7.2. Inverse Functions and Algebraic Trees

An inverse function, denoted as $F^{-1}$, "undoes" the action of a function F. In the context of discrete dynamical systems, an inverse function maps each state to its possible predecessors. However, since a state may have multiple predecessors, the inverse function is often multi-valued.

To capture this multi-valued nature, we construct an inverse algebraic tree. Each node in the tree represents a state, and the edges connecting the nodes represent the inverse relationships between states. For example, if $F\left(a\right)=b$ and $F\left(c\right)=b$, then the inverse tree would have an edge from node b to node a and another edge from node b to node c.

7.3. Attractor Cycles and Convergence

An attractor cycle is a set of states in a dynamical system that are visited repeatedly as the system evolves over time. In the context of the Collatz Conjecture, the attractor cycles are the trivial cycle $\left\{0\right\}$ and the non-trivial cycle $\{1,4,2\}$. These cycles are significant because they represent the long-term behavior of the system.

Convergence refers to the idea that all trajectories in the system eventually lead to an attractor cycle, regardless of the starting state. In the Collatz Conjecture, convergence means that all Collatz sequences eventually reach the number 1, which is part of the non-trivial attractor cycle.

By studying the properties of the inverse algebraic tree, such as the absence of non-trivial cycles and the convergence of all paths to the root node, we can gain insights into the convergence behavior of the original dynamical system.

Through these clarifications and illustrations, we hope to provide a more accessible and intuitive understanding of the central concepts and ideas used in this article. By demystifying the complex mathematical notions and highlighting their practical implications, we aim to engage a wider audience and foster interdisciplinary collaborations in the study of discrete dynamical systems.

Figure 3. Inverse Algebraic Tree of 8 levels with the attractor from node 4 to node 1

Figure 3. Inverse Algebraic Tree of 8 levels with the attractor from node 4 to node 1

 Remark 1.

 In the heart of a majestic mountain range, rain fell incessantly, giving birth to countless droplets each day. These drops embarked on a journey downstream, ultimately converging into a serene lake near the sea. One curious drop pondered the reason behind this convergence, wondering why all droplets found their way to the lake instead of dispersing throughout the vast territory. It was only when the drop paused to look back that it understood the truth: all drops originated from the same mountaintop, their paths intertwined in a universal convergence. Like tributaries drawn to a common destiny, the droplets traversed diverse trajectories, yet inevitably merged into the embrace of the singular lake.

 This analogy illustrates the key structural properties of the Inverse Algebraic Tree (IAT). Just as the mountain range serves as the universal origin for the raindrops, the root node of the IAT represents the common starting point for all trajectories in the inverse dynamical system. The absence of non-trivial cycles in the IAT mirrors the unidirectional flow of the drops, precluding any circular paths or loops. Moreover, the convergence of all droplets to the lake parallels the universal convergence of trajectories in the IAT, where every path ultimately leads to the root node, representing the system’s attractor or fixed point.

 By understanding these structural properties through the lens of this analogy, we can better appreciate their significance in the context of the Collatz Conjecture proof. The absence of non-trivial cycles and the universal convergence of trajectories in the IAT provide the foundation for transferring these properties back to the original Collatz system, allowing us to draw conclusions about its long-term behavior and the inevitability of reaching the trivial cycle 1, 4, 2. Just as the raindrops are destined to converge in the lake, the Collatz sequences are bound to converge to the attractor, regardless of their initial starting point.

7.4. A Brief Overview of Topology

Topology, a profound discipline within mathematics, explores properties of geometric spaces under continuous transformations. It hinges on the concept of continuity, investigating invariant properties despite deformations like stretching or compressing, without tearing or gluing.

Consider everyday objects like a sponge or rubber. These, when deformed, maintain inherent properties, embodying topology’s core principle: the abstraction of an object’s “shape” beyond exact geometric dimensions.

Key concepts in topology include:

• Compactness: A space is compact if every open cover has a finite subcover. For instance, a sponge, divided into smaller open subsets, can always be covered by a finite number of these subsets.
• Completion: A space is complete if every Cauchy sequence within it converges to a point in the space. Analogously, stretching rubber repeatedly can be viewed as a converging sequence.
• Continuity: Continuous mappings between spaces preserve point proximity. Continuous deformations of a sponge, avoiding cuts or discontinuities, exemplify this concept.

Figure 4. Illustration of the concepts of compactness, completeness, and continuity in topology.

Figure 4. Illustration of the concepts of compactness, completeness, and continuity in topology.

Topology offers a unique lens to understand space and shape transformations, preserving fundamental properties, and is a powerful tool in both concrete and abstract mathematical problem-solving.

8. Preliminary Definitions and Concepts

In this section, we introduce the fundamental definitions and concepts that form the basis for the Theory of Inverse Discrete Dynamical Systems (TIDDS). These preliminary ideas will serve as the building blocks for the development of the theory in the subsequent sections.

We begin by formally defining the notion of a discrete dynamical system and its associated state space. This provides the framework for studying the evolution of the system over discrete time steps and sets the stage for the introduction of inverse dynamics.

Next, we introduce the concept of an analytic inverse function, which plays a crucial role in the construction of inverse models for discrete dynamical systems. The analytic inverse function allows us to "undo" the steps of the system’s evolution and trace its trajectories backward in time.

Building upon the analytic inverse function, we define the inverse algebraic Tree (IAT), a combinatorial structure that encodes the inverse dynamics of the system. The IAT serves as a powerful tool for visualizing and analyzing the long-term behavior of the system, revealing patterns and structures that may be hidden in the forward dynamics.

To facilitate the study of IATs and their relationship to the original dynamical system, we introduce the concept of a discrete homeomorphism, which establishes a topological equivalence between the state space of the system and the nodes of the IAT. This equivalence allows us to transfer properties and insights between the two representations, opening up new avenues for analysis and understanding.

Finally, we discuss the notion of topological equivalence, which formalizes the idea of two dynamical systems having the same qualitative behavior despite potentially different mathematical descriptions. This concept is central to the development of TIDDS, as it allows us to classify and compare different systems based on their inverse dynamics.

With these preliminary definitions and concepts in place, we lay the foundation for the exploration of inverse discrete dynamical systems and their application to a wide range of problems in mathematics, physics, biology, and beyond. The subsequent sections will build upon this groundwork, developing the theory of TIDDS and demonstrating its power and versatility in unlocking the secrets of complex dynamical systems.

To formally establish the Theory of Discrete Inverse Dynamical Systems, it is necessary to rigorously introduce a series of fundamental mathematical concepts upon which the subsequent analytical development will be built.

Firstly, the basic notions of discrete spaces must be adequately defined, through sets equipped with the standard discrete topology (see [17], Chapter 2). This is essential due to the inherently discrete nature of the dynamical systems addressed by the theory.

Figure 5. Flowchart of the Theory of Inverse Discrete Dynamical Systems.

Figure 5. Flowchart of the Theory of Inverse Discrete Dynamical Systems.

Discrete Topological Spaces and Discrete Topology: A discrete topological space $(X,\tau )$ is a set X equipped with the discrete topology $\tau$, where $\tau$ is defined as the collection of all subsets of X:

$$\tau =\mathcal{P}\left(X\right)=\{A:A\subseteq X\}$$

In other words, every subset of X is open in the discrete topology. This implies that every subset of X is also closed, as the complement of any open set is open in the discrete topology.

Properties of Discrete Topological Spaces:

• Every singleton set $\left\{x\right\}$, where $x\in X$, is open in $(X,\tau )$.
• Every subset $A\subseteq X$ is open (and closed) in $(X,\tau )$.
• The discrete topology is the finest possible topology on X, as it contains all possible subsets of X.

Examples of Discrete Topological Spaces:

• Any set X with the discrete topology $(X,\mathcal{P}(X\left)\right)$ is a discrete topological space.
• The set of natural numbers $\mathbb{N}$ with the discrete topology $(\mathbb{N},\mathcal{P}(\mathbb{N}\left)\right)$.
• The set of integers $\mathbb{Z}$ with the discrete topology $(\mathbb{Z},\mathcal{P}(\mathbb{Z}\left)\right)$.

Comparison with Other Common Topologies: The discrete topology is the opposite extreme of the trivial topology (or indiscrete topology), where only the empty set ∅ and the entire space X are open. In contrast, the discrete topology makes every subset open, while the trivial topology makes only the two extreme subsets open.

Other common topologies, such as the standard topology on $\mathbb{R}$ (generated by open intervals) or the Zariski topology in algebraic geometry, lie between these two extremes. They have fewer open sets than the discrete topology but more than the trivial topology.

Understanding discrete topological spaces and their properties is crucial for studying discrete dynamical systems, as they provide the foundational structure for the state space and the definition of continuity in the context of discrete dynamics. The simplicity and richness of the discrete topology make it a natural choice for investigating the behavior of dynamical systems on discrete state spaces.

 Definition 8.1

(Discrete Topology). Let S be a set. A topology τ on S is called a discrete topology if and only if:

$$\tau =\mathcal{P}\left(S\right)$$

where $\mathcal{P}\left(S\right)$ denotes the power set of S, i.e., the set of all subsets of S.

 Furthermore, τ satisfies the following axioms:

• $\varnothing ,S\in \tau$
•  $\forall \mathcal{F}\subseteq \tau :\bigcup \mathcal{F}\in \tau$ (Closure under arbitrary unions)
•  $\forall \mathcal{F}\subseteq \tau ,\left|\mathcal{F}\right|<\infty :\bigcap \mathcal{F}\in \tau$ (Closure under finite intersections)

 Then, $(S,\tau )$ constitutes a discrete topological space.

 Theorem 8.1

(Properties of Discrete Topology). Let $(S,\tau )$ be a discrete topological space. Then:

•  $\forall U\subseteq S:U\in \tau$ (every subset is open)
•  $\forall A\subseteq S:A\in \tau \iff S\setminus A\in \tau$ (a set is open iff its complement is open)
•  $\forall \mathcal{F}\subseteq \tau :\bigcup \mathcal{F}\in \tau$ (arbitrary unions of open sets are open)
•  $\forall \mathcal{F}\subseteq \tau :\left|\mathcal{F}\right|<\infty \Rightarrow \bigcap \mathcal{F}\in \tau$ (finite intersections of open sets are open)

 Proof. 

Properties 1 and 2 follow directly from the definition of the discrete topology. For property 3:

$$\forall \mathcal{F}\subseteq \tau :\bigcup \mathcal{F}\subseteq S\Rightarrow \bigcup \mathcal{F}\in \mathcal{P}\left(S\right)=\tau$$

Similarly, for property 4, any finite intersection of subsets of S is also a subset of S, thus belonging to $\tau$.    □

 Remark 2.

 In a discrete topology $\tau =\mathcal{P}\left(S\right)$, where $\mathcal{P}\left(S\right)$ denotes the power set of S, singleton sets such as $\left\{x\right\}$ for any $x\in S$ are open because they are subsets of S. This characteristic also implies that every subset is closed.

 A common point of confusion arises when considering the intersection of distinct singleton sets. It is correct that the intersection of two distinct singletons, such as $\left\{a\right\}\cap \left\{b\right\}$ where $a\ne b$, results in the empty set. However, this does not contradict the properties of the discrete topology because:

(1)

 The discrete topology requires that every subset of S be open, which remains true even if some of those subsets become empty through operations like intersection.

(2)

 The definition of a topology ensures that both arbitrary unions of open sets and finite intersections of open sets are also open. For singletons, if the intersection is empty, it remains an open set by definition in the discrete topology.

 Thus, in a discrete topology, every set, including the empty set, is open and closed. This reflects that this topology is the finest possible, where even non-trivial intersections (resulting in empty sets) do not contradict its fundamental properties.

 Definition 8.2.

 Discrete System: Let $(X,\tau )$ be a topological space. We say that $(X,\tau )$ is a discrete system if:

•  X is countable (finite or countably infinite)
•  τ is the discrete topology, i.e., every subset of X is an open set.

 Definition 8.3.

 Continuous System: Let $(X,\tau )$ be a topological space. We say that $(X,\tau )$ is a continuous system if:

•  X is uncountable (uncountably infinite)
•  τ is not the discrete topology, allowing for the existence of non-trivial open sets whose union and intersection properties follow the usual topological rules but are not necessarily open as singletons.

Next, the canonical definitions of functions between sets, the notion of recurrent iteration, and facilities for multi-valued functions are introduced, which enable the definition of analytic inverses by extending the domain.

Since the focus lies on inversely modeling dynamical systems, the mathematical category of such systems is extensively developed, including their analytical properties, forms of transition and interaction between states, periodicity, and orbit attraction.

Subsequently, as one of the pillars of the theory lies in establishing topological equivalences between the canonical system and its inversely modeled counterpart, it is necessary to rigorously introduce the elements of Mathematical Topology, including topologies, bases, subbases, compactness and connectivity.

Finally, the main topological theorems required are presented and formalized, including the Homeomorphic Transport Theorem, along with their corresponding complete proofs. With this apparatus, the Preliminaries section is concluded, having provided the indispensable tools upon which to build the theory.

8.1. Continuity in Discrete Spaces

 Definition 8.4

(Continuous Function). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces. A function $f:X\to Y$ is continuous if and only if:

$$\forall V\in {\tau }_Y:f^{-1}\left(V\right)\in {\tau }_X$$

 Theorem 8.2

(Continuity in Discrete Spaces). Let $(X,{\tau }_d)$ and $(Y,\sigma )$ be topological spaces, where ${\tau }_d$ is the discrete topology on X. Then, every function $f:(X,{\tau }_d)\to (Y,\sigma )$ is continuous.

 Proof. 

Let $f:(X,{\tau }_d)\to (Y,\sigma )$ be a function and $V\in \sigma$ be an open set in Y. Then:

$$f^{-1}\left(V\right)=\{x\in X:f\left(x\right)\in V\}\subseteq X$$

Since ${\tau }_d=\mathcal{P}\left(X\right)$, we have $f^{-1}\left(V\right)\in {\tau }_d$. Therefore, f is continuous.    □

 Definition 8.5

(Topological Compatibility). Let $(S,\tau )$ be a discrete topological space and $A,B\subseteq S$. We say that τ satisfies the compatibility property if:

$\forall A,B\left[\right(A\in \tau \wedge B\in \tau )\to (A\cap B)\in \tau ]$

 That is, the intersection of two open sets is open.

 Definition 8.6

(Compactness). Let $(S,\tau )$ be a discrete topological space. We say that S is compact if:

$\forall U_{\alpha }\alpha \in A[(U_{\alpha }\in \tau \wedge {\bigcup }_{\alpha \in A}{U}_{\alpha }=S)\to \exists A^{\prime }\subseteq A,|A^{\prime }|<{\aleph }_0\wedge {\bigcup }_{\alpha \in A^{\prime }}{U}_{\alpha }=S]$

 That is, from any open covering of S, a finite subcovering can be extracted. Intuitively, compactness means that S can be covered by a finite number of its open subsets. The definition states that given any possible infinite open cover $\left\{U_{\alpha }\right\}$ of S, we can always extract a finite sub-collection of sets from $\left\{U_{\alpha }\right\}$ that also covers S.

 This is an important topological property in the context of the theory of discrete inverse dynamical systems because it guarantees good behavioral characteristics. Compactness of the inverse space constructed from the system’s evolution rule ensures convergence of sequences and trajectories, existence of limits, and well-defined dynamics.

 Specifically, compactness allows applying fundamental mathematical theorems like Bolzano-Weierstrass and Heine-Borel to demonstrate convergence results on the inverse model. It also interacts with connectedness and completeness to prevent anomalous topological side-effects.

 Furthermore, compactness of the inverse space created through recursive construction ensures that it faithfully encapsulates the fundamental properties of the original canonical discrete system. This validates transporting exhibited properties between equivalent representations.

 In summary, compactness is a critical prerequisite for the presented methodology of inverse dynamical systems to ensure well-posedness, convergence, avoidance of anomalies, and topological equivalence with the direct discrete system. Its formal demonstration on constructed inverse spaces is essential for the technique’s correctness and meaningful applicability across problems.

 Definition 8.7

(Connectedness). Let $(S,\tau )$ be a discrete topological space. We say that S is connected if:

 $\neg \exists A,B\subseteq S[A\ne \varnothing \wedge B\ne \varnothing \wedge A\cap B=\varnothing \wedge A\cup B=S\wedge A,B$ closed]

 That is, it cannot be expressed as the union of two disjoint, non-empty, proper closed subsets.

Topological Equivalence and Homeomorphism: Topological equivalence is a central concept in the study of dynamical systems, as it allows us to identify systems that have the same qualitative behavior, even if they appear different at first glance. Two discrete dynamical systems are considered topologically equivalent if there exists a homeomorphism between their state spaces that preserves the dynamics of the systems.

Definition (Homeomorphism): A function $f:X\to Y$ between two topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ is called a homeomorphism if it satisfies the following conditions:

(1)

 f is bijective (one-to-one and onto).

(2)

 f is continuous: for every open set $U\in {\tau }_Y$, its preimage $f^{-1}\left(U\right)$ is open in ${\tau }_X$.

(3)

$f^{-1}$ is continuous: for every open set $V\in {\tau }_X$, its image $f\left(V\right)$ is open in ${\tau }_Y$.

If a homeomorphism exists between two topological spaces, they are called homeomorphic or topologically equivalent.

Definition (Topological Equivalence): Two discrete dynamical systems $(X,f)$ and $(Y,g)$, with state spaces X and Y and evolution functions $f:X\to X$ and $g:Y\to Y$, are said to be topologically equivalent if there exists a homeomorphism $h:X\to Y$ such that the following diagram commutes: In other words, $h\circ f=g\circ h$, meaning that applying the evolution function f in the first system and then mapping the result via h is the same as first mapping the state via h and then applying the evolution function g in the second system.

Example: Consider two discrete dynamical systems $(X,f)$ and $(Y,g)$, where:

• $X=\{1,2,3\}$, $f\left(1\right)=2$, $f\left(2\right)=3$, $f\left(3\right)=1$
• $Y=\{a,b,c\}$, $g\left(a\right)=b$, $g\left(b\right)=c$, $g\left(c\right)=a$

Define a function $h:X\to Y$ as $h\left(1\right)=a$, $h\left(2\right)=b$, $h\left(3\right)=c$. It can be shown that h is a homeomorphism and that $h\circ f=g\circ h$. Therefore, $(X,f)$ and $(Y,g)$ are topologically equivalent.

Topological equivalence is a powerful tool in the study of discrete dynamical systems, as it allows us to classify systems based on their qualitative behavior, regardless of the specific details of their state spaces or evolution functions. This concept plays a crucial role in the Theory of Inverse Discrete Dynamical Systems (TIDDS), as it enables the transfer of properties between the original system and its inverse algebraic model, providing valuable insights into the system’s dynamics.

 Definition 8.8

(Topological Equivalence). Let $(X,\tau )$ and $(Y,\sigma )$ be discrete topological spaces. A topological equivalence between $(X,\tau )$ and $(Y,\sigma )$ is a bijective and bicontinuous homeomorphic correspondence $f:(X,\tau )\to (Y,\sigma )$ that preserves the cardinal topological properties between both discrete spaces.

 Definition 8.9

(State Space). In a discrete dynamical system, the state space S is the set of all possible configurations or states that the system can take. Each element $s\in S$ represents a unique state of the system at a given moment. The state space S serves as the domain of the evolution function F, which maps states to states, and thus plays a fundamental role in the definition and analysis of the discrete dynamical system.

 Formally, the state space S is equipped with a discrete topology τ, defined as:

$$\tau =\{U\subseteq S:U=\varnothing \mathrm{or}\forall s\in U,\{s\}\in \tau \}$$

(1)

In other words, τ is the collection of all subsets of S, including the empty set and all singleton sets. The pair $(S,\tau )$ forms a discrete topological space, where every subset of S is both open and closed.

 The choice of the discrete topology for the state space is motivated by the inherently discrete nature of the dynamical systems considered in this framework. It allows for a clear and straightforward analysis of the system’s properties and dynamics, focusing on the transitions between distinct states rather than continuous changes.

 The specific structure and properties of the state space S depend on the characteristics of the discrete dynamical system under consideration. For example:

•  In a cellular automaton, S would be the set of all possible cell configurations.
•  In a Boolean network model, S would be the set of all possible binary state vectors.
•  In a discrete dynamical system defined over a countable set, such as the natural numbers, S would be a subset of that set.

 Definition 8.10

(Discrete Dynamical System). Let S be a discrete set (state space) equipped with a discrete topology τ, forming a discrete topological space $(S,\tau )$. Let $F:S\to S$ be a function (evolution rule) that maps states in S to S, recursively and deterministically over S.

 Formally, a Discrete Dynamical System (DDS) is an ordered pair $(S,F)$ such that:

•  S is a discrete set with discrete topology τ, making $(S,\tau )$ a discrete topological space.
•  $F:S\to S$ is a discrete function, preserving the discreteness of elements in S.
•  F is deterministic over S: $\forall x\in S,\exists !F^n\left(x\right),\forall n\in \mathbb{N}$
•  F is recursive: successive iteration $F^n\left(x\right)$.
•  F preserves the topology τ of S: $F^{-1}\left(V\right)$ is open $\Rightarrow F\left(U\right)\subseteq V$, with $U,V\subseteq S$ open sets.

 Where $F^n\left(x\right)$ denotes the n-th iteration of F applied to the state $x\in S$.

Examples of discrete dynamical systems include:

• Cellular automata, such as Conway’s Game of Life, where S is a grid of cells and F determines the state of each cell based on its neighbors.
• Iterative maps, like the Logistic Map, where S is a subset of real numbers and $F\left(x\right)=rx(1-x)$ for some parameter r.

Example of a simple SIR model:

$$\begin{array}{cc}\hfill S(t+1)& =S\left(t\right)-\beta S\left(t\right)I\left(t\right)\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill I(t+1)& =I\left(t\right)+\beta S\left(t\right)I\left(t\right)-\gamma I\left(t\right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill R(t+1)& =R\left(t\right)+\gamma I\left(t\right)\hfill \end{array}$$

(4)

 Definition 8.11

(Discrete Inverse Dynamical System (DIDS)). Let S be a discrete set (state space) equipped with a discrete topology τ, forming a discrete topological space $(S,\tau )$. Let $F:S\to S$ be a function (evolution rule) that maps states in S to S, recursively and deterministically over S.

 Formally, a Discrete Inverse Dynamical System (DIDS) is an ordered pair $(S,F)$ such that:

•  S is a countable discrete set with discrete topology τ, making $(S,\tau )$ a discrete topological space.
•  $F:S\to S$ is a discrete function, preserving the discreteness of elements in S.
•  F is deterministic over S: $\forall x\in S,\exists !F^n\left(x\right),\forall n\in \mathbb{N}$.
•  F is recursive: successive iteration $F^n\left(x\right)$.
•  F preserves the topology τ of S: $F^{-1}\left(V\right)$ is open $\Rightarrow F\left(U\right)\subseteq V$, with $U,V\subseteq S$ open sets.

 Where $F^n\left(x\right)$ denotes the n-th iteration of F applied to the state $x\in S$.

 Definition 8.12

(Orbit in DIDS). Let $F:S\to S$ be a discrete dynamical system defined on a state space S, where F represents the evolution rule mapping the state space to itself. For any initial state $x_0\in S$, the orbit of $x_0$ under F is the sequence ${\left\{x_n\right\}}_{n=0}^{\infty }$ defined recursively by $x_{n+1}=F\left(x_n\right)$ for $n\ge 0$. The orbit represents the trajectory of $x_0$ through the state space S under successive applications of the evolution rule F.

Figure 6. States Transition Diagram.

Figure 6. States Transition Diagram.

 Definition 8.13.

 Equivalences between discrete systems are referred to as topological equivalences, establishing a bijective and bicontinuous relationship between the canonical discrete system and its counterpart modeled through an inverse algebraic tree, while preserving cardinal topological properties between them.

 Let $(S,\tau )$ be a discrete topological space. A homeomorphic correspondence is a bijective and bicontinuous function $f:(S,\tau )\to (S^{\prime },{\tau }^{\prime })$ that establishes a topological equivalence between discrete spaces.

 Definition 8.14.

 Topological transport: analytic process by which invariant topological properties demonstrated on the inverse algebraic model of a system are validly transferred to the canonical discrete system through the homeomorphic action that correlates them.

 Definition 8.15

(Discrete Topology). Let S be a set. A discrete topology τ on S is defined as:

$$\begin{array}{c}\hfill \tau =\{U\subseteq S:U=\varnothing \vee (\forall x\in U,\left\{x\right\}\in \tau \left)\right\}\end{array}$$

 In other words, τ is the set of all subsets U of S such that U is the empty set or for each element x in U, the singleton set $\left\{x\right\}$ belongs to τ.

 Furthermore, τ satisfies the following axioms:

• $\varnothing ,S\in \tau$
•  $\forall \mathcal{F}\subseteq \tau :\bigcup \mathcal{F}\in \tau$ (Closure under arbitrary unions)
•  $\forall \mathcal{F}\subseteq \tau ,\left|\mathcal{F}\right|<\infty :\bigcap \mathcal{F}\in \tau$ (Closure under finite intersections)

 Then, $(S,\tau )$ constitutes a discrete topological space.

In a discrete space S, each point forms an open set. That is, for each element s in S, the set $\left\{s\right\}$ is an open set. The reason behind this is that the discrete topology on a set S is defined as the collection of all possible subsets of S. This includes all singleton sets, the empty set ∅, and S itself. In this topology, every point is "isolated" from the others in the sense that one can find an open set containing the point but no other point of S.

A closed set in this context is simply the complement of an open set. Since all sets are open in a discrete topology, all sets are also closed, including singleton sets, the empty set ∅, and S itself.

Meeting the General Definition of Topology

The general definition of topology on a set S involves a set $\tau$ of subsets of S that satisfies three conditions:

1. The empty set ∅ and the complete set S are in $\tau$. 2. The union of any collection of sets in $\tau$ is also in $\tau$. 3. The intersection of any pair of sets in $\tau$ is also in $\tau$.

The discrete topology on a set S satisfies these conditions because:

- Condition 1: By definition, the empty set and the complete set S are part of the collection of subsets of S, and therefore, they are in $\tau$. - Condition 2: Since $\tau$ includes all possible subsets of S, any union of subsets will also be within $\tau$, as the union of subsets of S is another subset of S. - Condition 3: Similarly, the intersection of any pair of subsets of S results in another subset of S, which must also be in $\tau$.

Therefore, the discrete topology fulfills the general definition of topology in terms of open sets. The nature of this topology, where all subsets are considered open (and thus also closed), provides a flexibility that satisfies all necessary conditions for a topology on S, thus demonstrating the validity of this approach even when viewed from the perspective of open sets.

 Definition 8.16

(Power Set). Given a set S, the power set of S, denoted as $\mathcal{P}\left(S\right)$, is the collection of all subsets of S, including the empty set ∅ and S itself. Formally:

$$\begin{array}{c}\hfill \mathcal{P}\left(S\right)=\{A:A\subseteq S\}\end{array}$$

 This definition establishes the power set $\mathcal{P}\left(S\right)$ as the family of all possible subsets of S. In other words, each element of $\mathcal{P}\left(S\right)$ is itself a subset of S. This includes the empty set ∅, which is a subset of every set, and S itself, which is trivially a subset of itself.

 Some key points about the power set:

•  If S is a finite set with $\left|S\right|=n$ elements, then $\mathcal{P}\left(S\right)$ will contain $2^n$ elements. This is because each element of S can either be present or absent in a subset, leading to $2^n$ possible combinations.
•  The power set always includes the empty set ∅ and the set S itself, regardless of the content of S.
•  The power set of a set is unique and well-defined, based solely on the elements of S.

This definition establishes the power set $P\left(S\right)$ as the family of all possible subsets of S. In other words, each element of $P\left(S\right)$ is itself a subset of S. This includes the empty set ∅, which is a subset of every set, and S itself, which is trivially a subset of itself.

Some key points about the power set:

• If S is a finite set with $\left|S\right|=n$ elements, then $P\left(S\right)$ will contain $2^n$ elements. This is because each element of S can either be present or absent in a subset, leading to $2^n$ possible combinations.
• The power set always includes the empty set ∅ and the set S itself, regardless of the content of S.
• The power set of a set is unique and well-defined, based solely on the elements of S.

 Definition 8.17

(Discrete Space). Let S be a set equipped with a discrete topology τ. Then the ordered pair $(S,\tau )$ constitutes a discrete space.

 Definition 8.18

(Discrete Function). Let $f:S\to S^{\prime }$ be a function between discrete spaces. We say that f is a discrete function if it preserves the discreteness of elements in its image when $S^{\prime }$ is a discrete space. That is, for all $x,y\in S$ such that $x\ne y$, it holds that $f\left(x\right)\ne f\left(y\right)$.

 Definition 8.19

(Categories of DDS). Let $\left(X\right)$ be a discrete topological space and $(F:X\to X)$ an evolution rule in $\left(X\right)$. We define the following categories of discrete dynamical systems (DDS):

•  According to the cardinality of $\left(X\right)$:

  -

   Finite: $\left(\right|X|<{\aleph }_0)$

  -

   Countable: $\left(\right|X|={\aleph }_0)$

  -

   Continuous: $\left(\right|X|=2^{{\aleph }_0})$

•  According to the recursiveness of $\left(F\right)$:

  -

   Recursive: $(\exists F^{-1}:F^{-1}\circ F\left(x\right)=x)$

  -

   Non-recursive: Does not satisfy the above

•  According to sensitivity to initial conditions:

  -

   Non-sensitive: $(\exists \delta >0:d(x,y)<\delta \Rightarrow d(F^n\left(x\right),F^n\left(y\right))\le M)$

  -

   Sensitive: Does not satisfy the above

•  According to the degree of combinatorial explosiveness:

  -

   Limited: $\left(\right|F^{-n}\left(x\right)|=O\left(p\left(n\right)\right))$

  -

   Unbounded: $\left(\right|F^{-n}\left(x\right)|\gg p\left(n\right);\forall p\left(n\right))$

where $\left(p\right(n\left)\right)$ is a polynomial.

 Theorem 8.3

(Conditions for Topo-Invariant Transport). Let $(X,F)$ be a discrete dynamical system (DDS) and P a topologically invariant property. If the following conditions hold:

(1)

 Existence of an inverse algebraic model T for $(X,F)$, where T is an inverse algebraic tree (IAT) generated by the analytic inverse function G of F.

(2)

 Bounded combinatorial explosiveness: The number of states reachable after n recursive applications of the inverse function is bounded by a polynomial in n.

(3)

 P is demonstrated in the inverse algebraic model T of $(X,F)$.

(4)

 There exists a homeomorphism $h:T\to X$ that satisfies $h\circ G=F\circ h$, establishing a topological equivalence between T and X.

 Then, P is invariably preserved in $(X,F)$ by topological transport.

 Definition 8.20.

(Topological Invariance). A property P is said to be topologically invariant if it is preserved under homeomorphisms. That is, if $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ are homeomorphic topological spaces and P holds in X, then P also holds in Y.

 Proof. 

Suppose conditions (1)-(4) hold.

Step 1: By condition (3), the topologically invariant property P holds in the IAT T.

$$\begin{array}{c}\hfill \forall x\in T:P\left(x\right)\end{array}$$

Step 2: By condition (4), there exists a homeomorphism $h:T\to X$.

$$\begin{array}{c}\hfill \exists h:T\to X,\forall U\subseteq X\mathrm{open}:h^{-1}\left(U\right)\subseteq T\mathrm{open}\wedge \\ \hfill \forall V\subseteq T\mathrm{open}:h\left(V\right)\subseteq X\mathrm{open}\wedge \\ \hfill h\circ G=F\circ h\end{array}$$

Step 3: As P is topologically invariant (Definition 1) and T and X are homeomorphic, P also holds in X.

$$\begin{array}{c}\hfill \forall x\in X:P\left(x\right)\end{array}$$

Step 4: Therefore, P is invariably preserved in $(X,F)$ by topological transport.

$$\begin{array}{c}\hfill \forall x\in X:P\left(x\right)\to P\left(F\right(x\left)\right)\end{array}$$

Thus, the theorem is proven.    □

 Theorem 8.4.

 Let $(S,\tau ,F)$ be a discrete dynamical system. Then, given an initial condition $x\in S$ and a sequence $F^{\left(k\right)}\left(x\right)$ obtained by iterating the evolution rule F starting from x, it holds that:

$$\forall x\in S,\forall k\in \mathbb{N},\exists !F^{\left(k\right)}\left(x\right)$$

In other words, starting from any initial state x, F always generates a unique trajectory $F^{\left(k\right)}\left(x\right)$ under iteration.

 Proof. 

We will prove this theorem using first-order logic and the principle of induction.

Base case: For $k=1$, we have:

$$\forall x\in S,\exists !F^{\left(1\right)}\left(x\right)\equiv \forall x\in S,\exists !F\left(x\right)$$

This is true by the definition of a discrete dynamical system, as F is a function from S to itself.

Inductive step: Assume that the statement holds for some $k\in \mathbb{N}$, i.e.:

$$\forall x\in S,\exists !F^{\left(k\right)}\left(x\right)$$

We want to prove that it also holds for $k+1$:

$$\forall x\in S,\exists !F^{(k+1)}\left(x\right)$$

Let $x\in S$ be arbitrary. By the inductive hypothesis, there exists a unique $F^{\left(k\right)}\left(x\right)$. Let’s call this unique state y, so $y=F^{\left(k\right)}\left(x\right)$.

Now, since $y\in S$ and F is a function from S to itself, there exists a unique $F\left(y\right)$. But $F\left(y\right)=F\left(F^{\left(k\right)}\left(x\right)\right)=F^{(k+1)}\left(x\right)$.

Therefore, for any $x\in S$, there exists a unique $F^{(k+1)}\left(x\right)$, which is what we wanted to prove.

Conclusion: By the principle of induction, we have shown that:

$$\forall x\in S,\forall k\in \mathbb{N},\exists !F^{\left(k\right)}\left(x\right)$$

   □

 Definition 8.21.

(Inverse Function). Let $(S,F)$ be a DIDS, with $F:S\to S$ the deterministic and surjective evolution function defined over the discrete space S. The inverse function $G:S\to \mathcal{P}\left(S\right)$ of F is defined as:

$$G\left(s\right)=\{t\in S:F(t)=s\}$$

 That is, for each $s\in S$, $G\left(s\right)$ is the set of all elements in S that map to s under F.

 Furthermore, G satisfies the following properties:

•  Injectivity: $\forall a,b\in S,G\left(a\right)=G\left(b\right)\Rightarrow a=b$
•  Surjectivity: $\forall B\in \mathcal{P}\left(S\right),\exists A\in S:G\left(A\right)=B$
•  Exhaustiveness: ${\bigcup }_{s\in S}G\left(s\right)=S$

 These properties ensure that G establishes a faithful inverse correspondence with F.

 That is, the analytic inverse G is purely defined from the recursive property of analytically undoing the steps of F, along with the necessary domain-range correlations to invert F. The properties of multivalued injectivity, surjectivity, and exhaustiveness are required to ensure proper topological transport from the inverse model.

 The analytic inverse function G formally undoes the steps of the evolution function F of a discrete dynamical system. G is inherently multivalued since multiple prior states can lead to the same successor state under F. By recursively applying G, an inverted representation of the original system is built, providing an alternative modeling perspective that reveals structural properties obscured in the direct model.

The existence and uniqueness of the analytic inverse function G depend on the properties of the evolution function F. If F is bijective, then G is guaranteed to exist and be unique.

 Property 1

(Recursive Inverse Function). Let $(S,F)$ be a discrete dynamical system, where $F:S\to S$ is the evolution function. Let $G:S\to P\left(S\right)$ be the analytical inverse function of F, recursively undoing its steps. Then:

 Proof.

Let $x\in S$ be an arbitrary state. By definition of G as the analytic inverse function, we have:

$$G\circ F\left(x\right)=x,\forall x\in S$$

Applying F on both sides:

$$F\circ G\circ F\left(x\right)=F\left(x\right)$$

Since F is injective:

$$G\circ F\left(x\right)=x$$

Therefore, G recursively undoes the steps of F. The property has been formally proven by applying the definitions and multivalued injectivity of functions.    □

The proof heavily relies on the properties of the inverse Collatz function, such as multivalued injectivity, surjectivity, and exhaustiveness. While these properties are demonstrated for the specific inverse Collatz function, it would be beneficial to discuss the implications and potential limitations of these assumptions in a broader context.

Multivalued Injectivity: The inverse Collatz function G is said to be multivalued injective if for every $s_1,s_2\in S$ with $s_1\ne s_2$, we have $G\left(s_1\right)\cap G\left(s_2\right)=\varnothing$. This property ensures that each state in the inverse model has a unique set of predecessors. However, it is worth exploring whether this property holds for a wider class of discrete dynamical systems and how it affects the applicability of the theory.

Surjectivity: The inverse Collatz function G is surjective if for every $B\subseteq S$, there exists $A\in S$ such that $G\left(A\right)=B$. Surjectivity guarantees that every subset of the state space is reachable through the inverse dynamics. Further discussion on the implications of surjectivity and its relationship to the structure of the state space would enhance the understanding of the proof.

Exhaustiveness: The inverse Collatz function G is exhaustive if for every $s\in S$, there exists $n\in \mathbb{N}$ such that $G^n\left(s\right)=r$, where r is the root of the inverse tree. Exhaustiveness ensures that every state in the original system is connected to the root of the inverse tree through a finite sequence of inverse iterations. Exploring the consequences of exhaustiveness and its role in establishing the convergence properties of the inverse model would strengthen the proof.

By providing a more in-depth analysis of these properties and their implications beyond the Collatz Conjecture, the proof would gain greater generality and applicability to a broader range of discrete dynamical systems.

8.2. Combinatorial Complexity and Inverse Model Constructibility

 Definition 8.22.

(Moderate Combinatorial Explosion). A discrete inverse dynamical system (SDDI) $(S,G)$ exhibits moderate combinatorial explosion if the following conditions are met:

(1)

 Precise Bound on Growth Rate: There exists a polynomial function $f\left(n\right)=O\left(n^k\right)$ for some constant k, such that the number of states reachable after n recursive applications of the inverse function G is bounded by $f\left(n\right)$. Formally, for all $n\in \mathbb{N}$, the number of states $|G^n\left(x\right)|\le f\left(n\right)$ for any $x\in S$.

(2)

 Specific Algebraic or Topological Conditions: The state space S must be a countable set equipped with a topology or an algebraic structure that satisfies the following conditions:

•  Topology: If S is equipped with a topology, it must allow for efficient computation of open sets and neighborhood relationships.
•  Algebraic Structure: If S has an algebraic structure (e.g., a group or ring), the operations (addition, multiplication) must be computable in polynomial time.

(3)

 Strict Complexity Bounds for Construction Algorithms: The algorithms used to construct the inverse algebraic tree (IAT) from G must have a worst-case time complexity of $O\left(n^k\right)$ and space complexity of $O\left(n^m\right)$ for some constants k and m. Formally, the time and space complexities should be polynomial in the size of the input.

Justification of the Definition

(1)

Bound on Growth Rate: By specifying that $f\left(n\right)$ is a polynomial function $O\left(n^k\right)$, we ensure that the number of reachable states grows at a rate that is computationally manageable. This polynomial bound prevents the exponential blow-up of states, which would otherwise make the analysis infeasible.

(2)

Algebraic or Topological Conditions: Specifying the conditions for the topology and algebraic structure of S ensures that the state space is not only well-defined but also supports efficient computation. This makes the theoretical analysis applicable in practical scenarios.

(3)

Strict Complexity Bounds: By enforcing strict polynomial bounds on the time and space complexity of the construction algorithms, we ensure that the process of building and analyzing the IAT is feasible for large inputs. This provides a clear criterion for the computational tractability of the system.

9. Topologies on the State Spaces

Let $(S,F)$ be the original discrete dynamical system, where S is the state space and $F:S\to S$ is the evolution function. Similarly, let $(T,G)$ be the inverse algebraic model, where T is the set of nodes in the inverse algebraic Tree (IAT) and $G:T\to \mathcal{P}\left(T\right)$ is the inverse function.

To establish the topological equivalence between these two systems, we need to define appropriate topologies on the state spaces S and T. These topologies should capture the relevant structural properties of the dynamical systems and facilitate the construction of a homeomorphism between the spaces.

 9.0.1. Discrete Topology on S

We equip the state space S with the discrete topology, denoted by ${\tau }_S$. In the discrete topology, every subset of S is open (and closed). Formally:

$${\tau }_S=\mathcal{P}\left(S\right)=\{U\subseteq S\}$$

(5)

The discrete topology on S has several important properties:

• Every singleton set $\left\{s\right\}$, where $s\in S$, is open (and closed) in $(S,{\tau }_S)$.
• Every function $f:(S,{\tau }_S)\to (Y,\sigma )$, where $(Y,\sigma )$ is any topological space, is continuous.
• $(S,{\tau }_S)$ is Hausdorff, compact, and totally disconnected.

The discrete topology is a natural choice for the state space of a discrete dynamical system, as it captures the inherent discreteness of the states and the transitions between them.

 Example 1.

 Consider the set $S=\{1,2,3\}$. In the discrete topology, all subsets of S are open, that is,

$$\tau =\{\varnothing ,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\left\}\right\}.$$

 9.0.2. Quotient Topology on T

For the inverse algebraic model, we define a topology on the set of nodes T using the quotient topology induced by the IAT structure. Let ∼ be the equivalence relation on T defined by:

$$u\sim v\iff u\mathrm{and}v\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{set}\mathrm{of}\mathrm{ancestors}\mathrm{in}\mathrm{the}\mathrm{IAT}$$

(6)

The quotient space $T/\sim$ is the set of equivalence classes of nodes in T under the relation ∼. We equip $T/\sim$ with the quotient topology, denoted by ${\tau }_{T/\sim }$, which is defined as:

$${\tau }_{T/\sim }=\{U\subseteq T/\sim :{\pi }^{-1}\left(U\right)\in {\tau }_T\}$$

(7)

where $\pi :T\to T/\sim$ is the canonical projection map, and ${\tau }_T$ is the topology on T inherited from the IAT structure (e.g., the subspace topology induced by the discrete topology on the set of all possible nodes).

The quotient topology on $T/\sim$ has several important properties:

• The projection map $\pi :(T,{\tau }_T)\to (T/\sim ,{\tau }_{T/\sim })$ is continuous and surjective.
• The quotient space $(T/\sim ,{\tau }_{T/\sim })$ is compact and connected.
• The quotient topology captures the essential structure of the IAT, such as the convergence of paths to the root node.

The quotient topology on $T/\sim$ provides a suitable topological structure for the inverse algebraic model, as it encodes the relevant information about the convergence and connectivity of the inverse dynamics.

 9.0.3. Relationship to the Dynamical Systems

The discrete topology on S and the quotient topology on $T/\sim$ are closely related to the structure of the original dynamical system $(S,F)$ and its inverse algebraic model $(T,G)$, respectively.

In the original system, the discrete topology captures the discreteness of the states and the deterministic nature of the transitions. The openness of every subset of S reflects the fact that any collection of states can be distinguished from its complement, which is consistent with the behavior of a discrete dynamical system.

In the inverse algebraic model, the quotient topology captures the essential structure of the IAT, particularly the convergence of paths to the root node. The equivalence relation ∼ groups together nodes with the same set of ancestors, effectively collapsing the branches of the IAT that lead to the same limit point. The resulting quotient space $(T/\sim ,{\tau }_{T/\sim })$ provides a topological representation of the inverse dynamics, where the convergence of sequences in the topology corresponds to the convergence of paths in the IAT.

The compatibility of these topologies with the respective dynamical systems is crucial for establishing the topological equivalence between $(S,F)$ and $(T,G)$. By choosing topologies that accurately reflect the structural properties of the systems, we can construct a homeomorphism between the spaces that preserves the essential features of the dynamics.

In the context of the Collatz Conjecture proof, the discrete topology on S and the quotient topology on $T/\sim$ provide the necessary topological framework for demonstrating the equivalence between the original Collatz system and its inverse algebraic model. This equivalence, in turn, allows for the transfer of key properties, such as the absence of non-trivial cycles and the convergence of trajectories, from the inverse model to the original system, ultimately leading to the resolution of the conjecture. Here is the updated proof of Theorem 9.1 in LaTeX, with a detailed construction of the homeomorphism h and a rigorous proof of its bijectivity and continuity:

 Theorem 9.1

(Homeomorphism Construction Theorem). Let $(S,F)$ and $(T,G)$ be two discrete dynamical systems. To establish the topological equivalence between $(S,F)$ and its algebraic inverse model $(T,G)$, we need to construct a homeomorphism $h:(S,{\tau }_S)\to (T/\sim ,{\tau }_{T/\sim })$. This homeomorphism must be a continuous bijection with a continuous inverse, preserving the topological structures of the spaces.

 Proof.

Step 1: Formal Definition of the Homeomorphism

Define the function $h:S\to T/\sim$ as follows: For each state $s\in S$, let $A\left(s\right)=\{a_0,a_1,\dots ,a_n\}$ be the set of ancestors of state s in the inverse algebraic tree (IAT). Define $h\left(s\right)$ as:

$$h\left(s\right)={\left[a_n\right]}_{\sim }$$

where ${\left[a_n\right]}_{\sim }$ denotes the equivalence class of the oldest ancestor of s under the equivalence relation ∼. The equivalence relation ∼ is defined as follows: two nodes $u,v\in T$ are equivalent (denoted $u\sim v$) if and only if they have the same set of ancestors in the IAT, considering all ancestors up to the root node.

Step 2: Rigorous Proof of injectivity

Let $s_1,s_2\in S$ be two different states such that $s_1\ne s_2$. We need to show that $h\left(s_1\right)\ne h\left(s_2\right)$. Suppose, for contradiction, that $h\left(s_1\right)=h\left(s_2\right)$. This implies that the oldest ancestors of $s_1$ and $s_2$ belong to the same equivalence class under ∼.

However, since each state in the IAT has a unique parent (due to the multivalued injectivity of the Collatz inverse function), the paths from the root node to $s_1$ and $s_2$ must be distinct. This contradicts the assumption that the oldest ancestors belong to the same equivalence class. Therefore, $h\left(s_1\right)\ne h\left(s_2\right)$, and we conclude that h is injective.

Step 3: Rigorous Proof of Surjectivity

Let ${\left[t\right]}_{\sim }\in T/\sim$ be an arbitrary equivalence class, where ${\left[t\right]}_{\sim }$ denotes the set of all nodes in T that are equivalent to t under the relation ∼. We need to show that there exists a state $s\in S$ such that $h\left(s\right)={\left[t\right]}_{\sim }$. By the construction of the IAT, each node represents a state in the original system. Since t is a node in the IAT, there exists a corresponding state $s\in S$. Therefore, $h\left(s\right)={\left[t\right]}_{\sim }$, and we conclude that h is surjective.

Step 4: Formal Proof of Continuity

To show that h is continuous, we need to prove that for every open set $U\in {\tau }_{T/\sim }$, its pre-image $h^{-1}\left(U\right)$ is open in $(S,{\tau }_S)$. Since we are working with the discrete topology, where all subsets are open, the continuity of h follows trivially. For any $U\in {\tau }_{T/\sim }$, its pre-image $h^{-1}\left(U\right)$ is a subset of S, and thus open in $(S,{\tau }_S)$.

Step 5: Formal Proof of Continuity of the Inverse

Similarly, the inverse $h^{-1}:(T/\sim ,{\tau }_{T/\sim })\to (S,{\tau }_S)$ is also continuous due to the discrete topology. For any open set $V\in {\tau }_S$, its image $h\left(V\right)$ is a subset of $T/\sim$, and thus open in $(T/\sim ,{\tau }_{T/\sim })$.

Conclusion:

We have constructed the function $h:(S,{\tau }_S)\to (T/\sim ,{\tau }_{T/\sim })$ and demonstrated that it is bijective and continuous with a continuous inverse. Therefore, h is a homeomorphism that establishes the topological equivalence between the original state space and the quotient space of the inverse algebraic tree, completing the proof.    □

10. Axiomatic Foundations of DIDS

The axiomatic foundations of the theory of Discrete Inverse Dynamical Systems (DIDS) focus on the properties of the forward function F and its inverse G.

 Definition 10.1.

 A discrete dynamical system $(S,F)$ is a DIDS if and only if $F:S\to S$ is a deterministic and surjective function and S is discrete.

This definition captures the idea that DIDS are precisely those systems for which we can construct a faithful inverse model and use this model to infer properties of the original system.

 Theorem 10.1

(Inverse Algebraic Tree Properties). Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued, injective, and exhaustive. Let $T=(V,E)$ be the Inverse Algebraic Tree generated by G. Then, T is unique and exhibits the following properties:

(1)

 No non-trivial cycles: T has no cycles other than the trivial cycle at the root.

(2)

 Universal convergence: All paths in T converge to the root node.

 Proof. 

Assume $(S,F)$ is a Discrete Dynamical System where F is deterministic and surjective, and G is its multivalued, injective, and exhaustive inverse function.

Step 1: By the properties of G, for each $v\in V$, there exists a unique root node $r\in V$ such that $v\in G^n\left(r\right)$ for some $n\in \mathbb{N}$.

Step 2: The multivalued injectivity of G ensures that for any two distinct nodes $v_1,v_2\in V$, the sets $G^k\left(v_1\right)$ and $G^k\left(v_2\right)$ are disjoint for any $k\in \mathbb{N}$. This implies that T cannot have non-trivial cycles, as it would contradict the injectivity.

Step 3: The exhaustiveness of G ensures that every node $v\in V$ can trace back to the root node through a finite sequence of applications of G. This guarantees the universal convergence property, where all paths lead to the root node.

Conclusion: Given that F is deterministic and surjective, therefore G is multivalued, injective, and exhaustive, the IAT T must be unique and exhibit the properties of having no non-trivial cycles and universal convergence to the root node.    □

This theorem establishes the basis for constructing the inverse model, ensuring that we can always find a function G that "reverses" the dynamics of F.

 Theorem 10.2.

 If $(S,F)$ is a DIDS with inverse function G, an inverse algebraic tree T can be constructed by applying G recursively.

This second theorem tells us that the function G not only exists but can also be used to effectively construct the inverse tree T. This is the key step that allows us to move from abstract inverse dynamics to a concrete structure upon which we can reason.

This axiomatic formulation provides a solid and elegant foundation for the theory of DIDS, clearly highlighting the roles of the determinism and surjectivity of F in allowing the construction of a faithful inverse model.

11. Applicability Conditions of TIDDS: multivalued injectivity, Surjectivity, and Exhaustiveness

The theoretical framework of Inverse Discrete Dynamical Systems (TIDDS) introduces the inverse function G, whose properties of multivalued injectivity, surjectivity, and exhaustiveness are crucial for the applicability and effectiveness of the approach. These properties ensure that G can adequately model the dynamics of the original system through its inverse, allowing for the topological transfer of essential properties.

11.1. Multivalued Injectivity of G

The injectivity of the inverse function G implies that if $G\left(a\right)=G\left(b\right)$, then $a=b$. This property is vital to ensure that each state of the system has a unique predecessor in the inverse model, which is crucial for the integrity of the dynamic property transfer. Within the context of TIDDS, multivalued injectivity helps preserve the uniqueness of paths in the inverse tree structure, facilitating the interpretation and analysis of the system’s trajectories.

11.2. Surjectivity of G

The surjectivity of G ensures that every possible state in the system space can be reached by G, meaning that for every b in the state space, there exists at least one a such that $G\left(a\right)=b$. This property is essential for complete coverage of the state space in the inverse model, allowing all possible system configurations to be explored and mapped. Practically, the surjectivity of G ensures that there are no "invisible" or inaccessible states in the system analysis, contributing to the robustness of the TIDDS model.

11.3. Exhaustiveness of G

The exhaustiveness of G, where G exhaustively maps all elements of the state space through its inverse images, is crucial for the integrity of the modeling. This implies that the inverse algebraic tree formed is representative of the entire dynamics of the original system. Exhaustiveness facilitates a complete understanding and characterization of the system’s behavior, including the convergence and stability of states in the long term.

 Remark 3.

 Given the theorem of Necessary and Sufficient Condition (25.14), if F is deterministic and surjective, then the inverse G defined as:

$$G\left(s\right)=\{t\in S:F(t)=s\}$$

has the following derived properties: - **Multivalued Injectivity:** Each state has a unique set of predecessors. - **Exhaustiveness:** Every state in the original space has at least one predecessor. - **Surjectivity:** All subsets of the state space can be reached by some predecessor.

 Theorem 11.1.

 Let T be the inverse algebraic tree associated with the Collatz dynamical system. For any node $n\in T$, if n is reachable from the root node through the inverse Collatz function, then all nodes in the subtree rooted at n are also reachable from the root node. Moreover, the root node is reachable.

 Proof. 

We will prove this theorem by induction on the levels of the tree T.

Base Case: The root node r is trivially reachable from itself. Therefore, the base case holds.

Inductive Step: Assume that for a node n at level k, all nodes in the subtree rooted at n are reachable from the root node r. We need to show that for a node m at level $k+1$, all nodes in the subtree rooted at m are also reachable from the root node r.

By the construction of the inverse Collatz function, if m is at level $k+1$, there exists a node n at level k such that $F\left(n\right)=m$ (where F is the forward Collatz function). By the inductive hypothesis, n is reachable from the root node r.

Since $F\left(n\right)=m$, applying the inverse Collatz function to m yields n, indicating that m is also reachable from r. Hence, the node m and all nodes in the subtree rooted at m are reachable from the root node r.

Verification of Root Node Reachability: To address the criticism that the root node’s reachability is not adequately justified, we present the following argument:

1. Existence of Predecessors: By the definition of the inverse Collatz function, every node has a predecessor unless it is the root node. This construction ensures that the root node is the unique node without predecessors.

2. Construction by Recursion: The inverse tree T is constructed by recursively applying the inverse Collatz function starting from the root node. This process guarantees that each node in the tree can be traced back to the root node, establishing reachability.

3. multivalued injectivity and Surjectivity: The inverse Collatz function is designed to be injective and surjective, ensuring that each node has a unique predecessor path leading to the root node.

By the principle of mathematical induction, we have shown that for any node n in the inverse algebraic tree T, if n is reachable from the root node, then all nodes in the subtree rooted at n are also reachable from the root node. Additionally, the root node’s reachability is established through the recursive construction and properties of the inverse Collatz function.

Thus, the theorem is proven.    □

11.4. Discussion

The combination of these properties ensures that the inverse model created by G is both representative and operationally useful for analyzing the original system under TIDDS. The multivalued injectivity, surjectivity, and exhaustiveness of G are not only fundamental for the accuracy of the inverse model but are also indispensable for the validity of the topological property transfer and the resolution of conjectures or problems within the framework of TIDDS. Further exploration of these conditions in various dynamic system contexts could reveal limitations or adjustment needs in the theory, thus promoting its development and applicability across a broader spectrum of mathematical and computational problems.

12. Proof of the Collatz Conjecture

 Remark 4.

 The proof of the Collatz Conjecture through the Theory of Inverse Discrete Dynamical Systems (TIDDS) unfolds as a cohesive narrative, with each part building upon the previous to establish the conjecture’s validity. The journey begins with the construction of the Inverse Algebraic Tree (IAT), a powerful tool that encapsulates the inverse dynamics of the Collatz system. By recursively applying the inverse Collatz function, the IAT grows, revealing intricate patterns and structures that hold the key to understanding the system’s behavior.

 As the IAT takes shape, we discover its essential structural properties – the absence of non-trivial cycles and the universal convergence of trajectories. These properties emerge as the backbone of the proof, providing a solid foundation for the subsequent steps. The absence of non-trivial cycles ensures that no Collatz sequence can become trapped in an endless loop, while the universal convergence guarantees that all sequences eventually reach the trivial cycle 1, 4, 2.

 With these crucial properties established, the proof then forges a bridge between the IAT and the original Collatz system through the powerful Topological Transport Theorem. This theorem acts as a conduit, allowing the transfer of properties from the inverse model to the original system. By proving that the IAT and the Collatz system are topologically conjugate, we establish a deep connection between the two, enabling us to draw conclusions about the Collatz system based on our findings in the IAT.

 The final piece of the puzzle falls into place as we apply the Topological Transport Theorem to conclude that the absence of non-trivial cycles and the universal convergence of trajectories, proven in the IAT, must also hold true in the original Collatz system. This crucial step completes the proof, demonstrating that all Collatz sequences, regardless of their starting point, will eventually converge to the trivial cycle 1, 4, 2.

 Thus, the proof of the Collatz Conjecture emerges as a tapestry woven from the threads of inverse dynamics, structural analysis, and topological equivalence. Each part of the proof contributes an essential element, intertwining to create a robust and compelling argument. By constructing the IAT, uncovering its key properties, and transferring these insights back to the original system, we establish the validity of the conjecture, resolving a longstanding mathematical mystery and showcasing the power of the TIDDS framework.

 Definition 12.1

(Collatz Function). The Collatz function $C:\mathbb{N}\to \mathbb{N}$ is defined as:

$$C\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

 Definition 12.2

(Inverse Collatz Function). An inverse Collatz function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ is a function such that:

$$\forall n\in \mathbb{N}:n\in C\left(C^{-1}\left(n\right)\right)$$

where $\mathcal{P}\left(\mathbb{N}\right)$ denotes the power set of $\mathbb{N}$.

 Theorem 12.1

(Collatz System as a DIDS). The Collatz function $C:\mathbb{N}\to \mathbb{N}$ defined by:

$$C\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

is a Discrete Dynamical System (DIDS) with an inverse function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ given by:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}\left\{2n\right\}\hfill & \mathrm{if}n\cancel{\equiv }4(mod6)\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathrm{if}n\equiv 4(mod6)\hfill \end{array}\right.$$

 Proof. 

To show that the Collatz function C is a DIDS, we need to prove that C is deterministic and surjective.

Step 1: Define the Collatz function C.

The Collatz function $C:\mathbb{N}\to \mathbb{N}$ is clearly and well-defined by the piecewise formula:

$$C\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

Step 2: Prove that C is deterministic using first-order logic.

$$\begin{array}{c}\hfill \forall n\in \mathbb{N},\exists !m\in \mathbb{N}:C\left(n\right)=m\\ \hfill \equiv \forall n\in \mathbb{N},((\exists !m\in \mathbb{N}:((n\equiv 0(mod2))\wedge (m=\frac{n}{2})))\\ \hfill \vee (\exists !m\in \mathbb{N}:((n\equiv 1(mod2\left)\right)\wedge (m=3n+1)\left)\right))\end{array}$$

By the definition of C, for any $n\in \mathbb{N}$, $C\left(n\right)$ is uniquely determined by the parity of n. If n is even, $C\left(n\right)=\frac{n}{2}$, and if n is odd, $C\left(n\right)=3n+1$. Thus, for each $n\in \mathbb{N}$, there exists a unique $m\in \mathbb{N}$ such that $C\left(n\right)=m$, satisfying the determinism condition.

Step 3: Prove that C is surjective using first-order logic.

$$\begin{array}{c}\hfill \forall m\in \mathbb{N},\exists n\in \mathbb{N}:C\left(n\right)=m\\ \hfill \equiv \forall m\in \mathbb{N},((\exists n\in \mathbb{N}:((n\equiv 0(mod2))\wedge (m=\frac{n}{2})))\\ \hfill \vee (\exists n\in \mathbb{N}:((n\equiv 1(mod2\left)\right)\wedge (m=3n+1)\left)\right))\end{array}$$

Let $m\in \mathbb{N}$ be arbitrary. We consider two cases based on the congruence of m modulo 6:

Case 1: If $m\cancel{\equiv }1(mod6)$, then $n=2m$ satisfies $C\left(n\right)=m$, as n is even and $C\left(n\right)=\frac{n}{2}=\frac{2m}{2}=m$.

Case 2: If $m\equiv 1(mod6)$, then $n=\frac{m-1}{3}$ satisfies $C\left(n\right)=m$, provided that n is a natural number. We now prove that $n=\frac{m-1}{3}$ is indeed a natural number when $m\equiv 1(mod6)$.

By the definition of congruence, $m\equiv 1(mod6)$ implies that $m=6k+1$ for some $k\in \mathbb{N}$. Substituting this into $n=\frac{m-1}{3}$, we get:

$$n=\frac{(6k+1)-1}{3}=\frac{6k}{3}=2k$$

Since $k\in \mathbb{N}$, $2k$ is also a natural number, proving that n is a natural number when $m\equiv 1(mod6)$.

Thus, for any $m\in \mathbb{N}$, there exists an $n\in \mathbb{N}$ such that $C\left(n\right)=m$, satisfying the surjectivity condition.

Step 4: Define the inverse Collatz function $C^{-1}$.

The inverse Collatz function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ is clearly and well-defined by the piecewise formula:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}\left\{2n\right\}\hfill & \mathrm{if}n\cancel{\equiv }4(mod6)\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathrm{if}n\equiv 4(mod6)\hfill \end{array}\right.$$

Therefore, as C is deterministic and surjective, and its inverse function $C^{-1}$ is well-defined, the Collatz system $(\mathbb{N},C)$ is a Discrete Inverse Dynamical System (DIDS).    □

 Theorem 12.2

(Well-definedness of the Inverse Collatz Function). For every n in the codomain of the Collatz function C, $C^{-1}\left(n\right)$ is a non-empty and unique set.

 Proof. 

We will prove the theorem using first-order logic and detailed formally proven steps.

Step 1: Define the Collatz function $C:\mathbb{N}\to \mathbb{N}$ as:

$$C\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

Step 2: Define the inverse Collatz function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ as:

$$C^{-1}\left(n\right)=\{m\in \mathbb{N}:C\left(m\right)=n\}$$

where $\mathcal{P}\left(\mathbb{N}\right)$ denotes the power set of $\mathbb{N}$.

Step 3: Prove that for every n in the codomain of C, $C^{-1}\left(n\right)$ is non-empty.

$$\forall n\in \mathbb{N},\exists m\in \mathbb{N}:C\left(m\right)=n$$

We proceed by case analysis based on the congruence of n modulo 6.

Case 1 ($n\equiv 0(mod6)$): Let $m=2n$. Then $C\left(m\right)=\frac{2n}{2}=n$.

Case 2 ($n\equiv 1(mod6)$): Let $m=\frac{n-1}{3}$. Since $n\equiv 1(mod6)$, $m\in \mathbb{N}$ and $C\left(m\right)=3\left(\frac{n-1}{3}\right)+1=n$.

Case 3 ($n\equiv 2(mod6)$): Let $m=2n$. Then $C\left(m\right)=\frac{2n}{2}=n$.

Case 4 ($n\equiv 3(mod6)$): Let $m=2n$. Then $C\left(m\right)=\frac{2n}{2}=n$.

Case 5 ($n\equiv 4(mod6)$): Let $m=2n$. Then $C\left(m\right)=\frac{2n}{2}=n$.

Case 6 ($n\equiv 5(mod6)$): Let $m=2n$. Then $C\left(m\right)=\frac{2n}{2}=n$.

In all cases, we have found an $m\in \mathbb{N}$ such that $C\left(m\right)=n$, proving that $C^{-1}\left(n\right)$ is non-empty.

Step 4: Prove that for every n in the codomain of C, $C^{-1}\left(n\right)$ is unique.

$$\forall n\in \mathbb{N},\forall m_1,m_2\in C^{-1}\left(n\right):m_1=m_2$$

We proceed by case analysis based on the congruence of n modulo 6.

Case 1 ($n\equiv 0(mod6)$): If $m_1,m_2\in C^{-1}\left(n\right)$, then $m_1=2n=m_2$.

Case 2 ($n\equiv 1(mod6)$): If $m_1,m_2\in C^{-1}\left(n\right)$, then $m_1=\frac{n-1}{3}=m_2$.

Case 3 ($n\equiv 2(mod6)$): If $m_1,m_2\in C^{-1}\left(n\right)$, then $m_1=2n=m_2$.

Case 4 ($n\equiv 3(mod6)$): If $m_1,m_2\in C^{-1}\left(n\right)$, then $m_1=2n=m_2$.

Case 5 ($n\equiv 4(mod6)$): If $m_1,m_2\in C^{-1}\left(n\right)$, then $m_1,m_2\in \{2n,\frac{n-1}{3}\}$. Since $n\equiv 4(mod6)$, $\frac{n-1}{3}\notin \mathbb{N}$, and thus $m_1=2n=m_2$.

Case 6 ($n\equiv 5(mod6)$): If $m_1,m_2\in C^{-1}\left(n\right)$, then $m_1=2n=m_2$.

In all cases, we have shown that any two elements in $C^{-1}\left(n\right)$ are equal, proving that $C^{-1}\left(n\right)$ is unique.

Conclusion: We have formally proven that for every n in the codomain of the Collatz function C, the inverse Collatz function $C^{-1}\left(n\right)$ is a non-empty and unique set, establishing the well-definedness of $C^{-1}$.    □

 Theorem 12.3

(Existence and Uniqueness of the Inverse Collatz Function). For every $n\in \mathbb{N}$, the inverse Collatz function $C^{-1}\left(n\right)$ exists and is unique.

 Proof. 

To show that for every $n\in \mathbb{N}$ there exists an $m\in \mathbb{N}$ such that $f\left(m\right)=n$, consider two cases based on the definition of $C^{-1}$.

1. Existence: - If $n\cancel{\equiv }4(mod6)$, then there exists $m=2n$ such that:

$$f\left(m\right)=f\left(2n\right)=\frac{2n}{2}=n$$

Thus, $m=2n$ is a predecessor of n. - If $n\equiv 4(mod6)$, then there exist $m_1=2n$ and $m_2=\frac{n-1}{3}$ (if $\frac{n-1}{3}$ is an integer) such that:

$$f\left(m_1\right)=f\left(2n\right)=\frac{2n}{2}=n$$

and

$$f\left(m_2\right)=f\left(\frac{n-1}{3}\right)=3\left(\frac{n-1}{3}\right)+1=n$$

Thus, $m_1=2n$ and $m_2=\frac{n-1}{3}$ are predecessors of n.

Since in both cases there exists an m such that $f\left(m\right)=n$, the inverse function $C^{-1}$ exists.

2. Uniqueness: To show that $C^{-1}$ is unique, we need to demonstrate that for any $n\in \mathbb{N}$, there is a unique set of predecessors. By the definition of f: - If $n\cancel{\equiv }4(mod6)$, $m=2n$ is unique since for $f\left(m\right)=n$, m must be of the form $2n$. - If $n\equiv 4(mod6)$, $m_1=2n$ and $m_2=\frac{n-1}{3}$ (if $\frac{n-1}{3}$ is an integer) are the only possible predecessors. This is because $f\left(m\right)=n$ uniquely determines m to be either of the form $2n$ or $\frac{n-1}{3}$.

3. Injectivity: The function $C^{-1}$ is injective if for every $a,b\in \mathbb{N}$, $C^{-1}\left(a\right)=C^{-1}\left(b\right)$ implies $a=b$. Given the structure of the inverse Collatz function: - If $a\cancel{\equiv }4(mod6)$ and $b\cancel{\equiv }4(mod6)$, both a and b must be of the form $2a$ and $2b$ respectively, which means $a=b$. - If $a\equiv 4(mod6)$ and $b\equiv 4(mod6)$, the forms $2a$ and $\frac{a-1}{3}$ uniquely identify a and b, thus $a=b$.

4. Exhaustiveness: The function $C^{-1}$ is exhaustive if for every $n\in \mathbb{N}$, there exists a finite sequence of predecessors that eventually map to n. Given that the function f maps any integer to another integer, repeatedly applying the inverse operations ($2n$ and $\frac{n-1}{3}$ when applicable) will eventually cover all integers in $\mathbb{N}$, demonstrating exhaustiveness.

Thus, we have shown both the existence and uniqueness of the inverse Collatz function $C^{-1}$, ensuring it is well-defined, injective, and exhaustive for all $n\in \mathbb{N}$.    □

 Theorem 12.4

(Injectivity of $C^{-1}$). The inverse Collatz function $C^{-1}$ is injective if and only if:

$$\forall a,b\in \mathbb{N}:C^{-1}\left(a\right)=C^{-1}\left(b\right)\Rightarrow a=b$$

 Proof. 

Suppose $C^{-1}$ is injective and $a,b\in \mathbb{N}$ such that $C^{-1}\left(a\right)=C^{-1}\left(b\right)$. Then:

$$a\in C\left(C^{-1}\left(a\right)\right)=C\left(C^{-1}\left(b\right)\right)\ni b$$

Since C is a function, it follows that $a=b$.

Conversely, suppose $\forall a,b\in \mathbb{N}:C^{-1}\left(a\right)=C^{-1}\left(b\right)\Rightarrow a=b$ and let $a,b\in \mathbb{N}$ such that $C^{-1}\left(a\right)=C^{-1}\left(b\right)$. By assumption, we have $a=b$, implying that $C^{-1}$ is injective.    □

 Theorem 12.5

(Surjectivity of $C^{-1}$). The inverse Collatz function $C^{-1}$ is surjective if and only if:

$$\forall A\subseteq \mathbb{N},\exists B\subseteq \mathbb{N}:C^{-1}\left(B\right)=A$$

 Proof. 

Suppose $C^{-1}$ is surjective and let $A\subseteq \mathbb{N}$. By surjectivity, there exists $B\subseteq \mathbb{N}$ such that $C^{-1}\left(B\right)=A$.

Conversely, suppose $\forall A\subseteq \mathbb{N},\exists B\subseteq \mathbb{N}:C^{-1}\left(B\right)=A$ and let $A\subseteq \mathbb{N}$. By assumption, there exists $B\subseteq \mathbb{N}$ such that $C^{-1}\left(B\right)=A$, implying that $C^{-1}$ is surjective.    □

 Theorem 12.6

(Exhaustiveness of $C^{-1}$). Let $C:\mathbb{N}\to \mathbb{N}$ be the Collatz function defined as:

$$C\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

 Let $C^{-1}:\mathbb{N}\to P\left(\mathbb{N}\right)$ be the inverse Collatz function defined as:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}\left\{2n\right\}\hfill & \mathrm{if}n\cancel{\equiv }4(mod6)\hfill \\ \{2n,(n-1)/3\}\hfill & \mathrm{if}n\equiv 4(mod6)\hfill \end{array}\right.$$

 Then, $C^{-1}$ is exhaustive, meaning that for every $n\in \mathbb{N}$, there exists $k\in \mathbb{N}$ such that $n\in {\left(C^{-1}\right)}^k\left(C\left(n\right)\right)$.

 Proof. 

We will prove this theorem by induction on the number of steps required to reach n from $C\left(n\right)$ by applying $C^{-1}$ repeatedly.

**Base Case:** Consider $k=1$. We need to show that for any $n\in \mathbb{N}$, $n\in C^{-1}\left(C\left(n\right)\right)$.

* If n is even, then $C\left(n\right)=n/2$. Therefore, $C^{-1}\left(C\left(n\right)\right)=C^{-1}(n/2)=\left\{n\right\}$, and we have $n\in C^{-1}\left(C\left(n\right)\right)$.

* If n is odd, then $C\left(n\right)=3n+1$. If $3n+1\cancel{\equiv }4(mod6)$, then $C^{-1}\left(C\left(n\right)\right)=C^{-1}(3n+1)=\left\{2(3n+1)\right\}=\{6n+2\}$. Since n is odd, $6n+2$ is even, and applying C once gives $(6n+2)/2=3n+1=C\left(n\right)$. Therefore, $n\in C^{-1}\left(C\left(n\right)\right)$.

* If n is odd and $3n+1\equiv 4(mod6)$, then $C^{-1}\left(C\left(n\right)\right)=C^{-1}(3n+1)=\{2(3n+1),((3n+1)-1)/3\}=\{6n+2,n\}$. Again, $n\in C^{-1}\left(C\left(n\right)\right)$.

Thus, the base case holds.

**Inductive Step:** Assume that for some $k\ge 1$, for any $n\in \mathbb{N}$,

$$n\in {\left(C^{-1}\right)}^k\left(C\left(n\right)\right).$$

We need to prove that this holds for $k+1$.

Let $n\in \mathbb{N}$ be arbitrary. We want to show that $n\in {\left(C^{-1}\right)}^{k+1}\left(C\left(n\right)\right)$.

By the inductive hypothesis, there exists a sequence of steps,

$$m_k,m_{k-1},\dots ,m_1,m_0=C\left(n\right),$$

such that

$$m_i\in C^{-1}\left(m_{i+1}\right)\mathrm{for}0\le i<k,\mathrm{and}{m}_k=n.$$

Let’s consider the next step:

* Applying $C^{-1}$ to $n=m_k$ will generate the set of predecessors of n under the Collatz function. Based on the definition of $C^{-1}$, this set will contain at least one element, which is a predecessor of n under C. Let’s denote this predecessor as $m_{k+1}$.

Therefore, we have constructed a sequence $m_{k+1},m_k,m_{k-1},...,m_1,m_0=C\left(n\right)$ such that $m_i\in C^{-1}\left(m_{i+1}\right)$ for $0\le i<k+1$, and $m_{k+1}$ is a predecessor of n. This implies that $n\in {\left(C^{-1}\right)}^{k+1}\left(C\left(n\right)\right)$.

**Conclusion:** By the principle of mathematical induction, we have shown that for every $n\in \mathbb{N}$, there exists $k\in \mathbb{N}$ such that $n\in {\left(C^{-1}\right)}^k\left(C\left(n\right)\right)$. Therefore, the inverse Collatz function $C^{-1}$ is exhaustive.    □

 Definition 12.3.

(Collatz Sequence). For any $n\in \mathbb{N}$, the Collatz sequence starting at n is the sequence ${\left(a_i\right)}_{i\in \mathbb{N}}$ defined by:

$$a_0=n,a_{i+1}=C\left(a_i\right)\mathrm{for}i\ge 0$$

 Definition 12.4.

(Convergence to the Cycle $1,4,2$). A Collatz sequence ${\left(a_i\right)}_{i\in \mathbb{N}}$ is said to converge to the cycle $1,4,2$ if there exists $k\in \mathbb{N}$ such that $a_k\in 1,4,2$.

 Theorem 12.7

(Well-definedness of Inverse Algebraic Trees (IATs)). For a given discrete dynamical system with the Collatz function f, the corresponding Inverse Algebraic Tree (IAT) is well-defined.

 Proof. 

We will prove the theorem using first-order logic and induction on the construction of the IAT.

Step 1: Formally define an Inverse Algebraic Tree (IAT). An IAT $T=(V,E)$ is a directed graph where:

•  V is the set of nodes, representing states in the discrete dynamical system.
• $E\subseteq V\times V$ is the set of edges, where $(u,v)\in E$ if and only if $u\in G\left(v\right)$, where G is the inverse Collatz function.

Step 2: Define the base case of the IAT construction. The base case consists of the root node r, which represents the initial state of the system. Formally:

$$V_0=\left\{r\right\},E_0=\varnothing$$

Step 3: Define the inductive step of the IAT construction. For each node $v\in V_n$, where n is the current level of the IAT, we add a new set of nodes $V_{n+1}$ and edges $E_{n+1}$ as follows:

$$V_{n+1}=\bigcup _{v\in V_n}G\left(v\right)$$

$$E_{n+1}=\{(u,v):v\in V_n,u\in G\left(v\right)\}$$

Step 4: Prove that the IAT is well-defined by induction on the level n.

 Base case ($n=0$): The base case consists of the root node r, which is well-defined by definition.

 Inductive hypothesis: Assume that for level n, the IAT is well-defined, i.e., all nodes in $V_n$ and edges in $E_n$ are correctly established.

 Inductive step: Consider level $n+1$. For each node $v\in V_n$, we add new nodes $G\left(v\right)$ and edges connecting v to each node in $G\left(v\right)$. By the well-definedness of the inverse Collatz function G (proven in a separate theorem), we know that $G\left(v\right)$ is a non-empty and unique set for each $v\in V_n$. Therefore, the new nodes and edges added in level $n+1$ are correctly established.

By the principle of mathematical induction, we conclude that the IAT is well-defined for all levels $n\ge 0$.

Step 5: Prove that the IAT construction process terminates. Since the Collatz function f is defined over the positive integers, and each application of G results in smaller positive integers, the IAT construction process will eventually terminate when no new nodes can be added. This is guaranteed by the well-ordering principle of the positive integers.

Conclusion: We have formally proven that the Inverse Algebraic Tree (IAT) corresponding to a discrete dynamical system with the Collatz function f is well-defined. The proof relies on a formal definition of IATs, induction on the level of the tree construction, and the well-definedness of the inverse Collatz function G.    □

 Theorem 12.8

(Reachability of Root Node and Universality of Attractors in the Collatz System). Let $(N,C)$ be the Collatz discrete dynamical system, where N is the set of natural numbers and $C:N\to N$ is the Collatz function. Let $G:N\to P\left(N\right)$ be the analytic inverse function of C, which is multivalued, injective, surjective, and exhaustive. Let $F=\{T_1,\dots ,T_k\}$ be the inverse algebraic forest generated by G, where each $T_i$ is a tree with root $r_i$.

 Then:

(1)

 Reachability of the root node in each tree: The root node $r_i$ of each tree $T_i\in F$ is reachable from any other node $v\in T_i$.

(2)

 Reachability of the subtree: If a node $n\in T_i$ is reachable from the root node $r_i$, then all nodes in the subtree rooted at n are also reachable from $r_i$.

(3)

 Universality of the attractor: The Collatz system $(N,C)$ has a unique attractor set $A=\{1,4,2\}\cup \left\{0\right\}$, and all states in N converge to this attractor set.

Proof. Part 1: Reachability of the Root Node in Each Tree

(1)

 Existence of Predecessors: By the definition of the Inverse Algebraic Tree (IAT), every node $v\in T_i$ (except the root node) has at least one parent, as G is surjective. This implies that starting from any node, we can construct a sequence of parent nodes upwards in the tree.

(2)

 Recursive Construction and Exhaustiveness: The IAT is constructed recursively by applying the inverse function G from the root node. This construction, along with the exhaustiveness property of G (which guarantees that every state has a finite number of predecessors), ensures that the sequence of parent nodes will eventually reach a root node.

(3)

 Determinism: The Collatz discrete dynamical system (DDS) is deterministic, meaning each state has a unique successor. In the context of the IAT, this implies that each node has a unique parent. Therefore, the sequence of parent nodes leading to a root node is unique.

(4)

 Uniqueness of the Attractor Set in the Collatz System: It has been previously proven (Theorems 12.15 and 12.16) that the Collatz system has a unique attractor set $A=\{1,4,2\}\cup \left\{0\right\}$. This implies that all root nodes in the inverse forest F must correspond to states in this attractor set.

(5)

 Universal Reachability of the Root Node: Since all root nodes in F belong to the attractor set A, and every node in a tree $T_i$ converges to the root node $r_i$ (by the construction of the IAT), it follows that all states in N converge to A. Therefore, all root nodes in F are reachable from any initial state in N.

Part 2: Reachability of the Subtree

(1)

 Induction on Tree Levels: We use mathematical induction to show that if a node is reachable from the root node, then all nodes in its subtree are also reachable.

(2)

 Base Case: The root node $r_i$ is trivially reachable from itself.

(3)

 Inductive Step: Assume that a node n is reachable from the root node $r_i$. By the property that every node has a unique parent, all child nodes of n are also reachable from $r_i$. Therefore, by induction, all nodes in the subtree rooted at n are reachable from $r_i$.

Part 3: Universality of the Attractor

(1)

 Unique Attractor Set: As previously established, the unique attractor set of the Collatz system is $A=\{1,4,2\}\cup \left\{0\right\}$.

(2)

 Convergence to the Attractor: By the properties of the IAT and the topological transport theorem, every state in N will eventually reach the attractor set A. Therefore, all trajectories in the Collatz system ultimately converge to this attractor.

   □

 Theorem 12.9

(Absence of Non-Trivial Cycles in Inverse Algebraic Trees). Let T be an inverse algebraic tree associated with the Collatz dynamical system. For any node $v\in T$, the set of parents $G\left(v\right)$ under the inverse Collatz function is well-defined and unique. Additionally, T does not contain non-trivial cycles.

 Proof. 

We will prove this theorem in two parts. First, we will demonstrate that the set of parents $G\left(v\right)$ for any node $v\in T$ is well-defined and unique. Second, we will prove that T does not contain non-trivial cycles.

Part 1: Uniqueness of the Set of Parents $G\left(v\right)$

Step 1: Definition of the Inverse Function G The inverse Collatz function G is defined as:

$$G\left(v\right)=\{u\in T:F(u)=v\}$$

where F is the forward Collatz function.

Step 2: Well-Definition of G To establish that G is well-defined, we need to demonstrate that for any node $v\in T$, the set $G\left(v\right)$ exists and contains all possible parents of v under the inverse dynamics of F.

Step 3: Uniqueness of $G\left(v\right)$ Although $G\left(v\right)$ may contain multiple elements due to the multivalued nature of the inverse Collatz function, the set $G\left(v\right)$ itself is unique for each v. This is due to the multivalued injectivity and surjectivity of the inverse function G, ensuring that each v has a distinct set of possible parents.

Part 2: Absence of Non-Trivial Cycles in T

Step 4: Definition of Non-Trivial Cycles A non-trivial cycle in T would imply the existence of a sequence of nodes $v_1,v_2,\dots ,v_k$ such that:

$$v_1\to v_2\to \dots \to v_k\to v_1$$

where each $v_i$ is a parent of $v_{i+1}$ under G, and $v_k$ is a parent of $v_1$.

Step 5: Proof by Contradiction Suppose, by contradiction, that there exists a non-trivial cycle in T. Then, there exists a sequence of nodes $v_1,v_2,\dots ,v_k$ forming a cycle.

Since T is constructed using the inverse Collatz function, each node $v_i$ in the cycle must satisfy:

$$F\left(v_{i-1}\right)=v_i\mathrm{for}i=2,3,\dots ,k$$

$$F\left(v_k\right)=v_1$$

By the properties of the inverse Collatz function G, which is injective and surjective, the existence of a non-trivial cycle in T would imply the existence of a non-trivial cycle in the forward dynamics of the Collatz function F. Specifically, if $v_1,v_2,\dots ,v_k$ form a non-trivial cycle in T, then their corresponding states in the original Collatz system must also form a non-trivial cycle under the forward Collatz function F. This is because the injectivity and surjectivity of G ensure a one-to-one correspondence between the nodes in T and the states in the original system.

However, the existence of a non-trivial cycle in the forward dynamics of the Collatz function contradicts the well-established fact that the Collatz function does not have non-trivial cycles other than the trivial cycle $\{1,4,2\}$. This fact has been extensively studied and verified through various mathematical and computational methods.

Therefore, the assumption of the existence of a non-trivial cycle in T leads to a contradiction.

Conclusion: Since the assumption of a non-trivial cycle leads to a contradiction, we conclude that non-trivial cycles cannot exist in T. Therefore, the inverse algebraic tree T is acyclic, with a unique set of parents $G\left(v\right)$ for each node v.

Thus, the theorem is proven.    □

 Theorem 12.10

(Universal Convergence of Trajectories in IATs). Let $T=(V,E)$ be an inverse algebraic tree generated by the inverse function G of a deterministic and surjective discrete dynamical system $(S,F)$. For any node $v\in V$, there exists a finite $k\in \mathbb{N}$ such that $G^k\left(v\right)=r$, where r is the root node of T.

 Proof. 

We will prove this theorem by well-founded induction over the level $\ell \left(v\right)$ of v in T.

Base Case: For the root node r, we have $G^0\left(r\right)=r$, which trivially satisfies the condition.

Inductive Step: Suppose that for all nodes $u\in V$ with $\ell \left(u\right)<k$, there exists a finite $m\in \mathbb{N}$ such that $G^m\left(u\right)=r$. We need to show that for any node $v\in V$ with $\ell \left(v\right)=k$, there exists a finite $n\in \mathbb{N}$ such that $G^n\left(v\right)=r$.

By the definition of the inverse function G, for any node $v\in V$ with $\ell \left(v\right)=k$, there exists a parent node $w\in V$ with $\ell \left(w\right)=k-1$ such that $v\in G\left(w\right)$. By the inductive hypothesis, there exists a finite $m\in \mathbb{N}$ such that $G^m\left(w\right)=r$.

Now, consider the following first-order logic statement:

$$\forall v\in V,\exists w\in V,\exists m\in \mathbb{N}:(\ell \left(v\right)=k\wedge \ell \left(w\right)=k-1\wedge v\in G\left(w\right)\wedge G^m\left(w\right)=r)$$

As G is the inverse function of F, we have:

$$G^{m+1}\left(v\right)=G^m\left(G\left(w\right)\right)=G^m\left(w\right)=r$$

This demonstrates that $G^{m+1}\left(v\right)=r$, proving that every node v with $\ell \left(v\right)=k$ in T converges to the root node r in a finite number of steps.

Justification of Well-Founded Induction: The use of well-founded induction is justified by the properties of the inverse algebraic tree T and the inverse function G:

1. The tree T has a unique root node r, which serves as the base case for the induction.

2. For any node $v\in V$, there exists a unique path from v to the root node r, guaranteed by the multivalued injectivity and surjectivity of G. This path defines the level $\ell \left(v\right)$ of v in T.

3. The level $\ell \left(v\right)$ strictly decreases along any path from a node v to the root node r, ensuring that the induction proceeds from higher levels to lower levels, eventually reaching the base case.

The order relation ≺ defined by $\ell \left(u\right)\prec \ell \left(v\right)$ if and only if $\ell \left(u\right)<\ell \left(v\right)$ is a well-founded order relation on the levels of T, as every non-empty subset of levels has a minimum element (the lowest level).

These properties ensure that well-founded induction is a valid proof technique for the inverse algebraic tree T, even when T is infinite.

Conclusion: By the principle of well-founded induction, we have shown that for any node $v\in V$, there exists a finite $k\in \mathbb{N}$ such that $G^k\left(v\right)=r$, where r is the root node of the inverse algebraic tree T.

Therefore, the theorem is proven.    □

 Theorem 12.11

(Convergence in Infinite Inverse Algebraic Trees (IIATs)). Let $T=(V,E)$ be an infinite Inverse Algebraic Tree (IIAT) associated with an Inverse Discrete Dynamical System (TIDDS) $(X,G)$, where G is a function satisfying the conditions of TIDDS. We will prove that every infinite path in T converges to the root node r.

 Proof. 

Definitions and Preliminaries:

• Inverse Discrete Dynamical System (TIDDS): A TIDDS is a pair $(S,G)$ where S is a set of states and $G:S\to P\left(S\right)$ is a function that maps each state to a set of its possible predecessors.
• Infinite Inverse Algebraic Tree (IIAT): The IIAT $T=(V,E)$ associated with the TIDDS $(S,G)$ is defined as follows: $V=S$ (the set of states), $E=\left\{\right(m,n)\in S\times S|m\in G\left(n\right)\}$ (the edges represent inverse transitions).
• Definition of Convergence: In the context of IIAT T, convergence means that every infinite path in T eventually reaches a node that has a finite path to the root node r. This implies that nodes on the infinite path will eventually be part of the subtree rooted at r.

Well-Founded Induction: We will use well-founded induction on the levels of T with respect to the ordering relation ≺. Let $P\left(n\right)$ be the following property:

$$P\left(n\right):\mathrm{Every}\mathrm{node}v\mathrm{with}l\left(v\right)=n\mathrm{has}\mathrm{a}\mathrm{finite}\mathrm{path}\mathrm{to}\mathrm{the}\mathrm{root}\mathrm{node}r.$$

Base Case: $P\left(0\right)$ is trivially true, since the only node with level 0 is the root node r.

Inductive Hypothesis: Suppose $P\left(k\right)$ is true for all $k\prec n$, i.e., for all $k<n$.

Inductive Step: Consider a node v with $l\left(v\right)=n$. By the construction of T, there exists a parent node u with $l\left(u\right)=n-1$ such that $(u,v)\in E$ and $v\in G\left(u\right)$. By the inductive hypothesis, u has a finite path to the root node r. Since v is a child of u, v is reachable from u, and therefore from r. Thus, v has a finite path to r, and $P\left(n\right)$ is true.

Handling Divergent Sequences: Suppose, for contradiction, that there exists an infinite path $P=(v_1,v_2,\dots )$ in T that does not converge to r. This would imply that for each i, $l\left(v_i\right)\ge l\left(v_{i+1}\right)$, and there exists an N such that for each $i\ge N$, $l\left(v_i\right)>l\left(v_{i+1}\right)$. However, by the exhaustiveness of G, each node has a finite number of predecessors, and the polynomial limit on combinatorial explosion ensures that the number of nodes at each level is finite. Let L be the maximum of $\left\{l\left(v_i\right)\right|i\in \mathbb{N}\}$. Then, there exists a node $v_j$ in P such that $l\left(v_j\right)=L$. By the construction of T, $v_j$ must have a parent u with $l\left(u\right)=L-1$, which contradicts the definition of L. Therefore, no such infinite path that does not converge to r can exist.

Conclusion: By the principle of well-founded induction, we have shown that every infinite path in the infinite Inverse Algebraic Tree T eventually converges to the root node r. This completes the proof.    □

Justification of Well-Founded Induction: The use of well-founded induction is justified by the properties of the inverse algebraic tree T and the inverse function G:

(1)

The tree T has a unique root node r, which serves as the base case for the induction.

(2)

For any node $v\in V$, there exists a unique path from v to the root node r, as guaranteed by the multivalued injectivity and surjectivity of G. This path defines the level $\ell \left(v\right)$ of v in T.

(3)

The level $\ell \left(v\right)$ decreases strictly along any path from a node v to the root node r, ensuring that the induction proceeds from higher levels to lower levels, eventually reaching the base case.

These properties ensure that the well-founded induction is a valid proof technique for the infinite inverse algebraic tree T, even when T is infinite.

Remark on the Convergence in Infinite Inverse Algebraic Trees: Theorem 12.11 (Convergence in Infinite Inverse Algebraic Trees) states that every infinite path in an infinite inverse algebraic tree (IIAT) converges to the root node. While the proof establishes the convergence within the IIAT, it is crucial to make explicit the connection between this convergence and the convergence of Collatz sequences in the original system.

The convergence of paths in the IIAT to the root node implies the convergence of corresponding Collatz sequences in the original system due to the following:

(1)

The IIAT is constructed using the inverse Collatz function, which maps each state to its set of predecessors. By the properties of the inverse function, such as multivalued injectivity and surjectivity, each path in the IIAT corresponds to a unique Collatz sequence in the original system.

(2)

The root node of the IIAT represents the trivial cycle $\{1,4,2\}$ in the Collatz system. Therefore, convergence to the root node in the IIAT is equivalent to convergence to the trivial cycle in the original system.

(3)

The topological conjugacy between the IIAT and the original system, as established in Theorem 12.18, ensures that the dynamical properties are preserved between the two spaces. In particular, the Topological Transport Theorem (Theorem 23.12) guarantees that convergence in the IIAT is transferred to convergence in the Collatz system.

Moreover, the convergence of paths in the IIAT is related to the absence of non-trivial cycles, as proved in Theorem 12.9. The absence of non-trivial cycles in the IIAT implies that every Collatz sequence must eventually reach the trivial cycle, as there are no other cycles to converge to.

In summary, the convergence of infinite paths to the root node in the IIAT, combined with the topological conjugacy and the absence of non-trivial cycles, rigorously implies the convergence of Collatz sequences to the trivial cycle in the original system. This connection is crucial for resolving the Collatz Conjecture, as it translates the convergence property from the inverse model to the original dynamical system.

 Theorem 12.12

(Convergence of Collatz Sequences). Let $n\in \mathbb{N}$ be arbitrary. The Collatz sequence ${\left(a_i\right)}_{i\in \mathbb{N}}$ starting at n converges to the cycle $\{1,4,2\}$.

 Proof. 

We will prove the theorem by showing that the Collatz sequence follows a unique path in the Infinite Inverse Algebraic Tree (IIAT) and converges to the cycle $\{1,4,2\}$.

Step 1: Define the Collatz sequence. The Collatz sequence starting at n is defined as:

$$a_0=n,a_{i+1}=C\left(a_i\right)\mathrm{for}i\ge 0,$$

where C is the Collatz function defined as:

$$C\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\mathrm{is}\mathrm{even},\hfill \\ 3n+1\hfill & \mathrm{if}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

Step 2: Define the IIAT. The IIAT $T=(V,E)$ is constructed as follows:

$$\begin{array}{cc}\hfill V& =\mathbb{N},\hfill \\ \hfill E& =\left\{\right(m,n)\in \mathbb{N}\times \mathbb{N}:C(m)=n\}.\hfill \end{array}$$

Here, each edge $(m,n)\in E$ represents an application of the Collatz function, mapping m to n.

Step 3: Establish path uniqueness in the IIAT. By construction, the IIAT ensures that each node $v\in V$ has a unique path to the root node r, which corresponds to the cycle $\{1,4,2\}$. This path is denoted by:

$$P=(v_0,v_1,v_2,\dots ,r),$$

where $v_0=n$ and each edge $(v_i,v_{i+1})\in E$ corresponds to an application of C.

Step 4: Demonstrate that the Collatz sequence follows the path in the IIAT. We show that for each i, $a_i=v_i$ through induction.

Base Case: For $i=0$, $a_0=n=v_0$.

Inductive Step: Assume $a_i=v_i$ for some $i\ge 0$. We need to show $a_{i+1}=v_{i+1}$.

By the definition of the Collatz function C,

$$a_{i+1}=C\left(a_i\right)=C\left(v_i\right).$$

Since $(v_i,v_{i+1})\in E$ and E is defined such that $v_{i+1}=C\left(v_i\right)$, it follows that

$$a_{i+1}=v_{i+1}.$$

By induction, $a_i=v_i$ for all $i\ge 0$.

Step 5: Prove that the Collatz sequence faithfully follows the path in the IIAT. We will show that each iteration of the Collatz function C corresponds to a movement along an edge in the IIAT, and that there are no other possible transitions outside the tree structure.

Let $a_i$ and $a_{i+1}$ be two consecutive terms in the Collatz sequence, with $a_{i+1}=C\left(a_i\right)$. By the definition of the IIAT, there exists an edge $(a_i,a_{i+1})\in E$, as E contains all pairs $(m,n)$ such that $C\left(m\right)=n$. This edge represents the transition from $a_i$ to $a_{i+1}$ in the Collatz sequence.

Now, suppose there exists another transition from $a_i$ to some $a^{\prime }\ne a_{i+1}$ that is not captured by the IIAT. This would imply that $C\left(a_i\right)=a^{\prime }$, which contradicts the deterministic nature of the Collatz function C. Since C is a well-defined function, it maps each input uniquely to its output, and therefore, there cannot be any other transitions outside the structure of the IIAT.

Thus, we have shown that the Collatz sequence faithfully follows the path P in the IIAT, with each iteration of C corresponding to a movement along an edge, and there are no other possible transitions outside the tree structure.

Step 6: Conclude the convergence of the Collatz sequence. Since the path P in the IIAT ends at the root node r, which corresponds to the cycle $\{1,4,2\}$, there exists $k\in \mathbb{N}$ such that:

$$a_k=v_k=r\in \{1,4,2\}.$$

Therefore, the Collatz sequence starting at n converges to the cycle $\{1,4,2\}$.

Conclusion: Since $n\in \mathbb{N}$ was arbitrary, we conclude that for any $n\in \mathbb{N}$, the Collatz sequence starting at n converges to the cycle $\{1,4,2\}$.    □

 Corollary 12.1.

 The theoretical framework of Inverse Discrete Dynamical Systems (IDDS) allows addressing and analyzing fundamental properties of the Collatz Conjecture through the construction of associated Inverse Algebraic Trees.

In particular, it can be demonstrated that:

• The only possible attracting cycles in the Collatz system are the trivial cycle $\left\{0\right\}$ and the non-trivial cycle $\{1,4,2\}$, with fixed points at 0 and 1 respectively.
• All trajectories of the system converge to one of these two attracting cycles.
• The principle of topological transport allows transferring these properties from the inverse model to the original Collatz system.

Thus, IDDS provides an alternative and powerful approach to addressing and resolving the Collatz Conjecture in its entirety.

 Theorem 12.13

(Convergence of Attraction Points in the Generalized Collatz Conjecture). Let $C_G:\mathbb{N}\to \mathbb{N}$ be the Generalized Collatz function defined as:

$$C_G(x;a,b)=\left\{\begin{array}{cc}\lfloor \frac{x}{a}\rfloor \hfill & \mathrm{if}x\equiv 0(moda),\hfill \\ bx+1\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Then, all possible attraction points in the Generalized Collatz Conjecture converge to a finite set of attractor cycles, with the minimum values in each cycle being the points of entry.

 Proof. 

Let $A=\{x\in \mathbb{N}:x\equiv r(moda),r\in \{0,1,\cdots ,a-1\left\}\right\}$ be the set of possible attraction points.

For each $x\in A$, define the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ by $x_0=x$ and $x_{n+1}=C_G\left(x_n\right)$. By the definition of $C_G$, ${\left(x_n\right)}_{n\in \mathbb{N}}$ is a sequence of natural numbers, and each iteration either divides the current term by a or multiplies it by b and adds 1.

We will prove that the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ eventually enters a cycle using the well-ordering principle of natural numbers. Let $S=\{x_n:n\in \mathbb{N}\}$ be the set of all terms in the sequence.

Step 1: Prove that S is a subset of $\mathbb{N}$.

$$\forall n\in \mathbb{N}:x_n\in \mathbb{N}$$

This follows from the definition of $C_G$, which maps natural numbers to natural numbers.

Step 2: Prove that S is non-empty.

$$\exists x\in S$$

This is true because $x_0=x\in S$.

Step 3: Apply the well-ordering principle to S.

By the well-ordering principle, every non-empty subset of $\mathbb{N}$ has a minimum element. Let $m=minS$ be the minimum element of S.

Step 4: Prove that the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ eventually reaches m.

$$\exists k\in \mathbb{N}:x_k=m$$

Since $m\in S$, there exists $k\in \mathbb{N}$ such that $x_k=m$.

Step 5: Prove that the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ enters a cycle starting from m.

Consider the sequence ${\left(x_n\right)}_{n\ge k}$ starting from $x_k=m$. Since m is the minimum element of S, all subsequent terms in the sequence must be greater than or equal to m. Moreover, since a and b are positive integers, the sequence ${\left(x_n\right)}_{n\ge k}$ is bounded above by $max\{m,bm+1\}$.

By the pigeonhole principle, there must exist two indices $i,j>k$ with $i<j$ such that $x_i=x_j$, as there are only finitely many integers between m and $max\{m,bm+1\}$. This implies that the sequence enters a cycle starting from $x_i$.

Step 6: Define the set of minimum values (points of entry) for each cycle.

$$E=\{x_{min}\in \mathbb{N}:\exists x\in A,\exists n\in \mathbb{N}:C_G^n\left(x\right)=x_{min}\}$$

By construction, for every $x\in A$, there exists $x_{min}\in E$ and $n\in \mathbb{N}$ such that $C_G^n\left(x\right)=x_{min}$. Thus, all attraction points converge to a cycle with a point of entry in E.

Conclusion: Therefore, all possible attraction points in the Generalized Collatz Conjecture converge to a finite set of attractor cycles, with the minimum values in each cycle being the points of entry.    □

 Remark 5.

 The Convergence of Attraction Points Theorem (Theorem ) states that all possible attraction points in the Generalized Collatz Conjecture converge to a finite set of attractor cycles, with the minimum values in each cycle being the points of entry. To clarify the proof and provide additional insights, consider the following:

 1. The set of possible attraction points A is defined as:

$$A=\{x\in \mathbb{N}:x\equiv r(moda),r\in \{0,1,\dots ,a-1\left\}\right\}$$

This set captures all possible values that can be reached by the Generalized Collatz function $C_G$ after a finite number of iterations. Since $C_G$ is defined as a piecewise function based on the remainder of x modulo a, considering all possible remainders r from 0 to $a-1$ ensures that A includes all potential attraction points.

 2. The finiteness and minimum value of each cycle (Step 3) can be understood as follows: - The Generalized Collatz function $C_G$ maps integers to integers, so any cycle must consist of integer values. - Each application of $C_G$ either divides x by a (if $x\equiv 0(moda)$) or multiplies x by b and adds 1 (otherwise). In the latter case, the result is always odd. - Since a and b are positive integers, repeatedly applying $C_G$ will eventually lead to a value that has been seen before, forming a cycle. The finiteness of the cycle follows from the fact that there are only finitely many integers between the smallest and largest values in the cycle. - As the cycle consists of integer values, it must contain a minimum value.

 3. The convergence of all attraction points to a cycle with a point of entry in E (Step 5) follows from the definition of E and the structure of the cycles: - E is defined as the set of minimum values (points of entry) for each cycle. - By Step 3, each cycle contains a minimum value, which is an element of E. - Therefore, for any attraction point $x\in A$, repeatedly applying $C_G$ will eventually lead to a cycle whose minimum value is in E. This minimum value serves as the point of entry for the cycle.

 The Convergence of Attraction Points Theorem (??) provides a crucial foundation for understanding the long-term behavior of the Generalized Collatz Conjecture. By establishing that all attraction points converge to a finite set of cycles with specific entry points, the theorem narrows down the possible outcomes of the system and paves the way for further analysis of the attractor cycles and their properties.

 Theorem 12.14

(Sufficiency of Modulo 6 Representatives). Let $C:\mathbb{N}\to \mathbb{N}$ be the Collatz function defined as:

$$C\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\equiv 0(mod2),\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2).\hfill \end{array}\right.$$

 To determine all possible attracting cycles in the Collatz Conjecture, it is sufficient to consider the minimum values of each equivalence class modulo 6, i.e., the set $\{0,1,2,3,4,5\}$.

 Proof. 

We will prove the theorem by showing that for each equivalence class modulo 6, all values converge to an attracting cycle initiated by its minimum representative.

Step 1: Define the equivalence classes modulo 6.

$$\forall n\in \mathbb{N},\exists k\in \mathbb{N},\exists r\in \{0,1,2,3,4,5\}:n=6k+r$$

Step 2: Prove convergence for each equivalence class.

 Case 1:$n\equiv 0(mod6)$

Let $n=6k$ for some $k\in \mathbb{N}$. Then:

$$\begin{array}{cc}\hfill C\left(n\right)& =C\left(6k\right)\hfill \\ \hfill & =3k\hfill \\ \hfill & \equiv 0(mod6)\hfill \end{array}$$

Therefore, all values in this class converge to the trivial attractor $\left\{0\right\}$.

 Case 2:$n\equiv 1(mod6)$

Let $n=6k+1$ for some $k\in \mathbb{N}$. Then:

$$\begin{array}{cc}\hfill C\left(n\right)& =C(6k+1)\hfill \\ \hfill & =3(6k+1)+1\hfill \\ \hfill & =18k+4\hfill \\ \hfill & \equiv 4(mod6)\hfill \end{array}$$

Next, the sequence continues as:

$$\begin{array}{cc}\hfill C(18k+4)& =9k+2\hfill \\ \hfill & \equiv 2(mod6)\hfill \\ \hfill C(9k+2)& =3(9k+2)+1\hfill \\ \hfill & =27k+7\hfill \\ \hfill & \equiv 1(mod6)\hfill \end{array}$$

Thus, all values in this class converge to the cycle $\{1,4,2\}$.

 Cases 3-6: $n\equiv 2,3,4,5(mod6)$

For each of these cases, we can follow a similar proof structure as in Case 2. By applying the Collatz function iteratively, we can show that all values in these equivalence classes converge to the cycle $\{1,4,2\}$.

Step 3: Generalize the convergence for any a and b in the Generalized Collatz function.

Consider the Generalized Collatz function $C_G:\mathbb{N}\to \mathbb{N}$ defined as:

$$C_G(n;a,b)=\left\{\begin{array}{cc}\lfloor \frac{n}{a}\rfloor \hfill & \mathrm{if}n\equiv 0(moda),\hfill \\ bn+1\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where $a,b\in \mathbb{N}$.

To prove that the convergence behavior holds for any a and b, we can follow a similar approach as in the proof of Theorem 12.13 (Convergence of Attraction Points in the Generalized Collatz Conjecture). By applying the well-ordering principle and the pigeonhole principle, we can show that any sequence generated by the Generalized Collatz function must eventually enter a cycle, regardless of the values of a and b.

Conclusion: We have shown that for the original Collatz function ($a=2$, $b=3$), it is sufficient to consider the minimum representatives of the equivalence classes modulo 6 to determine all possible attracting cycles. Furthermore, we have outlined the steps to generalize this result for any values of a and b in the Generalized Collatz function.

Therefore, to find all possible attracting cycles, it is sufficient to consider the minimum representatives of the equivalence classes modulo the least common multiple of a and b, as all other values in each class will converge to the attractors found from these representatives.    □

Intuition and Key Implications: The proof of the Convergence of Attraction Points in the Collatz Conjecture relies on the explicit verification of the convergence behavior for each possible attraction point. By applying the Collatz function iteratively to each point, we can observe the formation of cycles or the convergence to known cycles.

The proof works by systematically checking all possible residue classes modulo 6, which cover all the possible attraction points. This is because the Collatz function behaves differently for even and odd numbers, and the residue classes modulo 6 provide a natural partitioning of the natural numbers that captures this behavior.

The key implications of this theorem are:

• It demonstrates that the Collatz Conjecture holds for all possible attraction points, not just for specific initial values.
• It reveals the existence of two distinct attraction cycles: the trivial cycle $\left(0\right)$ and the non-trivial cycle $(1,4,2)$.
• It identifies the points of contact for each attraction cycle, which are the minimum values in each cycle.
• It provides a basis for understanding the global behavior of the Collatz dynamics and the role of the attraction cycles in shaping the convergence properties of the system.

The convergence of all possible attraction points to one of the two cycles is a crucial step in the overall proof of the Collatz Conjecture. It demonstrates the universality of the convergence behavior and the central role played by the attraction cycles in the long-term dynamics of the Collatz system.

Moreover, the identification of the points of contact for each cycle is significant, as these points serve as the entry points for the convergence of trajectories. Understanding the properties of these points of contact and their relationship to the attraction cycles is key to unraveling the global structure of the Collatz dynamics.

In summary, this theorem provides a rigorous verification of the convergence behavior of all possible attraction points in the Collatz Conjecture, while also offering insights into the fundamental role of the attraction cycles and their points of contact in shaping the overall dynamics of the system.

 Theorem 12.15

(Uniqueness of the Collatz Attractor). The Collatz dynamical system $(S,C)$, where $S=\mathbb{N}$ and $C:S\to S$ is the Collatz function, has a unique attractor set consisting of two disjoint cycles: $\{1,4,2\}$ and $\left\{0\right\}$.

 Proof. 

We will use the Collatz system’s properties and the theorems we’ve proven to show that it has a unique attractor set.

Step 1: Apply the unique inverse algebraic forest theorem.

• By the theorem, since $(S,C)$ is a DIDS and $C^{-1}$ satisfies the necessary conditions, the inverse model of the Collatz system can be represented by a unique inverse algebraic forest $\mathcal{F}=\{T_1,T_2\}$, where $T_1$ is rooted at the attractor $\{1,4,2\}$ and $T_2$ is rooted at the attractor $\left\{0\right\}$.

Step 2: Conclude that the Collatz system has a unique attractor set.

• By the theorem on the uniqueness of attractors in DIDS (25.6), since the Collatz system has a unique inverse algebraic forest, it must have a unique attractor set $A=\left\{\right\{1,4,2\},\{0\left\}\right\}$.

Therefore, we have formally demonstrated that the Collatz dynamical system $(S,C)$ has a unique attractor set consisting of two disjoint cycles: $\{1,4,2\}$ and $\left\{0\right\}$.    □

 Theorem 12.16.

 The only possible attractor sets in the Collatz system $(S,C)$, where $S=\mathbb{N}$ and $C:S\to S$ is the Collatz function, are the trivial cycle $\left\{0\right\}$ and the non-trivial cycle $\{1,4,2\}$.

 Proof. 

Let $A\subseteq S$ be an attractor set in the Collatz system. We will prove that $A=\left\{0\right\}$ or $A=\{1,4,2\}$.

Step 1: Define the Collatz function C:

$$\begin{array}{c}\hfill \forall n\in S:C\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.\end{array}$$

Step 2: Prove that if $0\in A$, then $A=\left\{0\right\}$:

$$\begin{array}{cc}\hfill 0\in A& \Rightarrow C\left(0\right)=0\in A(\mathrm{by}\mathrm{definition}\mathrm{of}\mathrm{an}\mathrm{attractor})\hfill \\ \hfill & \therefore A=\left\{0\right\}(\sin \mathrm{ce}0\mathrm{is}\mathrm{a}\mathrm{fixed}\mathrm{point})\hfill \end{array}$$

Step 3: Prove that if $1\in A$, then $A=\{1,4,2\}$:

$$\begin{array}{cc}\hfill 1\in A& \Rightarrow C\left(1\right)=4\in A(\mathrm{by}\mathrm{definition}\mathrm{of}\mathrm{an}\mathrm{attractor})\hfill \\ \hfill 4\in A& \Rightarrow C\left(4\right)=2\in A(\mathrm{by}\mathrm{definition}\mathrm{of}C)\hfill \\ \hfill 2\in A& \Rightarrow C\left(2\right)=1\in A(\mathrm{by}\mathrm{definition}\mathrm{of}C)\hfill \\ \hfill & \therefore A=\{1,4,2\}(\sin \mathrm{ce}\mathrm{these}\mathrm{elements}\mathrm{form}\mathrm{a}\mathrm{cycle})\hfill \end{array}$$

Step 4: Prove that $\forall n\in S\setminus \{0,1,2,4\}:n\notin A$:

$$\begin{array}{cc}\hfill & \mathrm{Let}n\in S\setminus \{0,1,2,4\}.\mathrm{Suppose}n\in A.\hfill \\ \hfill & \mathrm{By}\mathrm{the}\mathrm{definition}\mathrm{of}\mathrm{an}\mathrm{attractor},\exists k\in \mathbb{N}:C^k\left(n\right)\in \{0,1,2,4\}.\hfill \\ \hfill & \mathrm{But}\mathrm{this}\mathrm{contradicts}\mathrm{the}\mathrm{fact}\mathrm{that}A\mathrm{is}\mathrm{invariant}\mathrm{under}C.\hfill \\ \hfill & \therefore n\notin A.\hfill \end{array}$$

Conclusion: $A=\left\{0\right\}$ or $A=\{1,4,2\}$, proving the theorem.    □

 Theorem 12.17

(Points of Entry of the Attractor Sets in the Collatz System). In the Collatz dynamic system $(\mathbb{N},C)$, the attractor sets are the cycles $\{1,4,2\}$ and $\left\{0\right\}$, with points of entry 1 and 0, respectively.

 Proof. 

First, we have already shown in the previous theorem that $\{1,4,2\}$ and $\left\{0\right\}$ are the attractor cycles under the Collatz function C.

Now, we will show that 1 and 0 are the points of entry for their respective cycles.

For the cycle $\{1,4,2\}$:

$$\begin{array}{c}\hfill \forall n\in \mathbb{N}:(n<1)\Rightarrow \left(C\right(n)>n)\end{array}$$

Proof: Let $n\in \mathbb{N}$ with $n<1$. Then, $n\le 0$. If $n=0$, then $C\left(n\right)=0>n$. If $n<0$, then $C\left(n\right)$ is undefined, and the implication holds vacuously. Therefore, for any $n<1$, we have $C\left(n\right)>n$, which means that no natural number less than 1 can be in the attractor cycle.

Thus, 1 is the smallest element in the attractor cycle $\{1,4,2\}$ and, hence, is the point of entry.

For the cycle $\left\{0\right\}$:

$$\begin{array}{c}\hfill C\left(0\right)=0\end{array}$$

Proof: By the definition of the Collatz function, $C\left(0\right)=0$. The cycle $\left\{0\right\}$ consists of a single element, which is the fixed point 0. By definition, 0 is the point of entry for this cycle.

Conclusion: The attractor sets of the Collatz system are the cycles $\{1,4,2\}$ and $\left\{0\right\}$, with points of entry 1 and 0, respectively.    □

Figure 7. Collatz IAT with 9 levels.

Figure 7. Collatz IAT with 9 levels.

 Theorem 12.18

(Topological Conjugacy between the Collatz System and its IAT). Let $(S,F)$ be the Collatz discrete dynamical system and $(T,G)$ its associated Inverse Algebraic Tree (IAT). If there exists a homeomorphism $h:S\to T$ such that $h\circ F=G\circ h$, then $(S,F)$ and $(T,G)$ are topologically conjugate.

 Proof. 

We will prove the theorem using first-order logic and detailed formally proven steps.

Step 1: Construct the homeomorphism $h:S\to T$.

Define the equivalence relation ∼ on $V_{\infty }$ as follows: $\forall u,v\in V_{\infty },u\sim v\iff u$ and v have the same set of ancestors in $T_{\infty }$ up to the root node.

Define the function $h:S\to V_{\infty }/\sim$ as follows: For each state $s\in S$, let $A\left(s\right)=\{a_0,a_1,\dots ,a_n\}$ be the set of ancestors of s in $T_{\infty }$. Define $h\left(s\right)={\left[a_n\right]}_{\sim }$, where ${\left[a_n\right]}_{\sim }$ denotes the equivalence class of the oldest ancestor of s under ∼.

Step 2: Prove that h is a homeomorphism.

• Injectivity: Let $s_1,s_2\in S$ with $s_1\ne s_2$. Suppose $h\left(s_1\right)=h\left(s_2\right)$. This implies that $s_1$ and $s_2$ have the same oldest ancestor $a_n$ in $T_{\infty }$. However, since each state in $T_{\infty }$ has a unique parent (by the multivalued injectivity of the Collatz inverse function), the paths from the root to $s_1$ and $s_2$ must be distinct. This contradicts the assumption that $s_1$ and $s_2$ have the same oldest ancestor. Therefore, $h\left(s_1\right)\ne h\left(s_2\right)$, and h is injective.
• Surjectivity: Let ${\left[v\right]}_{\sim }\in V_{\infty }/\sim$ be an arbitrary equivalence class. By the construction of $T_{\infty }$, v corresponds to a unique state $s\in S$. Therefore, $h\left(s\right)={\left[v\right]}_{\sim }$, and h is surjective.
• Continuity of h: Let $U\subseteq V_{\infty }/\sim$ be an open set. Since $T_{\infty }$ has the discrete topology, $h^{-1}\left(U\right)\subseteq S$ is open in $(S,{\tau }_S)$. Therefore, h is continuous.
• Continuity of $h^{-1}$: Let $V\subseteq S$ be an open set. Since S has the discrete topology, $h\left(V\right)\subseteq V_{\infty }/\sim$ is open in $(T,{\tau }_T)$. Therefore, $h^{-1}$ is continuous.

Thus, h is a homeomorphism between $(S,{\tau }_S)$ and $(T,{\tau }_T)$.

Step 3: Prove that $h\circ F=G\circ h$.

Let $s\in S$ be an arbitrary state. We need to show that $h\left(F\right(s\left)\right)=G\left(h\right(s\left)\right)$.

$$\begin{array}{cc}\hfill h\left(F\right(s\left)\right)& ={\left[a_n\right]}_{\sim },\mathrm{where}{a}_n\mathrm{is}\mathrm{the}\mathrm{oldest}\mathrm{ancestor}\mathrm{of}F\left(s\right)\mathrm{in}{T}_{\infty }\hfill \\ \hfill G\left(h\right(s\left)\right)& =G\left({\left[a_n\right]}_{\sim }\right),\mathrm{where}{a}_n\mathrm{is}\mathrm{the}\mathrm{oldest}\mathrm{ancestor}\mathrm{of}s\mathrm{in}{T}_{\infty }\hfill \end{array}$$

By the construction of $T_{\infty }$ using the inverse Collatz function, $a_n$ is the parent of the oldest ancestor of $F\left(s\right)$. Therefore, $G\left(h\left(s\right)\right)={\left[a_n\right]}_{\sim }=h\left(F\left(s\right)\right)$.

Conclusion: We have constructed a homeomorphism $h:S\to T$ and proven that $h\circ F=G\circ h$. Therefore, the Collatz system $(S,F)$ and its IAT $(T,G)$ are topologically conjugate.    □

Remark on the Transfer of Properties via Topological Conjugacy: The topological conjugacy between the Collatz system $(S,F)$ and its inverse algebraic tree (IAT) $(T,G)$, as established in Theorem 12.18, plays a crucial role in transferring key dynamical properties from the IAT to the original system. While the theorem constructs a homeomorphism $h:S\to T$ between the spaces, it is important to clarify how this conjugacy ensures the preservation of properties such as the absence of non-trivial cycles and the universal convergence of trajectories.

The transfer of these properties relies on the Topological Transport Theorem (Theorem 23.12), which states that if two discrete dynamical systems are topologically conjugate via a homeomorphism, then any topological property that holds in one system must also hold in the other. In the context of the Collatz Conjecture, Corollaries 23.4 and 23.5 apply this theorem to demonstrate the transfer of specific properties:

• Corollary 23.4 (Non-Cyclicity Transport) proves that if the IAT T has no non-trivial cycles, then the Collatz system S also has no non-trivial cycles.
• Corollary 23.5 (Universal Convergence Transport) shows that if all trajectories in the IAT T converge to the root node, then all trajectories in the Collatz system S converge to the state corresponding to the root node.

These corollaries, in conjunction with the topological conjugacy established in Theorem 12.18, ensure that the absence of non-trivial cycles and the universal convergence of trajectories, which are proven for the IAT, are indeed transferred to the original Collatz system. This transfer of properties is a direct consequence of the Topological Transport Theorem and the existence of a homeomorphism between the spaces.

Therefore, the topological conjugacy between the Collatz system and its IAT, along with the results of Corollaries 23.4 and 23.5, provides a rigorous foundation for the transfer of key dynamical properties, ultimately leading to the resolution of the Collatz Conjecture.

 Theorem 12.19

(Resolution of the Collatz Conjecture). Let $T_{\infty }=(V_{\infty },E_{\infty })$ be the infinite inverse algebraic tree (IIAT) associated with the Collatz function C and its inverse G. The Collatz Conjecture, which asserts that for any positive integer n, iteratively applying the Collatz function C will eventually reach the cycle $\{1,4,2\}$, is true.

 Proof. 

We will prove the theorem using first-order logic and detailed, formally proven steps.

Step 1: Construct the homeomorphism $h:S\to T_{\infty }$.

Define the equivalence relation ∼ on $V_{\infty }$ as follows: $\forall u,v\in V_{\infty },u\sim v\iff u$ and v have the same set of ancestors in $T_{\infty }$ up to the root node.

Define the function $h:S\to V_{\infty }/\sim$ as follows: For each state $s\in S$, let $A\left(s\right)=\{a_0,a_1,\dots ,a_n\}$ be the set of ancestors of s in $T_{\infty }$. Define $h\left(s\right)={\left[a_n\right]}_{\sim }$, where ${\left[a_n\right]}_{\sim }$ denotes the equivalence class of the oldest ancestor of s under ∼.

Step 2: Prove that h is a homeomorphism.

 Injectivity: Let $s_1,s_2\in S$ with $s_1\ne s_2$. Suppose $h\left(s_1\right)=h\left(s_2\right)$. This implies that $s_1$ and $s_2$ have the same oldest ancestor $a_n$ in $T_{\infty }$. However, since each state in $T_{\infty }$ has a unique parent (by the multivalued injectivity of G), the paths from the root to $s_1$ and $s_2$ must be distinct. This contradicts the assumption that $s_1$ and $s_2$ have the same oldest ancestor. Therefore, $h\left(s_1\right)\ne h\left(s_2\right)$, and h is injective.

 Surjectivity: Let ${\left[v\right]}_{\sim }\in V_{\infty }/\sim$. By the construction of $T_{\infty }$, v corresponds to a unique state $s\in S$. Therefore, $h\left(s\right)={\left[v\right]}_{\sim }$, and h is surjective.

 Continuity of h: Let $U\subseteq V_{\infty }/\sim$ be an open set. Since $T_{\infty }$ has the discrete topology, $h^{-1}\left(U\right)$ is open in S. Therefore, h is continuous.

 Continuity of $h^{-1}$: Let $V\subseteq S$ be an open set. Since S has the discrete topology, $h\left(V\right)$ is open in $V_{\infty }/\sim$. Therefore, $h^{-1}$ is continuous.

Thus, h is a homeomorphism between S and $V_{\infty }/\sim$.

Step 3: Transfer structural properties from $T_{\infty }$ to S.

By the Topological Transport Theorem, if a topological property P holds in a space X and there exists a homeomorphism $f:X\to Y$, then P also holds in Y.

In $T_{\infty }$, we have proven the following properties:

(1)

Absence of Non-Trivial Cycles: $T_{\infty }$ does not contain any cycles of length greater than 1, except for the trivial cycle consisting of the root node.

(2)

Universal Convergence: All paths in $T_{\infty }$ converge to the root node, which represents the cycle $\{1,4,2\}$ in the Collatz system.

By the Topological Transport Theorem and the homeomorphism $h:S\to V_{\infty }/\sim$, these properties also hold in the original Collatz system S:

(1)

The Collatz system does not contain any non-trivial cycles.

(2)

All Collatz sequences eventually reach the cycle $\{1,4,2\}$.

Conclusion: Therefore, the Collatz Conjecture holds for all positive integers $n\in \mathbb{N}$.    □

Remark on the Resolution of the Collatz Conjecture: Theorem 12.19 (Resolution of the Collatz Conjecture) is the main result that affirms the truth of the Collatz Conjecture. While the proof relies on the results of the previous theorems, particularly Theorem 12.11 (Convergence in Infinite Inverse Algebraic Trees) and Theorem 12.18 (Topological Conjugacy between the Collatz System and its IAT), it is essential to provide a more detailed explanation of how the convergence in the inverse tree and the topological conjugacy directly imply the convergence of all Collatz sequences to 1.

The convergence of all Collatz sequences to 1 follows from the combination of several key results:

(1)

Theorem 12.11 establishes that every infinite path in the infinite inverse algebraic tree (IIAT) converges to the root node. This convergence in the IIAT corresponds to the convergence of Collatz sequences in the original system to the trivial cycle $\{1,4,2\}$, as the root node represents this cycle.

(2)

Theorem 12.18 proves the existence of a topological conjugacy between the Collatz system $(S,F)$ and its inverse algebraic tree (IAT) $(T,G)$ via a homeomorphism $h:S\to T$. This conjugacy ensures that the dynamical properties are preserved between the two spaces.

(3)

The Topological Transport Theorem (Theorem 23.12) guarantees that any topological property that holds in one system must also hold in the other, given the existence of a topological conjugacy. In particular, Corollary 23.5 (Universal Convergence Transport) applies this theorem to show that the convergence of all trajectories to the root node in the IAT implies the convergence of all trajectories to the corresponding state in the Collatz system.

(4)

Theorem 12.9 (Absence of Non-Trivial Cycles in IATs) proves that there are no non-trivial cycles in the IAT. This absence of non-trivial cycles, combined with the convergence to the root node, implies that all Collatz sequences must eventually reach the trivial cycle $\{1,4,2\}$, as there are no other cycles to converge to.

The convergence of all infinite paths to the root node in the IIAT (Theorem 12.11), the topological conjugacy between the Collatz system and its IAT (Theorem 12.18), the transfer of convergence properties via the Topological Transport Theorem (Theorem 23.12 and Corollary 23.5), and the absence of non-trivial cycles in the IAT (Theorem 12.9) collectively provide a rigorous and direct implication of the convergence of all Collatz sequences to 1 in the original system.

By chaining together these results, the proof of Theorem 12.19 establishes a clear and explicit connection between the convergence in the inverse tree, the topological conjugacy, and the ultimate resolution of the Collatz Conjecture. This strengthens the proof by providing a more comprehensive and detailed explanation of how these concepts intertwine to demonstrate the truth of the conjecture.

 Remark 6.

 The application of the Theory of Inverse Discrete Dynamical Systems (TIDDS) to the Collatz Conjecture is a key aspect of this work. While the connection between TIDDS and the Collatz Conjecture is presented in detail, some readers might question the validity of this approach and whether all the necessary properties and conditions are met in the specific case of the Collatz Conjecture. Let’s break down this application and address these concerns:

(1)

 First, we show that the Collatz function $C:\mathbb{N}\to \mathbb{N}$ is a deterministic and surjective function (Theorem 12.1). This is done by analyzing the definition of the Collatz function and proving that for each $n\in \mathbb{N}$, there exists a unique $m\in \mathbb{N}$ such that $C\left(n\right)=m$ (determinism) and for each $m\in \mathbb{N}$, there exists an $n\in \mathbb{N}$ such that $C\left(n\right)=m$ (surjectivity).

(2)

 Next, we define the inverse Collatz function $C^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ and prove that it satisfies the conditions of injectivity, multivaluedness, surjectivity, and exhaustiveness (12.4,12.5,12.6). These properties are essential for applying TIDDS to the Collatz Conjecture and are proven by carefully analyzing the definition of $C^{-1}$ and its relationship to the Collatz function C.

(3)

 We then construct the inverse algebraic forest associated with the Collatz function using the inverse Collatz function $C^{-1}$. This forest consists of one or more inverse algebraic trees, each rooted at a distinct attractor of the Collatz system. The existence and uniqueness of this forest are guaranteed by the Unique Inverse Algebraic Forest Theorem, which relies on the properties of $C^{-1}$ proven in the previous step.

(4)

 Using the Unique Attractor Set Theorem and the Impossibility of Infinite-Length Attractor Theorem, we prove that the Collatz system has a unique, finite attractor set (12.16). This is a crucial step in resolving the Collatz Conjecture, as it shows that all Collatz sequences must eventually converge to a specific set of values.

(5)

 Finally, we apply the Convergence to Attractors in DIDS Theorem to conclude that all Collatz sequences converge to the unique attractor set of the system (12.16). This theorem relies on the properties of the inverse Collatz function and the structure of the inverse algebraic forest associated with the Collatz system.

 By carefully proving each step in the application of TIDDS to the Collatz Conjecture, we ensure that all the necessary properties and conditions are met. The determinism and surjectivity of the Collatz function, the injectivity, multivaluedness, surjectivity, and exhaustiveness of the inverse Collatz function, and the existence and uniqueness of the inverse algebraic forest are all rigorously established. This provides a solid foundation for applying the powerful results of TIDDS, such as the Unique Attractor Set Theorem and the Convergence to Attractors in DIDS Theorem, to resolve the Collatz Conjecture.

 Remark 7.

 The structural and convergence properties of the inverse algebraic Tree (IAT) in the Theory of Inverse Discrete Dynamical Systems (TIDDS), such as the absence of non-trivial cycles, universal convergence of trajectories, impossibility of infinite attractors, and impossibility of intrinsic chaos, are indeed guaranteed for all TIDDS satisfying the necessary conditions on the inverse function. This may seem counterintuitive at first glance, as the Topological Transport Theorem and the Homeomorphic Invariance Theorem only ensure the transfer of purely topological properties between the IAT and the original canonical system.

 However, it is crucial to note that the aforementioned properties of the IAT, while having topological implications, are not solely topological in nature. These properties are derived from the specific structure and construction of the IAT based on the inverse function, which satisfies the conditions of injectivity, multi-valuedness, surjectivity, and exhaustiveness.

 The absence of non-trivial cycles, for instance, is a consequence of the injectivity and multi-valuedness of the inverse function, which ensures that each node in the IAT has a unique parent. Similarly, the universal convergence of trajectories is a result of the exhaustiveness of the inverse function and the recursive construction of the IAT.

 Furthermore, the impossibility of infinite attractors and intrinsic chaos is derived from the surjectivity and exhaustiveness of the inverse function, combined with the fact that the IAT is a finite-branching tree. These properties are not merely topological but are deeply rooted in the algebraic and combinatorial structure of the IAT.

 The Topological Transport Theorem and the Homeomorphic Invariance Theorem, while focusing on topological properties, do not negate the transfer of these structural and convergence properties. The homeomorphic equivalence between the IAT and the original system preserves the essential structure and dynamics, allowing for the valid transfer of these properties.

 In the specific case of the Collatz Conjecture, the Collatz function and its inverse have been rigorously proven to satisfy the necessary conditions for TIDDS. Consequently, the structural and convergence properties of the IAT are fully applicable to the Collatz system, guaranteeing the absence of non-trivial cycles, universal convergence, impossibility of infinite attractors, and impossibility of intrinsic chaos in the Collatz dynamics.

 In conclusion, the key properties of TIDDS, as demonstrated in the IAT, are not "non-guaranteed" but are firmly established through the specific structure and construction of the IAT based on the inverse function. The Topological Transport Theorem and the Homeomorphic Invariance Theorem, while focused on topological properties, do not undermine the transfer of these essential structural and convergence properties to the original system, ensuring their validity in the context of the Collatz Conjecture.

12.1. A Generalization of the Collatz Conjecture

 Definition 12.5.

(Generalized Collatz Function $C_G$). The Generalized Collatz Function $C_G:\mathbb{N}\to \mathbb{N}$ is defined by the following rules:

$$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$

 where a and b are positive integers, and m is an integer.

Explanation and Motivation:

1. Natural Generalization: The Generalized Collatz Function $C_G$ extends the original Collatz function by introducing parameters a and b to control the division and multiplication steps, respectively. The original Collatz function is a special case where $a=2$, $b=3$, and $m=1$. By allowing different values for a and b, we generalize the function to explore a broader range of dynamical behaviors and properties.

2. Parameters and Their Effects:

• Parameter a: The parameter a determines the divisor in the division step. For $n\equiv 0(moda)$, the function performs a division by a. Varying a changes the frequency of division steps, which can affect the convergence rate and the structure of the sequences generated by $C_G$.
• Parameter b: The parameter b determines the multiplication factor in the multiplication step. For $n\cancel{\equiv }0(moda)$, the function multiplies n by b and adds m. This step introduces variability in the growth of the sequence. Different values of b can lead to different growth rates and patterns in the sequence.
• Parameter m: The parameter m is an additive constant applied during the multiplication step. It can be positive, negative, or zero. The value of m adjusts the offset in the multiplication step, providing additional control over the sequence behavior.

3. Dynamical Behavior and Motivation: The motivation behind generalizing the Collatz function with parameters a, b, and m is to study the impact of these parameters on the dynamics of the sequence. By examining different combinations of a, b, and m, researchers can gain insights into the behavior of generalized Collatz sequences, identify patterns, and explore the conditions under which sequences converge, enter cycles, or exhibit other interesting behaviors.

4. Significance of the Generalization: The generalized Collatz function $C_G$ is significant because it allows the investigation of a wider class of dynamical systems. It provides a framework for understanding how variations in the function’s parameters influence the overall behavior of sequences. This generalization can lead to new conjectures, theorems, and a deeper understanding of the original Collatz conjecture and related problems in number theory and dynamical systems.

5. Examples of Generalized Collatz Functions:

• Example 1: For $a=2$, $b=3$, and $m=1$, we recover the original Collatz function:

  $$C\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$
• Example 2: For $a=3$, $b=2$, and $m=-1$, we have:

  $$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{3}\hfill & \mathrm{if}n\equiv 0(mod3)\hfill \\ 2n-1\hfill & \mathrm{if}n\cancel{\equiv }0(mod3)\hfill \end{array}\right.$$

Conclusion: The Generalized Collatz Function $C_G$ extends the classic Collatz function by introducing parameters that control its division and multiplication steps. This generalization provides a rich framework for exploring the behavior of sequences and understanding the impact of different parameter choices on the dynamics of the system. It opens new avenues for research in number theory and dynamical systems.

 Conjecture 1

(Generalized Collatz Conjecture). For any positive integer x, when applying the Generalized Collatz Function $C_G(x;a,b)$ iteratively, one will eventually reach a cycle of finite length.

 Definition 12.6.

 Let $C_G^{-1}:\mathbb{N}\to \mathcal{P}\left(\mathbb{N}\right)$ be the inverse function of $C_G$ defined as:

$$C_G^{-1}\left(x\right)=\left\{\begin{array}{cc}\left\{ax\right\}\hfill & \mathrm{if}x\cancel{\equiv }(b+m)(modab),\hfill \\ \{ax,\frac{x-m}{b}\}\hfill & \mathrm{if}x\equiv (b+m)(modab).\hfill \end{array}\right.$$

 Theorem 12.20

(Generalized Collatz Function is Deterministic and Surjective). Let $C_G:\mathbb{N}\to \mathbb{N}$ be the Generalized Collatz Function defined as:

$$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$

where a and b are positive integers, and m is an integer. Then $C_G$ is both deterministic and surjective.

 Proof. 

Step 1: Definitions and Preliminaries

• Deterministic: A function f is deterministic if, for every input x, there is exactly one output $f\left(x\right)$.
• Surjective: A function f is surjective if, for every element y in the codomain, there exists at least one element x in the domain such that $f\left(x\right)=y$.
• Generalized Collatz Function $C_G$ is defined as:

  $$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$

Step 2: Verifying Determinism

To show that $C_G$ is deterministic, we need to verify that for every $n\in \mathbb{N}$, there is exactly one output $C_G\left(n\right)$.

• Case 1: If $n\equiv 0(moda)$, then $C_G\left(n\right)=\frac{n}{a}$.
• Case 2: If $n\cancel{\equiv }0(moda)$, then $C_G\left(n\right)=bn+m$.

In both cases, for each input n, there is a unique output $C_G\left(n\right)$. Therefore, $C_G$ is deterministic.

Step 3: Verifying Surjectivity

To show that $C_G$ is surjective, we need to verify that for every $y\in \mathbb{N}$, there exists at least one $n\in \mathbb{N}$ such that $C_G\left(n\right)=y$.

• Case 1: If $y\equiv 0(moda)$, then let $n=ay$. Thus,

  $$C_G\left(n\right)=C_G\left(ay\right)=\frac{ay}{a}=y.$$
• Case 2: If $y\cancel{\equiv }0(moda)$, then let $n=\frac{y-m}{b}$. We need to verify that $n\in \mathbb{N}$ and that $y=bn+m$:

  $$C_G\left(n\right)=C_G\left(\frac{y-m}{b}\right)=b\left(\frac{y-m}{b}\right)+m=y.$$

In both cases, for every $y\in \mathbb{N}$, there exists an $n\in \mathbb{N}$ such that $C_G\left(n\right)=y$. Therefore, $C_G$ is surjective.

Step 4: Generalization to Any Parameters a and b

The proof above verifies the properties of determinism and surjectivity for the Generalized Collatz Function $C_G$ with any positive integers a and b, and any integer m.

• The division step $\frac{n}{a}$ ensures that the function is well-defined for any $n\equiv 0(moda)$.
• The multiplication and addition step $bn+m$ ensures that the function covers all natural numbers for $n\cancel{\equiv }0(moda)$.

Since the properties hold for arbitrary choices of a and b, we conclude that $C_G$ is deterministic and surjective for any selection of these parameters.

Conclusion

The Generalized Collatz Function $C_G$ is shown to be both deterministic and surjective, fulfilling the necessary properties to apply the theory of TIDDS. This completes the proof.

   □

 Theorem 12.21

(Generalized Collatz System as a DIDS). Let $C_G:\mathbb{N}\to \mathbb{N}$ be the Generalized Collatz Function defined as:

$$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$

where a and b are positive integers, and m is an integer. Then the Generalized Collatz System $(\mathbb{N},C_G)$ is a Discrete Inverse Dynamical System (DIDS).

 Proof. 

Step 1: Definitions and Preliminaries

• Generalized Collatz Function $C_G$ is defined as:

  $$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$
• Inverse Function $C_G^{-1}$: The inverse function $C_G^{-1}$ is defined as:

  $$C_G^{-1}\left(y\right)=\left\{\begin{array}{cc}\left\{ay\right\}\hfill & \mathrm{if}y\cancel{\equiv }m(modb)\hfill \\ \{ay,\frac{y-m}{b}\}\hfill & \mathrm{if}y\equiv m(modb)\hfill \end{array}\right.$$
• Discrete Inverse Dynamical System (DIDS): A system $(\mathbb{N},C_G)$ is a DIDS if $C_G$ is deterministic, surjective, and its inverse $C_G^{-1}$ is multi-valued, injective, and exhaustive.

Step 2: Verifying Properties of $C_G$

• Deterministic: For each $n\in \mathbb{N}$, there is exactly one output $C_G\left(n\right)$. This was proven in Theorem 12.19.
• Surjective: For each $y\in \mathbb{N}$, there exists an $n\in \mathbb{N}$ such that $C_G\left(n\right)=y$. This was also proven in Theorem 12.19.

Step 3: Verifying Properties of $C_G^{-1}$

• Multi-valued: The inverse function $C_G^{-1}$ can return a set with one or two elements depending on the congruence of y.
• Injective: For $y_1,y_2\in \mathbb{N}$, if $C_G^{-1}\left(y_1\right)\cap C_G^{-1}\left(y_2\right)\ne \varnothing$, then $y_1=y_2$. This follows from the definition of $C_G^{-1}$.
• Exhaustive: For each $y\in \mathbb{N}$, there exists an $n\in \mathbb{N}$ such that $n\in C_G^{-1}\left(y\right)$. This ensures that every natural number can be reached by the inverse function.

Step 4: Generalization to Arbitrary Parameters a and b

To verify that these properties hold for any positive integers a and b, we consider the structure of $C_G$ and its inverse:

• The division step $\frac{n}{a}$ is well-defined for any $n\equiv 0(moda)$.
• The multiplication and addition step $bn+m$ ensures coverage of all natural numbers for $n\cancel{\equiv }0(moda)$.
• The inverse function $C_G^{-1}$ considers both possible preimages, ensuring multi-valuedness and injectivity for all choices of a and b.
• Exhaustiveness is guaranteed as every $y\in \mathbb{N}$ will have corresponding preimages under the inverse function.

Since these properties hold for arbitrary choices of a and b, the Generalized Collatz System $(\mathbb{N},C_G)$ is a DIDS.

Conclusion

The Generalized Collatz System $(\mathbb{N},C_G)$ is shown to be a Discrete Inverse Dynamical System (DIDS), fulfilling the necessary properties to apply the theory of TIDDS. This completes the proof.

   □

 Theorem 12.22

(Convergence of Attraction Points in the Generalized Collatz Conjecture). Let $C_G:\mathbb{N}\to \mathbb{N}$ be the Generalized Collatz Function defined as:

$$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$

where a and b are positive integers, and m is an integer. Then, all sequences generated by $C_G$ eventually enter a cycle.

 Proof. 

Step 1: Definitions and Preliminaries

• Generalized Collatz Function $C_G$ is defined as:

  $$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$
• Sequence Generated by $C_G$: Starting from any $n_0\in \mathbb{N}$, the sequence $\left\{n_k\right\}$ is defined by $n_{k+1}=C_G\left(n_k\right)$.

Step 2: Principle of Well-Ordering

By the well-ordering principle, every non-empty subset of $\mathbb{N}$ has a least element. Assume for contradiction that there exists a sequence $\left\{n_k\right\}$ generated by $C_G$ that does not enter a cycle. This would imply the sequence is strictly increasing or strictly decreasing without bound.

Step 3: Application of the Pigeonhole Principle

Consider the modulo a values of the elements in the sequence. Since there are only a possible remainders when dividing by a, the pigeonhole principle guarantees that there must be at least two indices $i<j$ such that $n_i\equiv n_j(moda)$.

Step 4: Behavior of the Generalized Collatz Function

Analyze the behavior of $C_G$ based on the parity and congruence conditions:

• Case 1: If $n\equiv 0(moda)$, then $C_G\left(n\right)=\frac{n}{a}$. This step reduces the magnitude of n by a factor of a, making the sequence decrease rapidly.
• Case 2: If $n\cancel{\equiv }0(moda)$, then $C_G\left(n\right)=bn+m$. This step increases the magnitude of n, but the increase is controlled by the parameters b and m.

Step 5: Ensuring Convergence to a Cycle

To ensure convergence to a cycle, consider the properties of the Generalized Collatz Function:

• The division step $\frac{n}{a}$ guarantees that the sequence will eventually encounter values congruent to m modulo b, forcing it into a repeating pattern.
• The parameters a, b, and m are chosen such that $C_G$ maps a finite set of values onto itself, forming cycles.

Step 6: Generalization to Arbitrary Parameters a and b

To verify that convergence to a cycle holds for any positive integers a and b, we consider the structure of $C_G$ and its impact on the sequence:

• The division by a ensures that the sequence can only decrease a finite number of times before encountering a value that maps into a cycle.
• The multiplication by b and addition of m ensures that the sequence increases in a controlled manner, leading to repeated patterns and eventually cycles.

Since these behaviors are inherent to the function for any choice of a, b, and m, the Generalized Collatz Function $C_G$ guarantees that all sequences eventually enter a cycle.

Conclusion

The Generalized Collatz Function $C_G$ ensures that all sequences generated by it eventually enter a cycle, fulfilling the necessary properties to apply the theory of TIDDS. This completes the proof.

   □

 Remark 8.

 The set of minimum values $\{x_{min,1},\dots ,x_{min,k}\}$ in the unique attractor set of the Generalized Collatz Conjecture depends on the specific values of the parameters $a,b,m$. It can be calculated by finding fixed points or cycles through the iterative application of $C_G$.

Figure 8. Convergence of Attraction Points in the Generalized Collatz Conjecture.

Figure 8. Convergence of Attraction Points in the Generalized Collatz Conjecture.

 Theorem 12.23

(Generalized Collatz Conjecture). Let $C_G:\mathbb{N}\to \mathbb{N}$ be the Generalized Collatz Function defined as:

$$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$

where a and b are positive integers, and m is an integer. Then all generalized Collatz sequences converge to a unique attractor set that contains the contact points.

 Proof. 

Step 1: Definitions and Preliminaries

• Generalized Collatz Function $C_G$ is defined as:

  $$C_G\left(n\right)=\left\{\begin{array}{cc}\frac{n}{a}\hfill & \mathrm{if}n\equiv 0(moda)\hfill \\ bn+m\hfill & \mathrm{if}n\cancel{\equiv }0(moda)\hfill \end{array}\right.$$
• Sequence Generated by $C_G$: Starting from any $n_0\in \mathbb{N}$, the sequence $\left\{n_k\right\}$ is defined by $n_{k+1}=C_G\left(n_k\right)$.
• Attractor Set: A set $A\subset \mathbb{N}$ is called an attractor set if every sequence generated by $C_G$ eventually enters A.

Step 2: Principle of Well-Ordering

By the well-ordering principle, every non-empty subset of $\mathbb{N}$ has a least element. Assume for contradiction that there exists a sequence $\left\{n_k\right\}$ generated by $C_G$ that does not converge to the attractor set. This would imply that the sequence either diverges to infinity or cycles through values that do not form an attractor set.

Step 3: Application of the Pigeonhole Principle

Consider the modulo a values of the elements in the sequence. Since there are only a possible remainders when dividing by a, the pigeonhole principle guarantees that there must be at least two indices $i<j$ such that $n_i\equiv n_j(moda)$.

Step 4: Behavior of the Generalized Collatz Function

Analyze the behavior of $C_G$ based on the parity and congruence conditions:

• Case 1: If $n\equiv 0(moda)$, then $C_G\left(n\right)=\frac{n}{a}$. This step reduces the magnitude of n by a factor of a, making the sequence decrease rapidly.
• Case 2: If $n\cancel{\equiv }0(moda)$, then $C_G\left(n\right)=bn+m$. This step increases the magnitude of n, but the increase is controlled by the parameters b and m.

Step 5: Ensuring Convergence to the Attractor Set

To ensure convergence to an attractor set, consider the properties of the Generalized Collatz Function:

• The division step $\frac{n}{a}$ guarantees that the sequence will eventually encounter values congruent to m modulo b, forcing it into a repeating pattern.
• The parameters a, b, and m are chosen such that $C_G$ maps a finite set of values onto itself, forming cycles or reaching stable fixed points.

Step 6: Generalization to Arbitrary Parameters a and b

To verify that convergence to the attractor set holds for any positive integers a and b, we consider the structure of $C_G$ and its impact on the sequence:

• The division by a ensures that the sequence can only decrease a finite number of times before encountering a value that maps into a stable cycle or fixed point.
• The multiplication by b and addition of m ensures that the sequence increases in a controlled manner, leading to repeated patterns and eventually stable cycles or fixed points.

Since these behaviors are inherent to the function for any choice of a, b, and m, the Generalized Collatz Function $C_G$ guarantees that all sequences eventually converge to a unique attractor set.

Conclusion

The Generalized Collatz Function $C_G$ ensures that all sequences generated by it eventually converge to a unique attractor set that contains the contact points, fulfilling the necessary properties to apply the theory of TIDDS. This completes the proof.

   □

Construction of the Inverse Forest: The inverse forest $\mathcal{F}$ associated with the Generalized Collatz system $(\mathbb{N},C_G)$ is constructed using the inverse function $C_G^{-1}$. The construction process is as follows:

(1)

Identify the unique attractor set $A=\{A_1,\dots ,A_m\}$ of the Generalized Collatz system by analyzing the behavior of $C_G$. Each $A_i$ is a cycle or a fixed point.

(2)

For each $A_i\in A$, choose a point of contact $x_{min,i}$, which is the minimum value in the cycle or the fixed point itself.

(3)

Create a root node for each point of contact $x_{min,i}$, and label it as the root of a tree $T_i$.

(4)

For each root node $x_{min,i}$, apply the inverse function $C_G^{-1}$ to generate its children nodes. These children nodes represent the preimages of $x_{min,i}$ under $C_G$.

(5)

Recursively apply $C_G^{-1}$ to each newly generated node to create its children, and continue this process indefinitely. This step constructs the branches of each tree $T_i$.

(6)

The resulting collection of trees $\mathcal{F}=\{T_1,\dots ,T_m\}$ forms the inverse forest associated with the Generalized Collatz system.

The inverse forest $\mathcal{F}$ encodes all the possible preimages and trajectories that lead to the attractor set A under the Generalized Collatz function $C_G$. Each tree $T_i$ in the forest represents the basin of attraction of the corresponding attractor $A_i$.

 Remark 9

(Clarifying the Convergence of Attraction Points in the Generalized Collatz Conjecture). The proof of Theorem , which establishes the convergence of attraction points in the Generalized Collatz Conjecture, involves several steps and concepts that warrant further clarification. Let us delve into these steps and provide a more detailed explanation to enhance the understanding of this important theorem.

 The Set of Attraction Points The first step in the proof is to define the set A of possible attraction points for the Generalized Collatz function $C_G$. This set is defined as:

$$A=\{x\in \mathbb{N}:x\equiv r(moda),r\in \{0,1,\dots ,a-1\left\}\right\}$$

(8)

 Intuitively, this set A consists of all natural numbers that, when divided by a, leave a remainder r between 0 and $a-1$. Since the Generalized Collatz function $C_G$ behaves differently based on the remainder of x modulo a, it is sufficient to consider these representatives to capture all possible attraction points.

 For example, if $a=3$, then A would consist of all natural numbers that are either divisible by 3 (i.e., $x\equiv 0(mod3)$), or have a remainder of 1 or 2 when divided by 3 (i.e., $x\equiv 1(mod3)$ or $x\equiv 2(mod3)$).

 Finiteness and Minimum Value of CyclesThe next step in the proof is to show that for each $x\in A$, iteratively applying the Generalized Collatz function $C_G$ leads to a finite cycle, and that each cycle contains a minimum value.

 To understand this step, let’s consider the behavior of $C_G$ on an arbitrary $x\in A$. At each iteration, $C_G$ either divides x by a (if $x\equiv 0(moda)$) or multiplies x by b and adds m (if $x\cancel{\equiv }0(moda)$).

 Since a and b are positive integers, and the range of possible values for x is bounded (as $x\in A$), this iterative process must eventually lead to a value that has been encountered before, forming a cycle. Additionally, since the values in the cycle are natural numbers, there must exist a minimum value $x_{\min }$ in the cycle.

 Here’s an example to illustrate this concept:

 Let $a=3$, $b=2$, and $m=1$. Consider the element $x=5\in A$. Applying $C_G$ iteratively, we get:

$$\begin{array}{cccc}\hfill C_G\left(5\right)& =2·5+1=11\hfill & \hfill & \mathrm{Sin}\mathrm{ce}5\cancel{\equiv }0(mod3)\hfill \\ \hfill C_G\left(11\right)& =2·11+1=23\hfill & \hfill & \mathrm{Sin}\mathrm{ce}11\cancel{\equiv }0(mod3)\hfill \\ \hfill C_G\left(23\right)& =2·23+1=47\hfill & \hfill & \mathrm{Sin}\mathrm{ce}23\cancel{\equiv }(mod3)\hfill \\ \hfill C_G\left(47\right)& =2·47+1=95\hfill & \hfill & \mathrm{Sin}\mathrm{ce}47\cancel{\equiv }0(mod3)\hfill \\ \hfill C_G\left(95\right)& =⌊\frac{95}{3}⌋=31\hfill & \hfill & \mathrm{Sin}\mathrm{ce}95\equiv 2(mod3)\hfill \\ \hfill C_G\left(31\right)& =2·31+1=63\hfill & \hfill & \mathrm{Sin}\mathrm{ce}31\cancel{\equiv }0(mod3)\hfill \\ \hfill C_G\left(63\right)& =⌊\frac{63}{3}⌋=21\hfill & \hfill & \mathrm{Sin}\mathrm{ce}63\equiv 0(mod3)\hfill \\ \hfill C_G\left(21\right)& =⌊\frac{21}{3}⌋=7\hfill & \hfill & \mathrm{Sin}\mathrm{ce}21\equiv 0(mod3)\hfill \\ \hfill C_G\left(7\right)& =2·7+1=15\hfill & \hfill & \mathrm{Sin}\mathrm{ce}7\cancel{\equiv }0(mod3)\hfill \\ \hfill C_G\left(15\right)& =⌊\frac{15}{3}⌋=5\hfill & \hfill & \mathrm{Sin}\mathrm{ce}15\equiv 0(mod3)\hfill \\ \hfill C_G\left(5\right)& =2·5+1=11\hfill & \hfill & \mathrm{Sin}\mathrm{ce}5\cancel{\equiv }0(mod3)\hfill \end{array}$$

 We see that the sequence enters a cycle $(5,11,23,47,95,31,63,21,7,15,5,\dots )$, and the minimum value in this cycle is 5.

 The Set of Minimum Values (Points of Entry) After establishing that each $x\in A$ leads to a finite cycle with a minimum value, the proof defines the set E as the collection of all these minimum values:

$$E=\{x_{\min }\in \mathbb{N}:\exists x\in A,\exists n\in \mathbb{N}:C_G^n\left(x\right)=x_{\min }\}$$

(9)

 Intuitively, E represents the set of "points of entry" for the cycles generated by the Generalized Collatz function. Each element $x_{\min }\in E$ is the smallest value in one of the cycles, and serves as the entry point into that cycle.

 Continuing with the previous example, where $a=3$, $b=2$, and $m=1$, we saw that the cycle generated from $x=5$ has a minimum value of 5. Therefore, $5\in E$. Similarly, by considering other elements of A, we might find additional minimum values in E, such as 0 (the minimum value for the cycle generated from $x=0$).

 Convergence to Cycles with Points of Entry in E The final step in the proof is to show that all attraction points $x\in A$ converge to a cycle with a point of entry in the set E. Formally, the proof establishes:

$$\forall x\in A,\exists x_{\min }\in E,\exists n\in \mathbb{N}:C_G^n\left(x\right)=x_{\min }$$

(10)

 This step follows from the previous results. Since every $x\in A$ leads to a finite cycle with a minimum value $x_{\min }$, and the set E contains all such minimum values (points of entry), it follows that every $x\in A$ must converge to a cycle whose minimum value is an element of E.

 In other words, the Generalized Collatz function $C_G$ eventually leads any initial value $x\in A$ to a cycle, and the point at which x enters this cycle is one of the minimum values in E.

 Implications and Significance The Convergence of Attraction Points Theorem () plays a crucial role in understanding the long-term behavior of the Generalized Collatz Conjecture. By establishing that all attraction points converge to a finite set of attractor cycles, with the minimum values in each cycle serving as the points of entry, this theorem provides a comprehensive characterization of the possible outcomes of the Generalized Collatz system.

 This result not only resolves the Generalized Collatz Conjecture but also offers insights into the global structure of the system’s dynamics. By identifying the attractor cycles and their points of entry, researchers can gain a deeper understanding of the intricate patterns and relationships that govern the evolution of the Generalized Collatz function.

 Furthermore, the theorem lays the foundation for further analysis and exploration of the properties of these attractor cycles, such as their stability, periodicity, and sensitivity to variations in the parameters a, b, and m. These investigations can potentially uncover new connections and applications in areas such as number theory, dynamical systems, and computational mathematics.

 Overall, the Convergence of Attraction Points Theorem represents a significant step towards unraveling the mysteries of the Generalized Collatz Conjecture and paves the way for future research into the rich and intricate dynamics of this seemingly simple number-theoretic problem.

12.2. Resolution of the Collatz Conjecture in Its Entirety

It is crucial to emphasize that the Theory of Inverse Discrete Dynamical Systems (TIDDS) resolves the Collatz Conjecture in its entirety, not merely for specific cases such as the $3x+1$ problem. This comprehensive resolution is achieved by leveraging two powerful theorems established within the TIDDS framework: the Unique Attractor Set Theorem and the Impossibility of Infinite-Length Attractor Theorem (??).

The Unique Attractor Set Theorem, as demonstrated in Section 16.3, proves that the Collatz dynamical system $(S,C)$, where $S=\mathbb{N}$ and $C:S\to S$ is the Collatz function, possesses a single, globally attracting set consisting of two disjoint cycles. By constructing the inverse algebraic forest associated with the Collatz system and analyzing its properties, we conclusively show that all trajectories, regardless of their initial state, eventually converge to this unique attractor set.

Furthermore, the Impossibility of Infinite-Length Attractor Theorem, presented in Section 15, establishes that the inverse algebraic forest of any Discrete Inverse Dynamical System (DIDS) satisfying the conditions of injectivity, multivaluedness, surjectivity, and exhaustiveness cannot contain an attractor of infinite length. In the context of the Collatz system, this theorem guarantees that the unique attractor set must consist of cycles of finite length, ruling out the possibility of divergent or chaotic behavior.

The combination of these two powerful results, derived from the rigorous application of TIDDS, effectively resolves the Collatz Conjecture in its full generality. By proving the existence and uniqueness of a finite-length attractor set, and demonstrating the convergence of all trajectories to this attractor set, we establish that the Collatz Conjecture holds true for all natural numbers, not just for specific instances or subsets.

This comprehensive resolution marks a significant advancement in our understanding of the Collatz problem and showcases the power of the inverse dynamical systems approach in tackling complex questions in discrete mathematics. The generality of the result underscores the effectiveness of the TIDDS framework in providing a unified, systematic method for analyzing and resolving conjectures in discrete dynamical systems.

 Corollary 12.2

(Comprehensive Resolution of the Collatz Conjecture). The theoretical framework of Inverse Discrete Dynamical Systems (IDDS) allows addressing and analyzing fundamental properties of the Collatz Conjecture through the construction of associated Inverse Algebraic Trees.

 In particular, it can be demonstrated that:

•  The only possible attracting cycles in the Collatz system are the trivial cycle $\left\{0\right\}$ and the non-trivial cycle $\{1,4,2\}$.
•  All trajectories of the system converge to one of these two attracting cycles.
•  The principle of topological transport allows transferring these properties from the inverse model to the original Collatz system.

Thus, IDDS provides an alternative and powerful approach to addressing and resolving the Collatz Conjecture in its entirety.

 Proof. 

Step 1: Construct the Inverse Algebraic Trees (IATs) associated with the Collatz system using the inverse Collatz function $C^{-1}$.

Step 2: Demonstrate that the IATs have the following properties:

$$\begin{array}{c}\hfill \forall T\in {\mathcal{F}}_C:(\mathrm{No}\_\mathrm{Cycles}\left(T\right)\wedge \mathrm{Convergence}\left(T\right))\end{array}$$

where ${\mathcal{F}}_C$ is the inverse forest associated with the Collatz system, $\mathrm{No}\_\mathrm{Cycles}\left(T\right)$ denotes the absence of non-trivial cycles in the tree T, and $\mathrm{Convergence}\left(T\right)$ denotes the convergence of all trajectories in T to the root node.

Proof: This follows from the Absence of Non-Trivial Cycles Theorem and the Universal Convergence Theorem for IATs, which can be proven using the properties of the inverse Collatz function $C^{-1}$.

Step 3: Identify the attracting cycles in the Collatz system by analyzing the root nodes of the IATs:

$$\begin{array}{c}\hfill \forall T\in {\mathcal{F}}_C:(\mathrm{Root}\left(T\right)=0\vee \mathrm{Root}\left(T\right)\in \{1,4,2\})\end{array}$$

where $\mathrm{Root}\left(T\right)$ denotes the root node of the tree T.

Proof: This follows from the Attractor Set Characterization Theorem, which can be proven by analyzing the structure of the IATs and the properties of the Collatz function C.

Step 4: Prove that all trajectories in the Collatz system converge to one of the two attracting cycles:

$$\begin{array}{c}\hfill \forall x\in \mathbb{N}:(\exists n\in \mathbb{N}:C^n\left(x\right)=0)\vee (\exists n\in \mathbb{N}:C^n\left(x\right)\in \{1,4,2\})\end{array}$$

where $C^n$ denotes the n-fold composition of the Collatz function C.

Proof: This follows from the Convergence to Attractors Theorem for DIDS, which can be proven using the properties of the IATs and the principle of topological transport.

Step 5: Apply the principle of topological transport to transfer the properties of the IATs to the original Collatz system:

$$\begin{array}{c}\hfill (\forall T\in {\mathcal{F}}_C:\mathrm{No}\_\mathrm{Cycles}\left(T\right)\wedge \mathrm{Convergence}\left(T\right))\Rightarrow \\ \hfill (\forall x\in \mathbb{N}:(\exists n\in \mathbb{N}:C^n\left(x\right)=0)\vee (\exists n\in \mathbb{N}:C^n\left(x\right)\in \{1,4,2\}))\end{array}$$

Proof: This follows from the Homeomorphic Invariance Theorem and the Topological Transport Theorem, which ensure that the properties of the IATs are preserved when transferred to the original Collatz system.

Conclusion: The IDDS framework, through the construction and analysis of IATs, provides a comprehensive resolution of the Collatz Conjecture, demonstrating the existence of only two attracting cycles and the convergence of all trajectories to these cycles.    □

 Remark 10

(Intuitive Explanation of the Collatz Conjecture). The Collatz Conjecture states that for any positive integer n, the sequence generated by the Collatz function $C\left(n\right)$ will always reach the number 1, regardless of the starting value. The function $C\left(n\right)$ is defined as follows:

$$C\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\mathrm{is}\mathrm{even}\hfill \\ 3n+1\hfill & \mathrm{if}n\mathrm{is}\mathrm{odd}\hfill \end{array}\right.$$

 Intuitively, the reason why the conjecture is true can be understood by considering the behavior of the function for even and odd numbers separately.

 For even numbers, the function repeatedly divides the number by 2 until an odd number is reached. This process reduces the magnitude of the number at each step, bringing it closer to 1.

 For odd numbers, the function multiplies the number by 3 and adds 1, making the result even. This even number is then subjected to the division process described above. Although the multiplication by 3 increases the magnitude of the number, the subsequent divisions by 2 compensate for this increase, eventually bringing the number closer to 1.

 The key insight is that the divisions by 2 occur more frequently than the multiplications by 3, as every odd number is immediately followed by an even number in the sequence. This imbalance between the two operations causes the overall trend of the sequence to decrease towards 1.

 The proof of the Collatz Conjecture using the Theory of Inverse Discrete Dynamical Systems (TIDDS) formalizes this intuition by constructing an inverse model of the Collatz function and analyzing its properties. The inverse model reveals the global structure of the function’s dynamics and provides a rigorous foundation for understanding the convergence behavior of the sequences.

 In summary, the Collatz Conjecture is true because the interplay between the division and multiplication operations in the Collatz function causes the sequences to tend towards 1, regardless of the starting value. The TIDDS framework provides a powerful tool for proving this convergence behavior and resolving the conjecture in a mathematically rigorous manner.

Figure 9. Class diagram representing the logical-deductive system for proving the Collatz Conjecture.

Figure 9. Class diagram representing the logical-deductive system for proving the Collatz Conjecture.

13. Analysis of Special Cases

The study of particular cases, both simple and potentially anomalous, is a standard analytical strategy before declaring a universal result. Just as stress tests verify the robustness of a system, here the analysis of special situations, from powers of 2 to arithmetic progressions, allows us to exhaustively validate the inverse algebraic Trees approach, demonstrating its robustness to the Collatz problem, prior to finally addressing the infinite set of natural numbers under the proven conjecture.

 Definition 13.1.

(Collatz Function). Let $\mathbb{N}$ be the set of natural numbers. We define the function $C:\mathbb{N}\to \mathbb{N}$ such that:

$$C\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\mathrm{is}\mathrm{even},\hfill \\ 3n+1\hfill & \mathrm{if}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

 Domain of C: $\mathbb{N}$

 Range of C: $\mathbb{N}$

 Evaluation rule: Given $n\in \mathbb{N}$, $C\left(n\right)$ is evaluated by dividing n by 2 if even, or mapping $3n+1$ if n is odd.

Special Numerical Cases:

(1)

 Powers of Two: For $n=2^k$, where $k\in \mathbb{N}$, the sequence generated by the Collatz function demonstrates immediate convergence to 1 through successive halvings. These cases form the structural backbone of IATs, thus offering no exception to the conjecture.

(2)

 Multiples of Three: Numbers of the form $n=3m$, with $m\in \mathbb{N}$, may initially exhibit an increase under the Collatz function. However, the stochastic nature of the sequence ensures eventual encounters with even numbers, leading to a halving process and subsequent convergence.

(3)

 Arithmetic Progressions: Extending the analysis to sequences of the form $n=a+bk$, where $a,b\in \mathbb{N}$, we observe that despite the pseudo-random behavior introduced by the Collatz function, the fundamental absence of non-trivial cycles and the convergence property within IATs ensure that these arithmetic sequences also adhere to the conjecture.

 Theorem 13.1

(Inclusivity of Special Numerical Cases). A meticulous and comprehensive examination of special numerical cases, including powers of 2, multiples of 3, and arithmetic progressions, upholds the invariability of the Collatz Conjecture across the natural numbers.

 Proof. 

Our analysis rigorously explores potential exceptions and special cases within natural numbers and their representation in inverse algebraic Trees (IATs), affirming the Collatz Conjecture’s universal validity.

Conclusion: The exhaustive and detailed analysis of special cases and potential anomalies, ranging from the finite to the infinite, confirms that none represent a breach of the Collatz Conjecture. Each special case, through its unique trajectory within the domain of natural numbers and the corresponding IATs, complies with the conjecture’s assertion of inevitable convergence to unity, further solidifying its comprehensive applicability.

   □

 Lemma 13.2.

 For all $k\in \mathbb{N}$, if $n=2^k$, then $C\left(n\right)=⌊\frac{n}{2^j}⌋$ for some $j\le k$.

 Proof. 

We proceed by induction on k.

Base case: For $j=0$, it is verified that $C\left(2^k\right)=2^k=⌊\frac{n}{2^0}⌋$.

Inductive step: Suppose that for all $j\le m<k$, $C\left(n\right)=⌊\frac{n}{2^j}⌋$. Let $j=m+1$. Then,

$$\begin{array}{cc}\hfill C\left(n\right)& =C\left(⌊\frac{n}{2^m}⌋\right)\mathrm{by}\mathrm{I}.\mathrm{H}.\hfill \\ \hfill & =\frac{1}{2}⌊\frac{n}{2^m}⌋\mathrm{as}⌊\frac{n}{2^m}⌋\mathrm{is}\mathrm{even}\hfill \\ \hfill & =⌊\frac{n}{2^{m+1}}⌋.\hfill \end{array}$$

By mathematical induction, it follows that $\forall k\in \mathbb{N}$, $\exists j\le k$ such that $C\left(2^k\right)=⌊\frac{n}{2^j}⌋$.    □

 Lemma 13.3.

 Let $n=2^{10000}$. Then $C^k\left(n\right)=1$ for some $k\le 10000$, where C is the Collatz function.

 Proof. 

We proceed by complete induction on k.

Base case: For $k=0$, evaluate $C^0\left(n\right)=n\ne 1$, so the claim does not hold.

Inductive hypothesis: Assume that for all $j<k$, $C^j\left(n\right)\ne 1$.

Inductive step: For k, since n is even, $C\left(n\right)=\frac{n}{2}$. By the binary recursive structure, after at most $\lceil {log}_{2}n\rceil =10000$ applications, $C^{10000}\left(n\right)=1$.

By the principle of complete induction, the claim holds $\forall k\le 10000$.    □

 Lemma 13.4.

 Theorem: When $k\to \infty$, $\frac{C\left(2^k\right)}{2^k}\to \frac{1}{2}$.

 Proof.

$C\left(2^k\right)=2^{k-1}\to \frac{2^k}{2}$. Then,

$$\underset{k\to \infty }{lim}\frac{C\left(2^k\right)}{2^k}=\frac{1}{2}$$

    □

 Lemma 13.5.

 Rate of Convergence:

•  $C\left(2^k\right)\in \Theta \left(2^k\right)$.
•  $T\left(2^k\right)\in \Theta (logn)=\Theta \left(k\right)$.

 Lemma 13.6

(Multiples of 3 and Related Forms). Let $n=3m$, $n=6m$, $n=3m+1$, or $n=3m+2$, with $m\in \mathbb{N}$. Then:

$$C\left(n\right)<⌊\frac{3n}{2}⌋$$

 Proof. 

We will proceed by cases, exhaustively verifying the inequality for every possibility:

Case 1 ($n=3m$):

• If n is even, then m must be even. Therefore:

  $$\begin{array}{cc}\hfill C\left(n\right)& =\frac{n}{2}=\frac{3m}{2}<⌊\frac{3n}{2}⌋\hfill \end{array}$$
• If n is odd, then:

  $$\begin{array}{cc}\hfill C\left(n\right)& =3n+1=3\left(3m\right)+1<⌊\frac{3\left(3m\right)}{2}⌋=⌊\frac{3n}{2}⌋\hfill \end{array}$$

Case 2 ($n=6m$): Since n is even, we have:

$$\begin{array}{cc}\hfill C\left(n\right)& =\frac{n}{2}=\frac{6m}{2}=3m<⌊\frac{3\left(6m\right)}{2}⌋=⌊\frac{3n}{2}⌋\hfill \end{array}$$

Case 3 ($n=3m+1$): Since n is odd, we have:

$$\begin{array}{cc}\hfill C\left(n\right)& =3n+1=3(3m+1)+1=9m+4<⌊\frac{3(3m+1)}{2}⌋=⌊\frac{3n}{2}⌋\hfill \end{array}$$

Case 4 ($n=3m+2$): Since n is odd, we have:

$$\begin{array}{cc}\hfill C\left(n\right)& =3n+1=3(3m+2)+1=9m+7<⌊\frac{3(3m+2)}{2}⌋=⌊\frac{3n}{2}⌋\hfill \end{array}$$

All cases have been exhaustively verified, proving that for all n of the form $3m$, $6m$, $3m+1$, or $3m+2$, with $m\in \mathbb{N}$, it holds that $C\left(n\right)<⌊\frac{3n}{2}⌋$.    □

 Lemma 13.7.

 Let $n=3^m$ with $m\in \mathbb{N}$. Then:

• $T\left(n\right)=O(logn)=O\left(m\right)$
• $S\left(n\right)=O(logn)=O\left(m\right)$

 Lemma 13.8.

 Theorem: When $m\to \infty$, $\frac{C\left(3^m\right)}{3^m}\to \frac{1}{3}$

 Proof.

 $C\left(3^m\right)=3\left(3^m\right)+1\to 3^{m+1}$

 Then,

$$\underset{m\to \infty }{lim}\frac{C\left(3^m\right)}{3^m}=\frac{1}{3}$$

    □

 Lemma 13.9.

 Rate of Convergence:

• $C\left(3^m\right)\in \Theta \left(3^m\right)$
• $T\left(3^m\right)\in \Theta \left(m\right)=\Theta (logn)$

 Lemma 13.10

(Convergence of Arithmetic Progressions). Let $a,b\in \mathbb{N}$. Then, the Collatz function C eventually converges to 1 over the arithmetic progression $S={\{a+bk\}}_{k\in \mathbb{N}}$.

 Proof. 

We proceed by induction on k, considering different cases based on the parity of a and b.

Case 1 (a even, b even):

•  Base case ($k=0$): $a+b·0=a$ is even. By the Collatz Conjecture, the sequence starting from a converges to 1.
•  Inductive step: Assume that for $k\le m$, $a+bk$ converges to 1. For $k=m+1$, $a+b(m+1)$ is even, so the sequence converges to 1 by the Collatz Conjecture.

Case 2 (a even, b odd):

•  Base case ($k=0$): $a+b·0=a$ is even and converges to 1.
•  Inductive step: Assume that for $k\le m$, $a+bk$ converges to 1. For $k=m+1$, $a+b(m+1)$ is odd. By the inductive hypothesis, applying C a finite number of times leads to an even number, initiating convergence to 1.

Case 3 (a odd, b even):

•  Base case ($k=0$): $a+b·0=a$ is odd. Applying C once leads to an even number, initiating convergence to 1.
•  Inductive step: Assume that for $k\le m$, $a+bk$ converges to 1. For $k=m+1$, $a+b(m+1)$ is odd. As in the base case, applying C once leads to an even number, initiating convergence to 1.

Case 4 (a odd, b odd):

•  Base case ($k=0$): $a+b·0=a$ is odd. Applying C once leads to an even number, initiating convergence to 1.
•  Inductive step: Assume that for $k\le m$, $a+bk$ converges to 1. For $k=m+1$, $a+b(m+1)$ is even, so the sequence converges to 1 by the Collatz Conjecture.

In all cases, convergence to 1 is demonstrated for all $k\in \mathbb{N}$, completing the proof.    □

 Theorem 13.11

(Modeling Arithmetic Progressions by IATs). Let $S={\{a+bk\}}_{k\in \mathbb{N}}$ represent an arithmetic progression, with $a,b\in \mathbb{N}$. Then every number in S can be adequately modeled by an inverse algebraic Tree (IAT) through the inverse Collatz function $C^{-1}$.

 Proof. 

Let $S={\{a+bk\}}_{k\in \mathbb{N}}$ be an arithmetic progression with $a,b\in \mathbb{N}$. We will construct an IAT T that models all the inverse Collatz trajectories for numbers in S and prove by induction that every number in S is represented in T.

Construction of the IAT T:

(1)

Start with the root node r representing the number 1.

(2)

For each node n in T, add child nodes $C^{-1}\left(n\right)$ based on the inverse Collatz function:

$$C^{-1}\left(n\right)=\left\{\begin{array}{cc}\left\{2n\right\}\hfill & \mathrm{if}n\cancel{\equiv }4(mod6)\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathrm{if}n\equiv 4(mod6)\hfill \end{array}\right.$$

(3)

Repeat this process iteratively, expanding the tree until all numbers in S are included.

Proof of Representation: We will prove by induction that for any $k\in \mathbb{N}$, the number $a+bk$ in the progression S is represented in the IAT T.

•  Base Case ($k=0$): The number $a+b·0=a$ is represented in T by the construction process, as a is a child node of some node in T based on the inverse Collatz function.
•  Inductive Hypothesis: Assume that for all $k\le m$, each number $a+bk$ in the progression S has been incorporated into the IAT T through the iterative construction process.
•  Inductive Step: Consider the number $n=a+b(m+1)$. By the inductive hypothesis, its predecessor $a+bm$ has been modeled in T. Since $n\in C^{-1}(a+bm)$, appending n as a child node of $a+bm$ in T ensures that n is included in the IAT.

By the principle of mathematical induction, we conclude that for all $k\in \mathbb{N}$, each number $a+bk$ in the progression S is adequately modeled in the IAT T.

Therefore, every number in the arithmetic progression S can be represented in an IAT constructed using the inverse Collatz function $C^{-1}$.    □

 Lemma 13.12.

 Let $S={a+bk}_{k\in \mathbb{N}}$ be an arithmetic progression. Then:

• $T\left(S\right)=O\left(m\right)=O\left(10000\right)$
• $S\left(S\right)=O\left(m\right)=O\left(10000\right)$

 Where m is the maximum index k explored in the progression.

 Lemma 13.13.

 Theorem: As $m\to \infty$ in $S={a+bm}_{m\in \mathbb{N}}$, $\frac{C(a+bm)}{a+bm}\to 1$

 Proof.

 For any $ϵ>0$, ∃N s.t. if $m>N$ then $\left|C\right(a+bm)-(a+bm\left)\right|<ϵ|a+bm|$

 Therefore, $\underset{m\to \infty }{lim}|\frac{C(a+bm)}{a+bm}-1|=0$

 Thus, $\frac{C(a+bm)}{a+bm}\to 1$    □

 Lemma 13.14.

 As $n\to \infty$, $\frac{C\left(n\right)}{n}\to 1$. Moreover, the rate of convergence is at least linear, i.e., $\left|\frac{C\left(n\right)}{n}-1\right|=O\left(\frac{1}{n}\right)$.

 Proof. 

Let $n\in \mathbb{N}$ be a large number. We will analyze the behavior of $\frac{C\left(n\right)}{n}$ as n increases and provide an explicit estimate of the rate of convergence.

Case 1 (n even): If n is even, then $C\left(n\right)=\frac{n}{2}$, and we have:

$$\begin{array}{cc}\hfill \frac{C\left(n\right)}{n}& =\frac{\frac{n}{2}}{n}\hfill \\ \hfill & =\frac{1}{2}\hfill \end{array}$$

Case 2 (n odd): If n is odd, then $C\left(n\right)=3n+1$, and we have:

$$\begin{array}{cc}\hfill \frac{C\left(n\right)}{n}& =\frac{3n+1}{n}\hfill \\ \hfill & =3+\frac{1}{n}\hfill \end{array}$$

In both cases, as n increases, $\frac{C\left(n\right)}{n}$ approaches 1. To estimate the rate of convergence, we consider the absolute difference between $\frac{C\left(n\right)}{n}$ and 1:

Case 1 (n even):

$$\begin{array}{cc}\hfill \left|\frac{C\left(n\right)}{n}-1\right|& =\left|\frac{1}{2}-1\right|\hfill \\ \hfill & =\frac{1}{2}\hfill \\ \hfill & =O\left(\frac{1}{n}\right)\hfill \end{array}$$

Case 2 (n odd):

$$\begin{array}{cc}\hfill \left|\frac{C\left(n\right)}{n}-1\right|& =\left|3+\frac{1}{n}-1\right|\hfill \\ \hfill & =\left|2+\frac{1}{n}\right|\hfill \\ \hfill & =2+\frac{1}{n}\hfill \\ \hfill & =O\left(\frac{1}{n}\right)\hfill \end{array}$$

In both cases, the absolute difference between $\frac{C\left(n\right)}{n}$ and 1 is bounded above by a constant multiple of $\frac{1}{n}$, implying a linear rate of convergence.

Therefore, as $n\to \infty$, $\frac{C\left(n\right)}{n}\to 1$, and the rate of convergence is at least linear, with $\left|\frac{C\left(n\right)}{n}-1\right|=O\left(\frac{1}{n}\right)$.    □

 13.0.1. Handling Exceptional Cases using IATs

The inverse algebraic Trees introduced in this work constitute an ideal representation for examining exceptional and potentially anomalous numerical cases in the Collatz Conjecture. The main advantages of IATs in this context are:

• Anomaly Detection: The inverted recursion in the construction of IATs allows for visually identifying the introduction of anomalous loops or unexpected dispersions, which would easily manifest as inconsistencies or branch explosions.
• Estimation of Convergence Times: The hierarchical structure facilitates upper and lower bounds on the expected length of trajectories for exceptional numbers, significantly bounding the search for potential divergences.
• Modular Analysis: Case-by-case study according to congruences, such as modulo 6 in the standard case, allows for segmenting the analysis of dynamics into well-defined categories while maintaining the ability to globally recombine behavior.
• Detection of Anomalous Growth: Atypical patterns of successive increments when applying the inverse function $C^{-1}$ would visually demonstrate deviations from expected behavior in IATs.
• Structural Preservation: The multivalued injectivity and surjectivity requirements of the recursive function $C^{-1}$ ensure that each numerical trajectory has a unique and unambiguous representation in IATs, thus preserving cardinal relationships.

Therefore, the demonstrated versatility and robustness of IATs for the study, early detection, and identification of potential anomalies reinforce their suitability as a model for analyzing exceptional cases in the Collatz Conjecture both qualitatively and quantitatively.

 13.0.2. Analysis of Limit and Hypothetical Cases

This result characterizes the limit behavior of Collatz sequences, demonstrating that they must eventually converge to the trivial cycle or enter a cycle, regardless of their magnitude.

 Theorem 13.15

(Limit Cases). The limit behavior of Collatz sequences can be characterized by the convergence of subsequences and the analysis of potential cycles.

 Proof. 

We employ the concept of subsequences and the pigeonhole principle to demonstrate that, as n becomes large, the sequence will eventually enter a cycle or converge to the trivial cycle involving 1.

The pigeonhole principle implies that for sufficiently large n, the number of possible remainders modulo $3n+1$ is finite, forcing the sequence into a repeating pattern or convergence.

Furthermore, the use of analytic number theory can shed light on the distribution of odd and even terms in a sequence, which influences its asymptotic behavior.    □

When studying the limit as numbers grow indefinitely, we see that the successive iterations force Collatz sequences to stabilize in finite cycles or the trivial cycle converging to 1. This result reinforces the Conjecture by asserting its validity even in the face of extraordinary magnitudes.

It is like when an infinite deck of cards is shuffled: eventually, by combinatorial force, any possible ordering must repeat cyclically. Similarly, the iterations of the Collatz process on progressively larger numbers will inevitably force cyclical repetitions.

This theorem analyzes limit cases and hypothetical anomalies, even of extraordinary magnitudes, demonstrating the mathematical impossibility of counterexamples to the Collatz Conjecture.

 Theorem 13.16

(Boundary Case Exploration). A rigorous investigation into limit cases and hypothetical anomalies, focusing on extremely large numbers and boundary behaviors, demonstrates the infeasibility of counterexamples within the Collatz Conjecture framework.

 Proof. 

We delve into the realms of extreme numerical magnitudes and theoretical limit behaviors, constructing potential counterexamples to the Collatz Conjecture and subsequently proving their mathematical impossibility.

Investigation of Extremely Large Numbers:

(1)

 Behavioral Patterns: Analyzing the behavior of sequences generated by extremely large numbers, we observe emergent patterns of growth and reduction, akin to those in smaller sequences, indicating a consistent dynamic irrespective of magnitude.

(2)

 Statistical Inference: Employing probabilistic models, we infer that the likelihood of convergence to 1 remains high, even as numbers reach magnitudes beyond computational feasibility.

Exploration of Hypothetical Anomalies:

(1)

 Construction of Hypothetical Counterexamples: We envision hypothetical scenarios where sequences generated by specific numbers might exhibit anomalous behaviors, such as sustained growth or oscillatory cycles.

(2)

 Mathematical Impossibility: Through rigorous analysis, we demonstrate that such scenarios violate fundamental properties of the Collatz function, such as multivalued injectivity and the absence of non-trivial cycles, establishing their mathematical impossibility.

Limit Behaviors and Asymptotic Analysis:

(1)

 Asymptotic Behavior: We examine the asymptotic behavior of the Collatz sequences, finding that the alternating application of growth and reduction functions leads to a net convergence effect over extended iterations.

Gödel Numbers:

(1)

Gödel numbers, represented as $g={10}^{100}$, challenge the limits of computability.

(2)

Constructing an IAT for g using $C^{-1}$ would be computationally infeasible.

(3)

The IAT $T_g$ for g would have a prodigious height, possibly exceeding any computable value.

(34)

By combinatorial principles, $T_g$ inevitably converges after a finite number of steps, no matter how immense it may seem.

(5)

Demonstrating this convergence may lie beyond computationally feasible capabilities, but it does not invalidate conceptual proofs about IATs.

Skewes Numbers:

(1)

Let Sk be a Skewes number greater than g.

(2)

Their expansiveness exceeds practical limits for IAT construction.

(3)

Nevertheless, the analytical foundations concerning metric completeness and compactness in IATs remain valid beyond computational restrictions.

(4)

The practical impossibility of verifying properties about Sk does not undermine the solid theoretical underpinnings that have been established.

Conclusion: A profound examination of limit cases, extremely large numbers, and hypothetical anomalies in the context of the Collatz Conjecture reveals the enduring validity of the conjecture. Despite the conceptual construction of potential counterexamples, their mathematical impossibility, validated through rigorous analysis and infinite considerations, reaffirms the conjecture’s robustness.

   □

The meticulous exploration of the limits of the behavior of the Collatz function reaffirms that, however extreme the magnitudes considered may be or however strange certain hypothetical anomalies may seem, their mathematical impossibility within this discrete dynamical system revalidates the Conjecture.

Just as astronauts perform spacewalks to reinforce the exterior of spacecraft, the analysis of limit cases reinforces that there are no cracks in the "Collatz spacecraft" that allow escapes to infinity or anomalous cycles.

14. Asymptotic Behavior

 Theorem 14.1

(Asymptotic Behavior). Let $C\left(n\right)$ be the Collatz function. Then, for every $ϵ>0$, there exists an $N\in \mathbb{N}$ such that for all $n>N$ with $n\in \mathbb{N}$, the following holds:

$$\left|\frac{C\left(n\right)}{n}\right|<1+ϵ$$

Proof of the previous theorem

 Proof.

Let $n>1$ and $ϵ>0$. We analyze two cases:

(i)

If n is even, then $C\left(n\right)=\frac{n}{2}$ and so $\left|\frac{C\left(n\right)}{n}\right|=\frac{1}{2}<1$.

(ii)

If n is odd, $C\left(n\right)=3n+1$ and then $\left|\frac{C\left(n\right)}{n}\right|=\left|3+\frac{1}{n}\right|$. For all $n>\frac{1}{ϵ}$, it follows that $\left|\frac{C\left(n\right)}{n}\right|<1+ϵ<3+ϵ$.

Taking $N=max\left(\frac{1}{ϵ},\frac{3}{ϵ}\right)$, ensures the inequality for all $n>N$.    □

Through this analysis, the asymptotic behavior of the Collatz function is formally demonstrated, establishing precise analytical bounds.

Conclusion: A profound examination of limit cases, extremely large numbers, and hypothetical anomalies in the context of the Collatz Conjecture reveals the enduring validity of the conjecture. Despite the conceptual construction of potential counterexamples, their mathematical impossibility, validated through rigorous analysis and infinite considerations, reaffirms the conjecture’s robustness.

 Lemma 14.2.

(Growth Rate). The growth rate of a Collatz sequence can be bounded by functions that represent the worst-case increase and the average-case behavior.

 Proof.

Let n be a natural number and $C\left(n\right)$ the Collatz function. We analyze the worst-case scenario where n is repeatedly multiplied by 3 and increased by 1 without intermediate halving steps. This is represented by the function $f\left(n\right)=3n+1$.

Conversely, we consider the average-case behavior assuming a random distribution of odd and even numbers in the sequence, leading to the heuristic function $g\left(n\right)=\frac{3n}{2}$.

The actual growth rate of a Collatz sequence is bounded by $f\left(n\right)$ and $g\left(n\right)$ for large values of n, which can be analyzed using logarithmic scales and probabilistic methods.    □

 Lemma 14.3.

(Limit Cases). The limit behavior of Collatz sequences can be characterized by the convergence of subsequences and the analysis of potential cycles.

 Proof.

We employ the concept of subsequences and the pigeonhole principle to demonstrate that, as n becomes large, the sequence will eventually enter a cycle or converge to the trivial cycle involving 1.

The pigeonhole principle implies that for sufficiently large n, the number of possible remainders modulo $3n+1$ is finite, forcing the sequence into a repeating pattern or convergence.

Furthermore, the use of analytic number theory can shed light on the distribution of odd and even terms in a sequence, which influences its asymptotic behavior.    □

 Lemma 14.4.

(Limits). Let $f\left(n\right)=(3n+1)mod{2}^k$. For sufficiently large n, the function $f\left(n\right)$ exhibits a cycle of length k.

 Proof.

We will prove the theorem using the pigeonhole principle and properties of modular arithmetic.

Step 1: Define the Function and Modular Space Consider the function $f:\mathbb{N}\to \{0,1,\dots ,2^k-1\}$ defined by $f\left(n\right)=(3n+1)mod{2}^k$.

Step 2: Pigeonhole Principle By the pigeonhole principle, if $n>2^k$, there must be at least two distinct integers m and n such that $f\left(m\right)=f\left(n\right)$. This is because there are n numbers but only $2^k$ possible residues when taken modulo $2^k$.

Step 3: Injective Property in the Range We need to show that the function is injective in the range considered. Suppose $f\left(m\right)=f\left(n\right)$ for some $m,n>2^k$.

$$(3m+1)\equiv (3n+1)(mod{2}^k)$$

This implies:

$$3m\equiv 3n(mod{2}^k)$$

Since 3 is coprime with $2^k$ (as $k\ge 1$), we can cancel out the 3, giving:

$$m\equiv n(mod{2}^k)$$

Thus, $m=n+j·2^k$ for some integer j. This shows that within each set of $2^k$ consecutive integers, f is injective.

Step 4: Cycle Formation Given $n>2^k$, consider the sequence generated by iterating f:

$$n_0=n,n_1=f\left(n_0\right),n_2=f\left(n_1\right),\dots$$

Since the sequence $n_i$ takes values in the finite set $\{0,1,\dots ,2^k-1\}$, there must be some $i\ne j$ such that $n_i\equiv n_j(mod{2}^k)$ (by the pigeonhole principle). Let $i<j$ be the smallest such indices. Then:

$$f^i\left(n\right)\equiv f^j\left(n\right)(mod{2}^k)$$

This implies that the sequence $n_i,n_{i+1},\dots ,n_{j-1}$ forms a cycle of length $j-i$.

Step 5: Cycle Length We need to show that the length of the cycle is exactly k. Given the injectivity within any segment of length $2^k$, the smallest cycle must cover all residues modulo $2^k$ at least once. Therefore, the cycle length must be exactly k.

Conclusion Thus, for sufficiently large n, the function $f\left(n\right)=(3n+1)mod{2}^k$ exhibits a cycle of length k, completing the proof.    □

Skewes Numbers Let S be the set of Skewes numbers, and C be the Collatz function. Skewes numbers are greater than Graham’s number, the largest number that can be expressed in ordinary language. In symbols:

$$\begin{array}{c}\hfill \forall s\in S,g<s\end{array}$$

Here, g represents Graham’s numbers. By definition, the function C maps natural numbers to natural numbers, so it is defined for Skewes numbers:

$$\begin{array}{c}\hfill \forall s\in S,\exists C\left(s\right)\end{array}$$

The objective is to study the behavior, particularly growth bounds, of the system when dealing with colossal magnitude numbers like Skewes numbers. Let $\varphi :\mathbb{N}\to \mathbb{N}$ be a computable and constructive function representing an upper bound on explosive growth under C:

$$\begin{array}{c}\hfill \forall n\in \mathbb{N},(C^k\left(n\right)<\varphi \left(n\right))\end{array}$$

Here, ${\left(C^k\left(n\right)\right)}_{k\in \mathbb{N}}$ denotes the k-th iteration of C starting from n.

Thus, by considering extremely large Skewes numbers as s, consecutive bounds on successive explosions are obtained. By analyzing asymptotically the behavior of these bounds in the face of variations in orders of colossal magnitude, the absolute limits of the system are analytically characterized.

Gödel Numbers Let G be the set of Gödel numbers, and T the set of IATs (Inverse Algebraic Trees). Gödel numbers transcend the algorithmic capacity for manipulation, while IATs are combinatorial structures constructed computationally.

Suppose, for the sake of contradiction, that it is possible to construct IATs for Gödel numbers:

$$\begin{array}{c}\hfill \exists g\in G,\exists T_g\in T\end{array}$$

Here, $T_g$ hypothetically represents an IAT associated with the number g.

By definition, every IAT represents its nodes using natural numbers:

$$\begin{array}{c}\hfill \forall T_{(.)}\in T,\forall v\in V\left(T_{(.)}\right),f\left(v\right)\in \mathbb{N}\end{array}$$

Here, $V\left(T\right)$ is the set of vertices/nodes of the IAT T, and $f:V\to \mathbb{N}$ is the bijective function that assigns values.

However, Gödel numbers by definition vastly exceed any computable natural number. Therefore, they cannot be contained within the image of any computational function f.

We arrive at a contradiction that reveals the theoretical impossibility of constructing IATs to encompass extraordinarily large numbers like Gödel numbers.

15. Inverse Modeling of Systems

Inverse modeling refers to the process of constructing an inverted representation of a discrete dynamical system through analytical means. Specifically, it involves building an inverse algebraic tree by recursively applying the inverse function that undoes the evolution rule of the original system.

Inverse modeling differs from direct modeling of dynamical systems in that it focuses on analytically inverting the system’s recursive function to achieve a reversed vantage point that reveals the inherent topology more clearly. This inverted perspective allows demonstrating structural properties that can then be mapped back to the canonical system via a correlating homeomorphism.

Therefore, inverse modeling provides an alternative framework for comprehending dynamical systems, overcoming limitations of direct modeling techniques that may struggle with explosions of complexity or transitions between intricate state spaces through a structured reformulation of the system’s dynamics.

After introducing the preliminary concepts, we are now in a position to formally develop the methodology of inverse modeling for discrete dynamical systems, which constitutes the core of the theory.

Given a canonical discrete dynamical system determined by a recurrence function F defined over a discrete space S, we begin by defining its analytical inverse G as the function that recursively undoes the steps of F.

Next, we introduce a combinatorial structure denoted as an inverse algebraic tree, which is constructed by recursively applying G starting from a root node associated with the initial or desired final state for the system (depending on whether modeling the direct or inverse evolution of the system is of interest).

It is shown how analytically iterating through the inverse of F, the resulting tree inversely replicates all inherent interrelations in the canonical discrete system, condensing the combinatorial explosion and structurally representing it entirely through the upward links in the acyclic tree structure.

Then, a homeomorphism is defined by bijectively associating nodes of the inverse tree with discrete states of the canonical system. This correlates both spaces, allowing the subsequent topological transport of cardinal structural properties between the canonical system and its inverted counterpart modeled through inverse analytical recursion in the combinatorial structure.

In this way, the determinant formal developments are completed, establishing the methodology provided by the theory to construct inverted representations of arbitrary discrete systems, facilitating their analytical treatment by repositioning the previously intractable combinatorial explosion under a manageable and transferable form to the original canonical system through topological-algebraic equivalences.

 Definition 15.1

(Discrete Topological Space). Let S be the discrete space over which a discrete dynamical system is defined. The discrete topology on S is defined as:

$\tau =\{\varnothing ,\left\{x_1\right\},\left\{x_2\right\},\cdots \}$

 where $x_i\in S$ and each element of S defines an open and closed set (a singleton).

 τ constitutes a discrete topology on S, where open sets are all subsets, and closed sets are the complements of the open sets. A basis for τ is given by the singletons, and a subbasis by the elements of S themselves.

 Then $(S,\tau )$ is said to be the relevant discrete topological space for the system.

 Definition 15.2

(Discrete Function). Let $f:S\to S^{\prime }$ be a function between discrete spaces. We say that f is a discrete function if it preserves the discreteness of elements in its image. That is, $\forall x,y\in S$ such that $x\ne y$, it holds that $f\left(x\right)\ne f\left(y\right)$.

 Definition 15.3.

 Let $(S,F)$ be a discrete dynamical system with a state space S and an evolution function $F:S\to S$. The inverse algebraic tree (IAT) associated with $(S,F)$ is a directed graph $T=(V,E)$ defined as follows:

 (1)

 Vertices (V): The set of vertices V is a subset of the state space S, formally represented as $V\subseteq S$. Each vertex $v\in V$ represents a state in the dynamical system.

 (2)

 Edges (E):The set of edges E is a subset of the Cartesian product $V\times V$, where an edge $(u,v)\in E$ exists if and only if v is a preimage of u under the evolution function F. Formally:

$$\forall u,v\in V:(u,v)\in E\iff u\in F^{-1}\left(v\right)$$

where $F^{-1}\left(v\right)$ denotes the preimage of v under F.

 (3)

 Root Node (r): The IAT has a designated root node $r\in V$ representing a specific state of interest in the dynamical system, often chosen to be an equilibrium state or a periodic orbit.

 (4)

 Tree Structure: The IAT is a rooted tree, meaning that there exists a unique path from the root node r to any other node $v\in V$. This path represents the sequence of inverse transitions under F that lead from the state represented by v to the state represented by r.

 Additional Properties:

•  Levels: The nodes in the IAT can be organized into levels based on their distance from the root node. The root node is at level 0, and the level of any other node is one more than the level of its parent.
•  Paths: A path in the IAT is a sequence of nodes connected by edges, starting from the root node and ending at a leaf node. Each path represents a possible trajectory of the dynamical system under the inverse dynamics defined by the inverse function of F.
•  Cycles: A cycle in the IAT is a path that starts and ends at the same node. The IAT associated with a deterministic dynamical system may contain cycles, including the trivial cycle consisting of a single node.

 Formal Representation of the Construction Process:

 The construction of the IAT can be formally represented using a recursive algorithm:

Algorithm 1 ConstructIAT(S, F, r)

 In this algorithm, the root node r is the starting point for the construction. The algorithm uses a queue Q to keep track of the nodes to be explored. At each iteration, a node is dequeued, and its preimages under F are added as child nodes. The process continues until all reachable nodes have been added to the IAT.

 Example 15.1

(Compactness of Inverse Algebraic Trees (IATs)). Let $(T,d)$ be an inverse algebraic tree (IAT) constructed from a Discrete Dynamical System $(S,F)$ with the analytic inverse function G. Then, $(T,d)$ is compact under the discrete topology.

 Proof. 

We will prove the compactness of $(T,d)$ by showing that every open cover of T has a finite subcover.

Step 1: Let $\mathcal{U}={\left\{U_{\alpha }\right\}}_{\alpha \in A}$ be an arbitrary open cover of T, where A is an indexing set.

$$\begin{array}{c}\hfill \forall \alpha \in A:U_{\alpha }\subseteq T\mathrm{is}\mathrm{open}\\ \hfill \bigcup _{\alpha \in A}{U}_{\alpha }=T\end{array}$$

Step 2: Construct a finite subcover ${\mathcal{U}}^{\prime }\subseteq \mathcal{U}$ of T.

Step 2.1: Define the set of nodes at depth n in T as $V_n=\{v\in V\left(T\right):d(v,r)=n\}$, where r is the root node of T and d is the graph distance.

Step 2.2: For each $n\in \mathbb{N}$, consider the set of nodes $V_n$ at depth n in T. Since T is locally finite (each node has a finite number of children), $V_n$ is finite.

$$\begin{array}{c}\hfill \forall n\in \mathbb{N}:|V_n|<\infty \end{array}$$

Step 2.3: For each $v\in V_n$, choose an open set $U_v\in \mathcal{U}$ such that $v\in U_v$. This is possible because $\mathcal{U}$ is an open cover of T.

$$\begin{array}{c}\hfill \forall n\in \mathbb{N},\forall v\in V_n,\exists U_v\in \mathcal{U}:v\in U_v\end{array}$$

Step 2.4: Define the finite subcover ${\mathcal{U}}^{\prime }$ as the union of the chosen open sets for each depth n up to some finite depth N:

$$\begin{array}{c}\hfill {\mathcal{U}}^{\prime }=\bigcup _{n=0}^N\{U_v:v\in V_n\}\end{array}$$

where N is chosen such that ${\bigcup }_{n=0}^N{V}_n=V\left(T\right)$. This is possible because T is connected and has finite depth.

Step 3: Prove that ${\mathcal{U}}^{\prime }$ is a finite subcover of T.

Step 3.1: ${\mathcal{U}}^{\prime }$ is finite, as it is a union of finitely many finite sets:

$$\begin{array}{c}\hfill |{\mathcal{U}}^{\prime }|=\left|\bigcup _{n=0}^N\{U_v:v\in V_n\}\right|\le \sum _{n=0}^N\left|V_n\right|<\infty \end{array}$$

Step 3.2: ${\mathcal{U}}^{\prime }$ covers T, as every node $v\in V\left(T\right)$ belongs to some $V_n$ with $n\le N$, and thus is covered by some $U_v\in {\mathcal{U}}^{\prime }$:

$$\begin{array}{c}\hfill \forall v\in V\left(T\right),\exists n\le N:v\in V_n\wedge \exists U_v\in {\mathcal{U}}^{\prime }:v\in U_v\\ \hfill \therefore \bigcup _{U\in {\mathcal{U}}^{\prime }}U=T\end{array}$$

Step 4: Conclude that $(T,d)$ is compact.

Since every open cover $\mathcal{U}$ of T has a finite subcover ${\mathcal{U}}^{\prime }$, $(T,d)$ is compact under the discrete topology.    □

Figure 10. This diagram illustrates an original system alongside its inverse algebraic tree. The nodes represent states within the system, with solid arrows depicting the progression or transformation between these states. The dashed arrows highlight the inverse relationships, mapping states back to their origins in the context of the algebraic tree, thereby visualizing the system’s underlying structure and the concept of inversion in algebraic terms.

Figure 10. This diagram illustrates an original system alongside its inverse algebraic tree. The nodes represent states within the system, with solid arrows depicting the progression or transformation between these states. The dashed arrows highlight the inverse relationships, mapping states back to their origins in the context of the algebraic tree, thereby visualizing the system’s underlying structure and the concept of inversion in algebraic terms.

 Theorem 15.2

(Existence and Uniqueness of the Inverse Algebraic Tree). Let $(S,F)$ be a discrete dynamical system, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued injective, surjective, and exhaustive. Let $F=\{T_1,\dots ,T_k\}$ be the Inverse Algebraic Forest generated by G, where each $T_i$ is a tree. Then, F is unique and each $T_i\in F$ is a single connected component.

 Proof. 

First, we prove that each $T_i$ is connected.

Step 1: Connectivity of Trees in the Inverse Algebraic Forest

 Lemma 15.3.

 For each $T_i\in F$, $T_i$ is a single connected component.

 Proof.

Suppose, for contradiction, that there exist two nodes $v_1,v_2\in V_i$ such that there is no sequence of edges connecting $v_1$ and $v_2$. This implies that $v_1$ and $v_2$ belong to two separate connected components, say $T_{i1}$ and $T_{i2}$, respectively.

By the exhaustiveness property of G, for each node $v\in V_i$, there exists a finite sequence of applications of G that leads to a root node $r_i$. Formally:

$$\forall v\in V_i,\exists n\in \mathbb{N},\exists r_i\in V_i:(\mathrm{Root}\left(r_i\right)\wedge v\in G^n\left(r_i\right))$$

where $\mathrm{Root}\left(r_i\right)$ denotes that $r_i$ is a root node, and $G^n$ represents the n-fold composition of G with itself. Let $r_{i1}$ and $r_{i2}$ be the root nodes of $T_{i1}$ and $T_{i2}$, respectively.

By the determinism of F, each node in $T_i$ has a unique child. By the surjectivity of F, each node in $T_i$, except for the root nodes, has a unique parent. Formally:

$$\forall v\in V_i\setminus \{r_{i1},r_{i2}\},\exists !u\in V_i:(u,v)\in E_i$$

The existence of separate components $T_{i1}$ and $T_{i2}$ leads to a contradiction when F is deterministic and surjective, and G is exhaustive. Therefore, each $T_i$ must be a single connected component.    □

Step 2: Uniqueness of the Inverse Algebraic Forest

 Lemma 15.4.

 If for every pair of nodes $v_1,v_2\in V_i$ in each tree $T_i$, there is at most one sequence of edges from $v_1$ to $v_2$, then F is unique.

 Proof.

Let $v_1,v_2\in V_i$ be any two nodes in $T_i$. Suppose there are two distinct sequences of edges from $v_1$ to $v_2$, denoted by $P_1$ and $P_2$.

Let u be the last common node of $P_1$ and $P_2$ before they diverge. Let $u_1$ and $u_2$ be the next nodes after u in $P_1$ and $P_2$, respectively.

By the determinism of F, u can have only one child. Therefore, $u_1=u_2$, contradicting the assumption that $P_1$ and $P_2$ are distinct paths. Thus, there can be at most one path between any two nodes in each $T_i$.

By the Path Uniqueness Theorem, F is unique.    □

Conclusion:

We have shown that the Inverse Algebraic Forest F generated by G is unique and each tree $T_i\in F$ is a single connected component, even when the state space S is countable.

Theorems on Absence of Non-Trivial Cycles and Universal Convergence: Two fundamental theorems in the Theory of Inverse Discrete Dynamical Systems (TIDDS) are the Absence of Non-Trivial Cycles and the Universal Convergence of Trajectories in inverse algebraic Trees (IATs). These theorems provide crucial insights into the structure and behavior of IATs and their corresponding discrete dynamical systems.

Theorem 1 (Absence of Non-Trivial Cycles): Let $(X,f)$ be a discrete dynamical system and $T=(V,E)$ be its associated inverse algebraic Tree (IAT). Then, there are no non-trivial cycles in T.

Proof Sketch: Assume, for contradiction, that there exists a non-trivial cycle $C=(v_1,v_2,\dots ,v_n,v_1)$ in T, with $n\ge 3$ and $v_1=v_n$. By the construction of the IAT, each vertex has a unique parent. However, the existence of the cycle C implies that $v_1$ has two distinct parents: $v_n$ (in the cycle) and its unique parent in T, leading to a contradiction. Therefore, T cannot contain any non-trivial cycles.

Theorem 2 (Universal Convergence of Trajectories): Let $(X,f)$ be a discrete dynamical system and $T=(V,E)$ be its associated inverse algebraic Tree (IAT). For any vertex $v\in V$, the unique path from v to the root of T corresponds to a trajectory in $(X,f)$ that converges to the equilibrium state or periodic orbit represented by the root.

Proof Sketch: Let $v\in V$ be an arbitrary vertex in T, and let $(v=v_1,v_2,\dots ,v_n)$ be the unique path from v to the root r of T. By the construction of the IAT, each edge $(v_i,v_{i+1})$ represents the application of the inverse dynamics, i.e., $f\left(v_{i+1}\right)=v_i$. Therefore, the sequence $(v_n,v_{n-1},\dots ,v_1)$ corresponds to a trajectory in $(X,f)$ that starts at the root r and ends at v. Since r represents an equilibrium state or periodic orbit, the trajectory converges to this state in the original system $(X,f)$.

Significance of the Theorems: The Absence of Non-Trivial Cycles and Universal Convergence of Trajectories theorems have important implications for understanding the dynamics of discrete systems:

• The absence of non-trivial cycles in IATs suggests that the inverse dynamics of the system are well-behaved and do not exhibit complex or chaotic behavior.
• The universal convergence of trajectories in IATs implies that the original system has a unique equilibrium state or periodic orbit that attracts all trajectories, regardless of their initial state.
• These properties can be used to classify discrete dynamical systems and study their long-term behavior, stability, and convergence properties.

The proofs of these theorems rely on the structural properties of IATs and the relationship between the inverse dynamics and the original system. By exploiting the unique parent property and the correspondence between paths in the IAT and trajectories in the original system, the theorems provide a strong foundation for the analysis of discrete dynamical systems using the tools and techniques of TIDDS.

 Theorem 15.5

(Properties of IATs). Let $T=(V,E)$ be an inverse algebraic Tree (IAT) constructed from a Discrete Dynamical System $(S,F)$ with the analytic inverse function G. Then:

•  T has no non-trivial cycles.
•  All paths in T converge to the root node r.

 Proof. 

We prove each property separately:

Property 1: Absence of Non-Trivial Cycles

• Define the notion of a non-trivial cycle:

  $$\forall v_1,\dots ,v_k\in V:\mathrm{NTC}(v_1,\dots ,v_k)\iff (k\ge 3)\wedge (v_1=v_k)\wedge (\forall i\in \{1,\dots ,k-1\}:(v_i,v_{i+1})\in E)$$
• Prove that any non-trivial cycle leads to a contradiction:

  $$\forall v_1,\dots ,v_k\in V:\mathrm{NTC}(v_1,\dots ,v_k)\Rightarrow \perp$$
  Proof. Assume, for contradiction, that there exists a non-trivial cycle $v_1,\dots ,v_k$.
  By the recursive construction of T using the injective function G, each node has a unique parent. Consider two consecutive nodes $v_i$ and $v_{i+1}$ in the cycle. By the unique parent property, $v_{i+1}$ must have $v_i$ as its unique parent.
  However, $v_{i+1}$ also has a unique parent outside the cycle, as the tree extends infinitely upwards from each node. This leads to a contradiction, as $v_{i+1}$ cannot have two distinct parents due to the multivalued injectivity of G.
  Therefore, there cannot exist any non-trivial cycle in T.    □

Property 2: Convergence of Paths to Root Node

(1)

Let $P\subseteq V$ be a path in T. We say P converges to the root node r if following P from any node $v\in P$ leads directly to r without cycles or deviations.

Proof. Consider any node $v\in V$ and the unique path P from v to r (due to the tree structure and multivalued injectivity of G). Since there are no cycles, P must terminate at r. This holds for all nodes v, hence every path in T converges to r.    □

Figure 11. Representation of a system with and without cycles, showing how the system’s structure can significantly vary with the introduction of cycles. On the left, an IAT without cycles demonstrates a linear progression of states, while on the right, the system with cycles illustrates the added complexity by closed loops.

Figure 11. Representation of a system with and without cycles, showing how the system’s structure can significantly vary with the introduction of cycles. On the left, an IAT without cycles demonstrates a linear progression of states, while on the right, the system with cycles illustrates the added complexity by closed loops.

 Theorem 15.6

(Uniqueness of Paths). Let $T=(V,E)$ be an inverse algebraic Tree (IAT) constructed from a Discrete Dynamical System $(S,F)$ with the analytic inverse function G. For any two nodes $u,v\in V$, there exists a unique path from u to v in T.

 Proof. 

We will prove the uniqueness of paths by contradiction using first-order logic.

(1)

Define the existence of a path between two nodes in T.

$$\begin{array}{cc}\hfill \forall u,v\in V:& \exists P\subseteq E:\mathrm{Path}(P,u,v)\hfill \\ & \iff (P=\{(w_1,w_2),(w_2,w_3),\dots ,(w_{n-1},w_n)\}\wedge w_1=u\wedge w_n=v\hfill \\ & \wedge \forall i\in \{1,\dots ,n-1\}:(w_i,w_{i+1})\in E)\hfill \end{array}$$

(2)

Assume, for contradiction, that there exist two distinct paths between nodes u and v in T.

$$\exists u,v\in V,\exists P_1,P_2\subseteq E:(\mathrm{Path}(P_1,u,v)\wedge \mathrm{Path}(P_2,u,v)\wedge P_1\ne P_2)$$

(3)

Let w be the first node at which the paths $P_1$ and $P_2$ differ.

$$\begin{array}{cc}\hfill \exists w\in V,\exists i,j\in \mathbb{N}:& (w\in P_1\wedge w\in P_2\wedge P_1\left[i\right]=w\wedge P_2\left[j\right]=w\hfill \\ & \wedge \forall k<min(i,j):P_1\left[k\right]=P_2\left[k\right]\wedge P_1[i+1]\ne P_2[j+1])\hfill \end{array}$$

(4)

By the construction of T using the injective function G, each node has a unique parent. Therefore, w cannot have two distinct children in T.

$$\forall w\in V,\forall x,y\in V:\left(\right(w,x)\in E\wedge (w,y)\in E\to x=y)$$

(5)

The existence of two distinct paths $P_1$ and $P_2$ contradicts the unique parent property of T. Therefore, the assumption in Step 2 must be false.

(6)

We conclude that for any two nodes $u,v\in V$, there exists a unique path from u to v in T.

$$\forall u,v\in V,\exists !P\subseteq E:\mathrm{Path}(P,u,v)$$

Thus, the uniqueness of paths in the inverse algebraic Tree T is formally proven by contradiction. □   □

 Theorem 15.7

(Uniqueness of Non-Trivial Cycles in DIDS). Let $G:S\to \mathcal{P}\left(S\right)$ be the inverse function of a generic DIDS $(S,F)$, where S is the state space and $F:S\to S$ is the evolution function. Then:

(1)

 If a non-trivial cycle exists in the inverse algebraic tree of $(S,F)$, it must have a specific structure:

$$\begin{array}{c}\hfill \exists k\in \mathbb{N},\exists x_1,\dots ,x_k\in S:(x_1=x_k)\wedge \\ \hfill (\forall i\in \{1,\dots ,k-1\}:x_{i+1}\in G\left(x_i\right))\end{array}$$

 where k is a constant specific to the system.

(2)

 There exists at most one non-trivial cycle in the inverse algebraic tree of $(S,F)$.

 Proof. 

Let $G:S\to \mathcal{P}\left(S\right)$ be the inverse function of a generic DIDS $(S,F)$, where S is the state space and $F:S\to S$ is the evolution function.

Step 1: Define the notion of a non-trivial cycle.

$$\begin{array}{c}\hfill \forall x_1,\dots ,x_n\in S:\mathrm{NTC}(x_1,\dots ,x_n)\iff (n\ge 3)\wedge (x_1=x_n)\wedge \\ \hfill (\forall i\in \{1,\dots ,n-1\}:x_{i+1}\in G\left(x_i\right))\end{array}$$

Step 2: Prove that any non-trivial cycle must have a specific structure.

$$\begin{array}{c}\hfill \forall x_1,\dots ,x_n\in S:\mathrm{NTC}(x_1,\dots ,x_n)\Rightarrow \\ \hfill (\exists k\in \mathbb{N}:n=k\wedge \forall i\in \{1,\dots ,k-1\}:x_{i+1}\in G\left(x_i\right))\end{array}$$

Proof: Let $x_1,\dots ,x_n\in S$ be a non-trivial cycle. By the definition of a non-trivial cycle, we have $n\ge 3$, $x_1=x_n$, and $x_{i+1}\in G\left(x_i\right)$ for all $i\in \{1,\dots ,n-1\}$. Setting $k=n$ satisfies the claimed structure.

Step 3: Prove that there exists at most one non-trivial cycle in the inverse algebraic tree of $(S,F)$.

$$\begin{array}{c}\hfill (\exists x_1,\dots ,x_k\in S:\mathrm{NTC}(x_1,\dots ,x_k))\Rightarrow \\ \hfill (\forall x_1^{\prime },\dots ,x_{k^{\prime }}^{\prime }\in S:\mathrm{NTC}(x_1^{\prime },\dots ,x_{k^{\prime }}^{\prime })\Rightarrow \\ \hfill (k=k^{\prime }\wedge \forall i\in \{1,\dots ,k\}:x_i=x_i^{\prime }))\end{array}$$

Proof: Suppose, for contradiction, that there exist two distinct non-trivial cycles $x_1,\dots ,x_k$ and $x_1^{\prime },\dots ,x_{k^{\prime }}^{\prime }$ in the inverse algebraic tree of $(S,F)$.

By Step 2, both cycles must have the structure:

$$\begin{array}{cc}\hfill x_1& =x_k\wedge \forall i\in \{1,\dots ,k-1\}:x_{i+1}\in G\left(x_i\right)\hfill \\ \hfill x_1^{\prime }& =x_{k^{\prime }}^{\prime }\wedge \forall i\in \{1,\dots ,k^{\prime }-1\}:x_{i+1}^{\prime }\in G\left(x_i^{\prime }\right)\hfill \end{array}$$

Since G is a function, $x_2\in G\left(x_1\right)$ and $x_2^{\prime }\in G\left(x_1^{\prime }\right)$ imply that $x_1=x_1^{\prime }$. By induction, this implies $x_i=x_i^{\prime }$ for all $i\in \{1,\dots ,min(k,k^{\prime })\}$. If $k<k^{\prime }$, then $x_1=x_k=x_k^{\prime }\in G\left(x_{k-1}^{\prime }\right)=G\left(x_{k-1}\right)$, contradicting the fact that $x_{k-1}$ has a unique successor in the cycle $x_1,\dots ,x_k$. Similarly, if $k^{\prime }<k$, we obtain a contradiction. Therefore, $k=k^{\prime }$, and the two cycles are identical.

Thus, we have shown that there can be at most one non-trivial cycle in the inverse algebraic tree of a generic DIDS.    □

 Theorem 15.8

(Convergence of Distinct Trajectories). Let $(S,F)$ be a discrete dynamical system and $T=(V,E)$ be the associated inverse algebraic tree generated by the inverse analytic function $G:S\to \mathcal{P}\left(S\right)$. For any two distinct trajectories $P_1,P_2\subset V$ in the same tree T, both trajectories converge to a common node $u\in V$, which is ultimately the root node of T.

 Proof. 

Let $(S,F)$ be a discrete dynamical system and $T=(V,E)$ be the associated inverse algebraic tree generated by the inverse analytic function $G:S\to \mathcal{P}\left(S\right)$. Consider two distinct trajectories $P_1,P_2\subset V$ in the same tree T.

Step 1: Define the notion of a trajectory in T.

$$\begin{array}{c}\hfill \forall P\subseteq V:\mathrm{Trajectory}\left(P\right)\iff (\forall u,w\in P:(u,w)\in E\vee (w,u)\in E)\end{array}$$

Step 2: Define the convergence of a trajectory to a node.

$$\begin{array}{c}\hfill \forall P\subseteq V,\forall u\in V:\mathrm{Converges}(P,u)\iff \\ \hfill (\forall \epsilon >0,\exists N\in \mathbb{N},\forall n\ge N,\forall w\in P:d(w,u)<\epsilon )\end{array}$$

where d is the graph distance in T.

Step 3: Prove that every node in T has a unique path to the root node.

$$\begin{array}{c}\hfill \forall v\in V,\exists !P\subseteq V:\left(\mathrm{Trajectory}\right(P)\wedge v\in P\wedge \exists r\in V:\\ \hfill \left(\mathrm{Root}\right(r)\wedge r\in P\wedge \forall u\in P\setminus \{r\}:(u,r)\notin E))\end{array}$$

Proof: By the recursive construction of T using the injective function G, each node has a unique parent. Therefore, for any node $v\in V$, there exists a unique path from v to the root node r, which is obtained by following the parent nodes until reaching r.

Step 4: Prove that if $P_1$ and $P_2$ are in the same tree T, they must share a common node.

$$\begin{array}{c}\hfill \mathrm{Trajectory}\left(P_1\right)\wedge \mathrm{Trajectory}\left(P_2\right)\wedge P_1,P_2\subset V\\ \hfill \Rightarrow \exists v\in V:(v\in P_1\wedge v\in P_2)\end{array}$$

Proof: Assume, for contradiction, that $P_1$ and $P_2$ do not share any common node. Then, there exists a node $w\in P_1$ such that $w\notin P_2$. By Step 3, there is a unique path from w to the root node r. This path must intersect $P_2$ at some node v, as both paths end at r. Therefore, $v\in P_1$ and $v\in P_2$, contradicting the assumption that $P_1$ and $P_2$ do not share any common node.

Step 5: Let v be a common node of $P_1$ and $P_2$, and let $P_v$ be the unique path from v to the root node r. Prove that $P_1$ and $P_2$ converge to r.

$$\begin{array}{c}\hfill \mathrm{Trajectory}\left(P_1\right)\wedge \mathrm{Trajectory}\left(P_2\right)\wedge v\in P_1\cap P_2\\ \hfill \Rightarrow \exists P_v\subseteq V:(v\in P_v\wedge P_v\subseteq P_1\wedge P_v\subseteq P_2)\\ \hfill \Rightarrow \mathrm{Converges}(P_1,r)\wedge \mathrm{Converges}(P_2,r)\end{array}$$

Proof: By Step 4, there exists a common node $v\in P_1\cap P_2$. By Step 3, there is a unique path $P_v$ from v to the root node r. Since $v\in P_1$ and $v\in P_2$, and $P_v$ is the unique path from v to r, we have $P_v\subseteq P_1$ and $P_v\subseteq P_2$. Therefore, both $P_1$ and $P_2$ converge to the root node r via the common subpath $P_v$.

Therefore, if $P_1$ and $P_2$ are in the same inverse algebraic tree T, they necessarily converge to a common node, which is ultimately the root node r of T, completing the proof.    □

Figure 12. This diagram illustrates the concept of universal convergence of trajectories in a system, showing how different paths (represented in various colors) converge towards a common root or state ($s_0$). Each path, despite starting from distinct states and undergoing unique transitions, ultimately merges into the unified structure, symbolizing a fundamental property of the system’s dynamics.

Figure 12. This diagram illustrates the concept of universal convergence of trajectories in a system, showing how different paths (represented in various colors) converge towards a common root or state ($s_0$). Each path, despite starting from distinct states and undergoing unique transitions, ultimately merges into the unified structure, symbolizing a fundamental property of the system’s dynamics.

 Theorem 15.9

(Convergence in Infinite IATs (Generalized)). Let $(T=(V,E\left)\right)$ be an infinite inverse algebraic tree (IAT) associated with a discrete inverse dynamical system (TIDDS) $\left(\right(S,G\left)\right)$. If the following conditions hold:

 Well-defined Inverse: For every $n\in S$, $G\left(n\right)$ is a non-empty and finite set.

 Exhaustive Inverse: There exists a finite set of states $(R\subset S)$ (the "root set") such that for every $n\in S$, there exists a $k\in \mathbb{N}$ such that $G^k\left(n\right)\cap R\ne \varnothing$.

 Finite Branching Factor: The maximum number of children of any node in T is finite.

 Then, every infinite path in T eventually enters and remains within the subtree rooted at R.

 Proof. 

Step 1: Definitions and Preliminaries

Discrete Inverse Dynamical System (TIDDS): A TIDDS is a pair $\left(\right(S,G\left)\right)$ where S is a set of states and $G:S\to \mathcal{P}\left(S\right)$ is a function that maps each state to a set of its possible predecessors.

Infinite Inverse Algebraic Tree (IIAT): The IIAT $(T=(V,E\left)\right)$ associated with the TIDDS $\left(\right(S,G\left)\right)$ is defined as follows: $V=S$ (the set of states), $E=\left\{\right(m,n)\in S\times S\mid m\in G(n\left)\right\}$ (the edges represent inverse transitions).

Step 2: Definition of Convergence to Subtree

An infinite path $P=(v_0,v_1,v_2,\dots )$ in the IIAT T converges to the subtree rooted at R if and only if:

$$\exists k\in \mathbb{N}:\forall i\ge k,v_i\in T_R$$

where $T_R$ is the subtree of T rooted at the set of nodes R.

Step 3: Proof by Contradiction

Assume, for the sake of contradiction, that there exists an infinite path $P=(v_0,v_1,v_2,\dots )$ in the IIAT T that does not converge to the subtree rooted at R.

Since P does not converge to the subtree rooted at R, there must be infinitely many nodes in P that are not in $T_R$. Let’s denote this infinite set of nodes as $V_{\mathrm{out}}$.

By the well-ordering principle, $V_{\mathrm{out}}$ has a minimum element, say $v_m$. Consider the node $v_{m+1}$. Since P is a path in the IAT, we have $v_m\in G\left(v_{m+1}\right)$. By the exhaustiveness of G, there exists a finite sequence of inverse steps from $v_{m+1}$ to a node in R. This means that $v_{m+1}$ must eventually reach a node in R, and since $v_m\in G\left(v_{m+1}\right)$, $v_m$ must also eventually reach a node in R. This contradicts our assumption that $v_m$ is not in $T_R$.

Step 4: Conclusion

Therefore, our initial assumption must be false, and every infinite path in the IIAT T must eventually enter and remain within the subtree rooted at R.

   □

 Theorem 15.10.

 Let $F:S\to S$ be the function defining the dynamics of the Collatz system on a set S of positive integers. Suppose there exists a function $G:S\to \mathcal{P}\left(S\right)$ such that $\forall s\in S:G\left(s\right)=\{x\in S:F(x)=s\}$. Then, the inverse algebraic tree constructed using G completely captures the inverse dynamics of F without introducing anomalies or losing critical information.

 Proof. 

We prove the theorem using the following formal steps:

(1)

Construction of G: We define the function G as follows:

$$\begin{array}{c}\hfill \forall s\in S:G\left(s\right)=\{x\in S:F(x)=s\}\end{array}$$

By definition, G undoes the steps of F by assigning to each state s the set of all states x that map to s under F. Formally:

$$\begin{array}{c}\hfill \forall s\in S,\forall x\in S:(x\in G(s)\leftrightarrow F(x)=s)\end{array}$$

This ensures that all inverse dynamics of F are represented in G.

(2)

Integrity of the Inverse Tree: We demonstrate that the inverse tree constructed using G includes all possible backward paths in the system’s dynamics.

Let $T=(V,E)$ be the inverse algebraic tree, where $V=S$ and $E=\left\{\right(s,x):s\in S,x\in G(s\left)\right\}$. For any state $s\in S$ and any predecessor $x\in S$ such that $F\left(x\right)=s$, we have $x\in G\left(s\right)$ by the definition of G. Therefore, $(s,x)\in E$, meaning that the edge connecting s to its predecessor x is included in the tree.

Since this holds for all states and their predecessors, the inverse tree T captures all possible backward paths in the system’s dynamics, ensuring that no information about the inverse dynamics is lost.

(3)

Absence of Anomalies: We show that the inverse tree does not introduce non-existent cycles or bifurcations in the actual dynamics.

Suppose the local multivalued injectivity of G holds, i.e., $\forall s_1,s_2\in S:(s_1\ne s_2\to G\left(s_1\right)\cap G\left(s_2\right)=\varnothing )$. This means that no two distinct states share a common predecessor under G. Consequently, the inverse tree cannot contain cycles, as each state has a unique set of predecessors that do not overlap with those of other states.

Furthermore, suppose that each state has a finite number of predecessors, i.e., $\forall s\in S:\left|G\right(s\left)\right|<\infty$. This ensures that each node in the inverse tree has a manageable number of backward connections, preventing the introduction of spurious bifurcations that do not exist in the actual dynamics.

Therefore, under the assumptions of local multivalued injectivity of G and finiteness of predecessors, the inverse tree maintains its acyclic structure and avoids introducing anomalies.

In conclusion, we have formally demonstrated that, under the given assumptions, the inverse algebraic tree constructed using G effectively represents the entirety of the inverse dynamics of F without introducing structural errors or losing critical dynamics details. This allows the tree to be used for further analyses or validations on the behavior of the system under varying initial conditions.    □

 Remark 11

(Observations on the Convergence of Trajectories and Universal Convergence). The convergence of distinct trajectories to a common node and the universal convergence of all trajectories towards the root node are both supported by the theorem of uniqueness of non-trivial cycles in Discrete Inverse Dynamical Systems (DIDS). This theorem plays a crucial role in establishing the overall convergence behavior of the system.

 Firstly, the uniqueness of non-trivial cycles theorem ensures that there are no additional cycles beyond the trivial cycle and the unique non-trivial cycle that includes the point of contact $pc$. This absence of additional cycles guarantees that trajectories cannot become trapped in any other cycles, allowing them to converge towards the root node without being diverted or oscillating indefinitely.

 Secondly, the theorem establishes the existence of a unique non-trivial cycle that includes the point of contact $pc$. This cycle acts as an attractor, drawing trajectories towards it due to its intrinsic attracting nature. Consequently, all trajectories in the system, regardless of their initial conditions, will eventually converge towards this non-trivial cycle, and subsequently, towards the root node.

 The convergence of distinct trajectories to a common node is ensured because there are no other cycles that could divert or trap these trajectories separately. Instead, they all converge to the same non-trivial cycle and, ultimately, to the root node.

 Moreover, the universal convergence of all trajectories towards the root node is a direct consequence of the attracting nature of the unique non-trivial cycle and the absence of any other cycles that could prevent trajectories from reaching the root node.

 In summary, the theorem of uniqueness of non-trivial cycles in DIDS plays a fundamental role in establishing the convergence properties of the system by eliminating the possibility of additional cycles that could disrupt convergence and by identifying the unique non-trivial cycle as the attractor towards which all trajectories eventually converge. This theoretical foundation supports the observations on the convergence of trajectories and the universal convergence towards the root node, providing a rigorous mathematical basis for understanding the system’s dynamics.

 Corollary 15.1.

 The properties of absence of non-trivial cycles and universal convergence to the root hold for any IAT constructed from a DDS with an analytic inverse satisfying multivalued injectivity and surjectivity.

 Proof. 

Let $T=(V,E)$ be an IAT constructed from a DDS $(S,\tau ,F)$ with an analytic inverse G that satisfies multivalued injectivity and surjectivity.

To show that T has no non-trivial cycles, suppose for contradiction that there exists a non-trivial cycle $C=v_1,\dots ,v_k$ with $k\ge 3$. By the multivalued injectivity of G, each node has a unique parent. But then $v_1$ would have two distinct parents: $v_k$ (in the cycle) and its unique parent by recursion. This leads to a contradiction, so no such cycle exists.

To show that all paths in T converge to the root node r, let $P=(v_1,v_2,\dots )$ be an arbitrary infinite path in T. By the surjectivity of G, each node has a child. By multivalued injectivity, the sequence of depths $d\left(v_i\right)$ is strictly decreasing. As natural numbers are well-ordered, there exists an n such that $d\left(v_n\right)=0$, i.e., $v_n=r$. By the uniqueness of paths, P converges to r.

Therefore, the properties of absence of non-trivial cycles and universal convergence to the root hold for any IAT constructed from a DDS with an analytic inverse satisfying multivalued injectivity and surjectivity.    □

16. Construction of the inverse algebraic Tree and Topological Equivalence

inverse algebraic Trees (IATs): An inverse algebraic Tree (IAT) is a discrete structure that represents the inverse dynamics of a discrete dynamical system. It captures the relationships between states in the system by mapping each state to its preimage under the evolution function. The properties and structure of the IAT provide valuable insights into the behavior of the original dynamical system.

Definition (inverse algebraic Tree): Given a discrete dynamical system $(X,f)$, where X is the state space and $f:X\to X$ is the evolution function, an inverse algebraic Tree (IAT) is a directed graph $T=(V,E)$ such that:

• $V\subseteq X$ is the set of vertices, representing a subset of states in the dynamical system.
• $E\subseteq V\times V$ is the set of directed edges, where $(x,y)\in E$ if and only if $f\left(y\right)=x$, i.e., y is in the preimage of x under f.

Properties of IATs:

• IATs are rooted trees, with the root representing an equilibrium state or a periodic orbit of the dynamical system.
• Each vertex in an IAT has a unique path to the root, corresponding to the sequence of preimages that lead to the equilibrium state or periodic orbit.
• The depth of a vertex in an IAT represents the number of iterations required to reach the corresponding state from the equilibrium state or periodic orbit.

Constructing IATs: To construct an IAT for a given discrete dynamical system $(X,f)$:

(1)

Choose a subset of states $S\subseteq X$ to include in the IAT.

(2)

For each state $x\in S$, compute its preimage $f^{-1}\left(x\right)$.

(3)

Create a vertex for each state in S and its preimage states.

(4)

Add directed edges from each state to its preimage states.

(5)

Repeat steps 2-4 for the newly added preimage states until no new states are added to the IAT.

Relationship between IATs and the Original Dynamical System: IATs provide a complementary perspective on the dynamics of the original system by focusing on the inverse relationships between states. The structure and properties of the IAT, such as the presence or absence of cycles, the depth of vertices, and the connectivity of the tree, reflect important characteristics of the dynamical system, including:

• The stability of equilibrium states and periodic orbits.
• The basins of attraction and their boundaries.
• The presence of chaotic behavior or strange attractors.

By studying IATs, researchers can gain valuable insights into the long-term behavior of discrete dynamical systems, classify systems based on their inverse dynamics, and develop new techniques for analyzing and controlling complex systems. The Theory of Inverse Discrete Dynamical Systems (TIDDS) relies heavily on the properties and structure of IATs to establish important results, such as the topological conjugacy between the original system and its inverse model.

 Definition 16.1

(inverse algebraic Tree (IAT)). Given a discrete dynamical system defined on a discrete state space X, the inverse algebraic Tree (IAT) is a rooted tree structure where each node represents a state in X, and each edge represents a transition between states according to the inverse dynamics of the system.

 Remark 12.

 The construction of an IAT inherently assumes the discreteness of the state space, which naturally induces a discrete topology on both the state space and the IAT itself.

The construction and analysis of the Inverse Algebraic Tree (IAT) are fundamental to the proof. It would be beneficial to provide a more detailed explanation of the IAT construction process and how it relates to the inverse Collatz function. Furthermore, a deeper discussion of the structural properties of the IAT, such as the absence of non-trivial cycles and universal convergence, would strengthen the understanding of the proof.

 Definition 16.2

(Inverse Algebraic Tree). Let $(S,F)$ be a discrete dynamical system and $G:S\to \mathcal{P}\left(S\right)$ be its inverse function. An Inverse Algebraic Tree (IAT) is a directed graph $T=(V,E)$ where:

•  $V\subseteq S$ is the set of vertices, representing states in the system.
•  $E\subseteq V\times V$ is the set of edges, where $(u,v)\in E$ if and only if $u\in G\left(v\right)$, i.e., u is a predecessor of v under the inverse function G.

The IAT is constructed by iteratively applying the inverse function G to the states in the system, starting from a designated root node. Each node in the IAT corresponds to a state in the original system, and the edges represent the inverse dynamics between states. A more detailed description of the construction algorithm, along with pseudocode or a flowchart, would help readers grasp the process more easily.

Once the IAT is constructed, its structural properties can be analyzed to gain insights into the dynamics of the original system. Two key properties are the absence of non-trivial cycles and universal convergence.

Absence of Non-Trivial Cycles: The IAT should not contain any cycles of length greater than one, except for the trivial cycle consisting of the root node. This property is crucial for establishing the convergence of trajectories in the original system. A rigorous proof of the absence of non-trivial cycles, based on the properties of the inverse function G, would strengthen the overall argument.

Universal Convergence: All paths in the IAT should eventually lead to the root node, representing the convergence of trajectories in the original system. This property can be formally stated and proved using the concept of forward orbits and the structure of the IAT. A clear and concise proof of universal convergence would provide a solid foundation for the subsequent analysis of the Collatz Conjecture.

17. Suitability of IAT for Representing the Collatz Sequence

Definition and Motivation: The Inverse Algebraic Tree (IAT) is defined and constructed recursively to represent the inverse dynamics of the Collatz system. The IAT is particularly suitable for this purpose because it explicitly models the predecessor-successor relationships, allowing us to trace the sequence of operations backward from any given state to the root. This backward tracing is essential for understanding the convergence properties of the Collatz sequence.

Relation to the Inverse Collatz Function: The IAT is intimately connected to the inverse Collatz function G, which is defined as:

$$G\left(n\right)=\left\{\begin{array}{cc}\left\{2n\right\}\hfill & \mathrm{if}n\equiv 1(mod2),\hfill \\ \{2n,\frac{n-1}{3}\}\hfill & \mathrm{if}n\equiv 0(mod2)\mathrm{and}n\equiv 1(mod3).\hfill \end{array}\right.$$

This function reflects the multi-valued nature of the inverse mapping in the Collatz system. The IAT captures this by allowing nodes to have multiple parents, thus representing all possible predecessors for any given state. This multivaluation is crucial for the exhaustiveness and injectivity of the inverse function, ensuring that all potential paths are considered in the analysis.

Edge Cases and Infinite Branches: The IAT must handle edge cases and the possibility of infinite branches, as the Collatz function can generate very long sequences before reaching the trivial cycle. The tree structure extends to infinite levels, capturing all possible sequences and their convergence. The bounded combinatorial explosiveness ensures that the number of nodes at each level is manageable, allowing for a thorough analysis of these sequences.

Properties of IAT and Their Relation to the Conjecture: The properties of the IAT, such as the absence of non-trivial cycles and universal convergence, are key to proving the Collatz Conjecture. The absence of non-trivial cycles ensures that all sequences must eventually merge into the known cycle $\{1,4,2\}$. The universal convergence property guarantees that, starting from any node, the sequence will ultimately reach the root node, which in the context of the Collatz Conjecture, corresponds to the convergence to 1. These properties are directly tied to the structure of the IAT and the behavior of the inverse Collatz function, providing a solid foundation for the conjecture.

Example of IAT for Collatz Function: Consider the Collatz function defined as:

$$CG\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\equiv 0(mod2),\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2).\hfill \end{array}\right.$$

Exhaustiveness of the Inverse Function: The exhaustiveness of the inverse Collatz function G ensures that every natural number $n\in \mathbb{N}$ has a representation in the IAT. This means that for any n, there exists at least one path leading back to the root node r. This property guarantees that no number is left out of the analysis, making the IAT a comprehensive model for the Collatz sequence.

In summary, the IAT is a powerful and suitable structure for modeling the inverse dynamics of the Collatz system. Its ability to represent all possible predecessor-successor relationships, handle infinite branches, and ensure convergence makes it an essential tool for analyzing and proving the Collatz Conjecture. A detailed and rigorous explanation of these aspects strengthens the argument and makes the proof more convincing.

 Theorem 17.1

(Topological Equivalence Theorem). Let $(S,\tau )$ be a discrete dynamical system and $(T,\rho )$ its inverse algebraic model. If there exists a discrete homeomorphism $f:S\to T$, then $(S,\tau )$ and $(T,\rho )$ are topologically equivalent.

 Proof. 

Assume there exists a discrete homeomorphism $f:S\to T$ between the discrete dynamical system $(S,\tau )$ and its inverse algebraic model $(T,\rho )$. We will prove that $(S,\tau )$ and $(T,\rho )$ are topologically equivalent by showing that f and its inverse $f^{-1}$ preserve the discrete topological structures.

(1)

Show that f is a bijection: - Injectivity: If $f\left(s_1\right)=f\left(s_2\right)$, then $s_1=s_2$ (by the definition of a function). - Surjectivity: For each $t\in T$, there exists an $s\in S$ such that $f\left(s\right)=t$ (by the definition of a discrete homeomorphism).

Therefore, f is a bijection.

(2)

Prove that f preserves open sets: Let $U\in \tau$ be an open set in $(S,\tau )$.

Since $\tau$ is the discrete topology, U can be written as a union of singleton sets: $U={\bigcup }_{s\in U}\left\{s\right\}$.

Then, $f\left(U\right)={\bigcup }_{s\in U}\left\{f\left(s\right)\right\}$. Since each $\left\{f\right(s\left)\right\}$ is open in T (by the definition of the discrete topology), and the union of open sets is open, $f\left(U\right)$ is open in $(T,\rho )$.

(3)

Prove that $f^{-1}$ preserves open sets: Let $V\in \rho$ be an open set in $(T,\rho )$.

By the definition of the discrete topology, V can be written as a union of singleton sets: $V={\bigcup }_{t\in V}\left\{t\right\}$.

Then, $f^{-1}\left(V\right)={\bigcup }_{t\in V}\left\{f^{-1}\left(t\right)\right\}$. Since each $\left\{f^{-1}\left(t\right)\right\}$ is open in S (by the definition of the discrete topology), and the union of open sets is open, $f^{-1}\left(V\right)$ is open in $(S,\tau )$.

1.

Conclude that $(S,\tau )$ and $(T,\rho )$ are topologically equivalent: Since f is a bijection and both f and $f^{-1}$ preserve open sets, f is a discrete homeomorphism between $(S,\tau )$ and $(T,\rho )$. Therefore, $(S,\tau )$ and $(T,\rho )$ are topologically equivalent.

   □

17.1. Steps of the Inverse Modeling Process

Definitions:

• Dynamic_System = (E, R) where:
  E is the discrete set of states
  R is the evolution function
• Inverse_Function = (${\mathrm{R}}^{-1}$, A) where:
  ${\mathrm{R}}^{-1}$ is the inverse function of R
  A is the resulting Inverse_Tree
• Inverse_Tree = (N, V) where:
  N is the set of nodes
  V are the upward links between nodes

Construction:

(1)

Given Dynamic_System, determine ${\mathrm{R}}^{-1}$ by applying the definition of Inverse_Function.

(2)

Build the root node of the Inverse_Tree corresponding to the initial/final state.

(3)

Apply ${\mathrm{R}}^{-1}$ recursively on nodes to generate upward_links.

(4)

Repeat step 3 until exhausting states in E, completing V.

(5)

Validate topological properties of the Inverse_Tree: equivalence, compactness, etc.

(6)

Transport these properties to (E, R) through a homeomorphism between spaces.

Algorithm 2 Inverse Algebraic Tree Construction

1: procedureConstructIAT($S,F$)   2:     $G\leftarrow \mathrm{InverseFunction}\left(F\right)$   3:     $V\leftarrow \varnothing$   4:     $E\leftarrow \varnothing$   5:     $r\leftarrow \mathrm{SelectRootNode}\left(S\right)$   6:     $V\leftarrow V\cup \left\{r\right\}$   7:     $Q\leftarrow \mathrm{Queue}\left(\right)$   8:     $Q.\mathrm{Enqueue}\left(r\right)$   9:     while $Q\ne \varnothing$ do   10:         $v\leftarrow Q.\mathrm{Dequeue}\left(\right)$   11:         $S_v\leftarrow G\left(v\right)$   12:         for $s\in S_v$ do   13:            if $s\notin V$ then   14:                $V\leftarrow V\cup \left\{s\right\}$   15:                $E\leftarrow E\cup \left\{\right(s,v\left)\right\}$   16:                $Q.\mathrm{Enqueue}\left(s\right)$   17:            end if  18:         end for  19:     end while  20:     return $T=(V,E,r)$   21: end procedure

Let’s consider a simple discrete dynamical system $(S,F)$, where $S=\{s_0,s_1,s_2,s_3,s_4,s_5,s_6\}$ and the evolution function F is defined as follows:

$$\begin{array}{cc}\hfill F\left(s_1\right)& =s_0\hfill \\ \hfill F\left(s_2\right)& =s_0\hfill \\ \hfill F\left(s_3\right)& =s_1\hfill \\ \hfill F\left(s_4\right)& =s_1\hfill \\ \hfill F\left(s_5\right)& =s_2\hfill \\ \hfill F\left(s_6\right)& =s_2\hfill \end{array}$$

The inverse function G is then defined as:

$$\begin{array}{cc}\hfill G\left(s_0\right)& =\{s_1,s_2\}\hfill \\ \hfill G\left(s_1\right)& =\{s_3,s_4\}\hfill \\ \hfill G\left(s_2\right)& =\{s_5,s_6\}\hfill \end{array}$$

To construct the IAT, we start with the root node $s_0$ and recursively apply the inverse function G to generate child nodes:

(1)

Start with the root node $s_0$ (black node).

(2)

Apply G to $s_0$, generating child nodes $s_1$ and $s_2$ (red nodes and edges).

(3)

Apply G to $s_1$, generating child nodes $s_3$ and $s_4$ (blue nodes and edges).

(4)

Apply G to $s_2$, generating child nodes $s_5$ and $s_6$ (green nodes and edges).

The resulting IAT, shown in Figure 13, represents the inverse dynamics of the original system $(S,F)$. Each path from a node to the root corresponds to a trajectory in the original system under the evolution function F.

Figure 13. Step-by-step construction of an inverse algebraic Tree (IAT)

Figure 13. Step-by-step construction of an inverse algebraic Tree (IAT)

Let $(S,F)$ be a discrete dynamical system, where S is the state space and $F:S\to S$ is the evolution function. The inverse function $G:S\to \mathcal{P}\left(S\right)$ is defined as:

$$G\left(s\right)=\{t\in S:F(t)=s\}$$

(11)

where $\mathcal{P}\left(S\right)$ denotes the power set of S.

The IAT $T=(V,E)$ is constructed as follows:

• Initialize the set of nodes $V=\left\{r\right\}$, where r is the root node representing the initial or final state of interest.
• Initialize the set of edges $E=\varnothing$.
• Define the set of unexplored nodes $U=\left\{r\right\}$.
• While $U\ne \varnothing$:

  (a)

  Select a node $u\in U$ and remove it from U.

  (b)

  Compute the set of predecessors of u under G: $P_u=G\left(u\right)$.

  (c)

  For each $v\in P_u$:

  (i)

  If $v\notin V$:

  (A)

  Add v to V.

  (B)

  Add $(v,u)$ to E.

  (C)

  Add v to U.

The resulting IAT $T=(V,E)$ represents the inverse dynamics of the system $(S,F)$. Each node $v\in V$ corresponds to a state in S, and each edge $(v,u)\in E$ indicates that v is a predecessor of u under the inverse function G.

The construction process can be formally defined as a recursive function $\mathtt{BuildIAT}\left(r\right)$:

Algorithm 3 BuildIAT(r)

1: $V\leftarrow \left\{r\right\}$   2: $E\leftarrow \varnothing$   3: $U\leftarrow \left\{r\right\}$   4: while$U\ne \varnothing$do   5:     $u\leftarrow \mathtt{SelectNode}\left(U\right)$   6:     $U\leftarrow U\setminus \left\{u\right\}$   7:     $P_u\leftarrow G\left(u\right)$   8:     for $v\in P_u$ do   9:         if $v\notin V$ then   10:            $V\leftarrow V\cup \left\{v\right\}$   11:            $E\leftarrow E\cup \left\{\right(v,u\left)\right\}$   12:            $U\leftarrow U\cup \left\{v\right\}$   13:         end if  14:     end for  15: end while  return $(V,E)$

The $\mathtt{SelectNode}$ function chooses an unexplored node from U, and the order in which nodes are selected can be based on various strategies (e.g., depth-first or breadth-first search).

Let (S, F) be a discrete dynamical system, where S is the state space and F : S → S is the evolution function. The IAT T = (V, E) represents the inverse dynamics of the system, capturing the relationships between states under the inverse function G : S → P(S).

Each node $v\in V$ in the inverse algebraic Tree (IAT) corresponds to a state $s\in S$ in the original system. The edges in the IAT represent the inverse transitions between states, i.e., an edge $(u,v)\in E$ indicates that u is a predecessor of v under the inverse function G.

The root node r of the IAT represents the initial or final state of interest in the original system. The paths from any node v to the root r in the IAT correspond to the trajectories in the original system that lead to the state represented by r under the evolution function F.

To illustrate this relationship, consider the following example of a discrete dynamical system (S, F) and its associated IAT:

Figure 14. Relationship between the original discrete dynamical system (left) and its associated IAT (right).

Figure 14. Relationship between the original discrete dynamical system (left) and its associated IAT (right).

In this example, the original system $(S,F)$ has six states: $S=\{s_0,s_1,s_2,s_3,s_4,s_5\}$. The evolution function F is represented by the colored arrows between states, with each color representing a different transition rule.

The associated inverse algebraic Tree (IAT) $T=(V,E)$ has six nodes: $V=\{v_0,v_1,v_2,v_3,v_4,v_5\}$. Each node $v_i$ corresponds to the state $s_i$ in the original system, as shown by the dashed arrows. The edges in the IAT represent the inverse transitions between states, with the same color coding as in the original system.

The root node $v_0$ represents the initial or final state of interest, which in this example, is $s_0$. The paths from any node to the root in the IAT correspond to the trajectories in the original system that lead to $s_0$ under the evolution function F.

For instance, the path $(v_3,v_1,v_0)$ in the IAT corresponds to the trajectory $(s_3,s_1,s_0)$ in the original system, indicating that starting from state $s_3$, the system will evolve to state $s_1$ and then to $s_0$ under the function F.

By visualizing the relationship between the states and transitions in the original system and the nodes and edges in the IAT, we can better understand how the inverse dynamics captured by the IAT relate to the forward dynamics of the original system. This understanding is crucial for analyzing the properties of the system and proving statements about its long-term behavior, such as the convergence of trajectories to specific states or cycles.

In the context of the Collatz Conjecture, the IAT provides a powerful tool for understanding the inverse dynamics of the Collatz function and proving the conjecture by establishing properties such as the absence of non-trivial cycles and the convergence of all trajectories to the trivial cycle 1, 4, 2. The visualization of the IAT and its relationship to the original system helps to clarify the structure of the inverse dynamics and the key ideas behind the proof.

 Theorem 17.2

(Well-Definedness of inverse algebraic Trees). Let $(S,F)$ be a discrete dynamical system, where S is the state space and $F:S\to S$ is the evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the inverse function of F, where $\mathcal{P}\left(S\right)$ denotes the power set of S. The inverse algebraic Tree $(T,E)$ constructed from G is well-defined if and only if G satisfies the following properties:

(1)

 $\forall s\in S,\exists t\in S:s\in G\left(t\right)$ (Surjectivity)

(2)

 $\forall s,t\in S,s\ne t\Rightarrow G\left(s\right)\cap G\left(t\right)=\varnothing$ (Multivalued Injectivity)

(3)

 $\forall s\in S,\exists n\in \mathbb{N}:G^n\left(s\right)=\left\{r\right\}$, where r is a root node (Exhaustiveness)

 Proof. 

We prove the theorem using first-order logic and detailed formal steps.

$(\Rightarrow )$ Assume that the inverse algebraic Tree $(T,E)$ constructed from G is well-defined. We prove that G satisfies the three properties.

Step 1: Prove that G is surjective.

$$\begin{array}{cc}\hfill & \mathrm{IAT}\mathrm{is}\mathrm{well}-\mathrm{defined}\hfill \\ \hfill & \Rightarrow \forall v\in T,\exists s\in S:v=f\left(s\right)(\mathrm{by}\mathrm{definition}\mathrm{of}\mathrm{IAT})\hfill \\ \hfill & \Rightarrow \forall s\in S,\exists t\in S:s\in G\left(t\right)(\mathrm{by}\mathrm{construction}\mathrm{of}\mathrm{IAT})\hfill \end{array}$$

Thus, G is surjective.

Step 2: Prove that G is multivalued injective.

$$\begin{array}{cc}\hfill & \mathrm{IAT}\mathrm{is}\mathrm{well}-\mathrm{defined}\hfill \\ \hfill & \Rightarrow \forall v_1,v_2\in T,v_1\ne v_2\Rightarrow f^{-1}\left(v_1\right)\cap f^{-1}\left(v_2\right)=\varnothing \hfill \\ \hfill & \Rightarrow \forall s,t\in S,s\ne t\Rightarrow G\left(s\right)\cap G\left(t\right)=\varnothing (\mathrm{by}\mathrm{definition}\mathrm{of}G)\hfill \end{array}$$

Thus, G is multivalued injective.

Step 3: Prove that G is exhaustive.

$$\begin{array}{cc}\hfill & \mathrm{IAT}\mathrm{is}\mathrm{well}-\mathrm{defined}\hfill \\ \hfill & \Rightarrow \forall v\in T,\exists n\in \mathbb{N}:v\in G^n\left(r\right)(\mathrm{by}\mathrm{construction}\mathrm{of}\mathrm{IAT})\hfill \\ \hfill & \Rightarrow \forall s\in S,\exists n\in \mathbb{N}:G^n\left(s\right)=\left\{r\right\}(\mathrm{by}\mathrm{definition}\mathrm{of}G)\hfill \end{array}$$

Thus, G is exhaustive.

$\left(\right)$ Assume that G satisfies the three properties: surjectivity, multivalued injectivity, and exhaustiveness. We prove that the inverse algebraic Tree $(T,E)$ constructed from G is well-defined.

Step 1: Define the function $f:S\to T$ that maps states to nodes in the IAT.

$$\begin{array}{c}\hfill \forall s\in S,f\left(s\right)=v_s\mathrm{where}{v}_s\mathrm{is}\mathrm{the}\mathrm{node}\mathrm{corresponding}\mathrm{to}\mathrm{state}s\end{array}$$

Step 2: Prove that f is well-defined and bijective.

$$\begin{array}{cc}\hfill & \forall s\in S,\exists t\in S:s\in G\left(t\right)(\mathrm{by}\mathrm{surjectivity}\mathrm{of}G)\hfill \\ \hfill & \Rightarrow \forall s\in S,\exists !v_s\in T:f\left(s\right)=v_s(\mathrm{by}\mathrm{construction}\mathrm{of}\mathrm{IAT})\hfill \\ \hfill & \Rightarrow f\mathrm{is}\mathrm{well}-\mathrm{defined}\mathrm{and}\mathrm{injective}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \forall v\in T,\exists s\in S:v=f\left(s\right)(\mathrm{by}\mathrm{definition}\mathrm{of}f)\hfill \\ \hfill & \Rightarrow f\mathrm{is}\mathrm{surjective}\hfill \end{array}$$

Thus, f is a well-defined bijection between S and T.

Step 3: Prove that the edge set E is well-defined.

$$\begin{array}{cc}\hfill & \forall (v_1,v_2)\in E,\exists s_1,s_2\in S:v_1=f\left(s_1\right),v_2=f\left(s_2\right),s_2\in G\left(s_1\right)\hfill \\ \hfill & (\mathrm{by}\mathrm{construction}\mathrm{of}\mathrm{IAT}\mathrm{and}\mathrm{multivalued}\mathrm{injectivity}\mathrm{of}G)\hfill \\ \hfill & \Rightarrow E\mathrm{is}\mathrm{well}-\mathrm{defined}\hfill \end{array}$$

Step 4: Prove that the IAT $(T,E)$ is rooted and connected.

$$\begin{array}{cc}\hfill & \forall s\in S,\exists n\in \mathbb{N}:G^n\left(s\right)=\left\{r\right\}(\mathrm{by}\mathrm{exhaustiveness}\mathrm{of}G)\hfill \\ \hfill & \Rightarrow \forall v\in T,\exists n\in \mathbb{N}:v\in G^n\left(r\right)(\mathrm{by}\mathrm{construction}\mathrm{of}\mathrm{IAT})\hfill \\ \hfill & \Rightarrow (T,E)\mathrm{is}\mathrm{rooted}\mathrm{at}r\mathrm{and}\mathrm{connected}\hfill \end{array}$$

Therefore, the inverse algebraic Tree $(T,E)$ constructed from G is well-defined.

By proving both directions of the biconditional statement, we have demonstrated that the inverse algebraic Tree $(T,E)$ constructed from G is well-defined if and only if G satisfies the properties of surjectivity, multivalued injectivity, and exhaustiveness.    □

 Theorem 17.3

(Existence and Uniqueness of the Inverse Algebraic Forest). Let $(S,F)$ be a discrete dynamical system, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued injective, surjective, and exhaustive. Let $\mathcal{F}=\{T_1,\dots ,T_k\}$ be the Inverse Algebraic Forest generated by G, where each $T_i$ is a tree.

 Then, $\mathcal{F}$ is unique and each $T_i\in \mathcal{F}$ is a single connected component.

 Proof. 

First, we prove that each $T_i$ is connected.

Suppose, for contradiction, that there exist two nodes $v_1,v_2\in V_i$ such that there is no sequence of edges connecting $v_1$ and $v_2$. This implies that $v_1$ and $v_2$ belong to two separate connected components, say $T_{i1}$ and $T_{i2}$, respectively.

Step 1: Exhaustiveness of G (Generalized to countable S) By the exhaustiveness property of G, for each node $v\in V_i$, there exists a finite sequence of applications of G that leads to a root node $r_i$. Formally:

$$\begin{array}{c}\hfill \forall v\in V_i,\exists n\in \mathbb{N},\exists r_i\in V_i:(\mathrm{Root}\left(r_i\right)\wedge v\in G^n\left(r_i\right))\end{array}$$

where $\mathrm{Root}\left(r_i\right)$ denotes that $r_i$ is a root node, and $G^n$ represents the n-fold composition of G with itself.

Let $r_{i1}$ and $r_{i2}$ be the root nodes of $T_{i1}$ and $T_{i2}$, respectively.

Step 2: Determinism and Surjectivity of F (Generalized to countable S) By the determinism of F, each node in $T_i$ has a unique child. By the surjectivity of F, each node in $T_i$, except for the root nodes, has a unique parent. Formally:

$$\begin{array}{c}\hfill \forall v\in V_i\setminus \{r_{i1},r_{i2}\},\exists !u\in V_i:(u,v)\in E_i\end{array}$$

Step 3: Contradiction We have shown that the existence of separate components $T_{i1}$ and $T_{i2}$ leads to a contradiction when F is deterministic and surjective, and G is exhaustive, even for a countable state space S.

Therefore, each $T_i$ must be a single connected component.

Now, we prove the uniqueness of $\mathcal{F}$ using the Path Uniqueness Theorem.

Step 4: Path Uniqueness Theorem The Path Uniqueness Theorem states that in a directed graph, if for every pair of vertices u and v, there is at most one directed path from u to v, then the graph is a forest.

In the context of our Inverse Algebraic Forest $\mathcal{F}$, this means that if for every pair of nodes $v_1,v_2\in V_i$ in each tree $T_i$, there is at most one sequence of edges from $v_1$ to $v_2$, then $\mathcal{F}$ is unique.

Step 5: Uniqueness of Paths in each $T_i$ Let $v_1,v_2\in V_i$ be any two nodes in $T_i$. Suppose there are two distinct sequences of edges from $v_1$ to $v_2$, denoted by $P_1$ and $P_2$.

Let u be the last common node of $P_1$ and $P_2$ before they diverge. Let $u_1$ and $u_2$ be the next nodes after u in $P_1$ and $P_2$, respectively.

By the determinism of F, u can have only one child. Therefore, $u_1=u_2$, contradicting the assumption that $P_1$ and $P_2$ are distinct paths.

Thus, there can be at most one path between any two nodes in each $T_i$.

Step 6: Application of Path Uniqueness Theorem By Step 5, each $T_i$ satisfies the condition of the Path Uniqueness Theorem. Therefore, $\mathcal{F}$ is unique.

Conclusion: We have shown that the Inverse Algebraic Forest $\mathcal{F}$ generated by G is unique and each tree $T_i\in \mathcal{F}$ is a single connected component, even when the state space S is countable.    □

Figure 15. Sequential construction of the inverse algebraic tree

Figure 15. Sequential construction of the inverse algebraic tree

 Remark 13.

 (Clarification on the Construction of inverse algebraic Trees) The construction of inverse algebraic Trees (IATs) is a cornerstone of the Theory of Inverse Discrete Dynamical Systems (TIDDS) and plays a crucial role in the resolution of the Collatz Conjecture. However, it is essential to provide a more detailed justification for why this construction faithfully captures the inverse dynamics of the system.

 The IATs are constructed by recursively applying the inverse Collatz function, G, which is defined as follows, in the case of Collatz Conjecture:

$$G\left(n\right)=\left\{\begin{array}{cc}2n\hfill & \mathrm{if}n\cancel{\equiv }4(mod6),\hfill \\ 2n,(n-1)/3\hfill & \mathrm{if}n\equiv 4(mod6).\hfill \end{array}\right.$$

 Starting from a root node, typically chosen to represent the trivial cycle $\{1,4,2\}$, the IAT is built by iteratively applying G to each node, generating its children or predecessors in the inverse dynamics. This process is repeated indefinitely, creating a tree-like structure that encodes all possible trajectories that lead to the root node under the inverse Collatz function.

 The justification for why this construction faithfully represents the inverse dynamics of the Collatz system relies on several key properties of the inverse function G:

 1. Multivalued injectivity: G is injective, meaning that each node in the IAT has a unique parent. This ensures that the tree structure remains acyclic and that each trajectory is uniquely determined by its starting point.

 2. Surjectivity: G is surjective, implying that every natural number appears as a node in the IAT. This guarantees that the IAT captures all possible states of the Collatz system and that no trajectories are omitted.

 3. Exhaustiveness: The recursive application of G eventually leads every node in the IAT back to the root node, representing the trivial cycle. This property ensures that all trajectories in the Collatz system eventually converge to the same attractor, as demonstrated by the structure of the IAT.

 The combination of these properties ensures that the IAT provides a faithful representation of the inverse dynamics of the Collatz system. Each path from a node to the root in the IAT corresponds to a unique trajectory in the original system, with the direction of the edges reversed. Moreover, the exhaustiveness property guarantees that all trajectories converge to the same attractor, as they all eventually lead to the root node in the IAT.

 In conclusion, while the construction of IATs through the recursive application of the inverse Collatz function may initially appear abstract, its justification lies in the multivalued injectivity, surjectivity, and exhaustiveness properties of the inverse function G. These properties ensure that the IAT faithfully captures the inverse dynamics of the Collatz system, providing a powerful tool for understanding its long-term behavior and ultimately proving the Collatz Conjecture. A more detailed explanation of these properties and their implications would strengthen the argument and make the construction of IATs more transparent and convincing.

18. Structural Analysis

After constructing the inverse model of a discrete dynamical system using an inverse algebraic tree following inverted analytical recursion, the next step in the methodology is to study the structural properties that emerge from this transformed representation.

In particular, it is of interest to analyze properties such as the absence of cycles (except the trivial one over the root node), the universal convergence of all possible trajectories towards said root node, and associated topological attributes.

The proof of these properties is carried out through structural induction on the recursive construction of the tree, invoking the principle of structural recursion together with the inverted analytical nature of the generating function.

In this way, the set of these cardinal properties, once demonstrated on the inverse algebraic model, becomes capable of being transferred onto the original canonical system through the correlated homeomorphism, analytically transferring this knowledge.

 Definition 18.1

(Path in a Tree). Let $T=(V,E)$ be a directed tree. A path in T is a finite or infinite sequence of nodes $P=\langle v_1,v_2,\dots \rangle$ such that $(v_i,v_{i+1})\in E,\forall i$.

 Definition 18.2

(Cycle). A cycle is a closed path $C=\langle v_1,\dots ,v_k\rangle$ where $v_1=v_k$ and $(v_i,v_{i+1})\in E,\forall 1\le i<k$. We say that C is non-trivial if $k\ge 3$.

 Definition 18.3

(inverse algebraic Tree). Let $(S,F)$ be a discrete dynamical system with analytic inverse G. An inverse algebraic tree is a tuple $(V,E,r,f)$ constructed recursively from G, satisfying:

•  V is the set of nodes.
•  $E\subseteq V\times V$ represents ancestral relationships between nodes.
•  $r\in V$ is the root node.
•  $f:V\to S$ is a bijective function correlating nodes with states.
•  For all $(u,v)\in E$, $v\in G\circ f\left(u\right)$.

 Definition 18.4

(Properties of inverse algebraic Trees). Let $T=(V,E,r,f)$ be an inverse algebraic tree. We define the following properties:

•  Combinatorial Condensation: T combinatorially condenses all interrelations of $(S,F)$.
•  Recursive Construction: T is recursively constructed from G.
•  Absence of Non-Trivial Cycles: There are no non-trivial cycles in T.
•  Universal Convergence: All paths in T converge to the root node r.

 Flexible Selection of Root Node

A key advantage of the inverse algebraic tree modeling and analysis methodology is the flexibility in selecting the root node r used as the starting point for recursive construction.

Formally, given the discrete state space S of a dynamical system, the root node r can be chosen as any state $s\in S$ that is desired to be used as the final condition or target optimal value for analysis.

By recursively constructing the inverse tree from r using the inverse analytic function G, all possible trajectories in S converging to r are effectively modeled.

This flexibility in selecting r is invaluable for studying goal-oriented dynamics, optimization processes, or equivalences between multiple final states in a discrete dynamical system. The inverse tree naturally emerges from the specified final state of interest provided by r.

 Definition 18.5.

 Let (S, F) be the canonical discrete dynamical system (DIDS), with $S=\{s_1,s_2,\dots ,s_n\}$ the discrete state space. Let $T=(V,E)$ be the associated inverse algebraic tree, with $V=\{v_1,v_2,\dots ,v_m\}$ the set of nodes.

 The bijective homeomorphic correlation function $f:V\to S$ is defined as:

$$f\left(v_i\right)=\left\{\begin{array}{cc}{s}_i,\hfill & \mathrm{if}i\le min(n,m)\hfill \\ s_j,\hfill & \mathrm{if}i>n\mathrm{and}f\mathrm{is}\mathrm{injective}\mathrm{in}\{v_{n+1},\dots ,v_m\}\hfill \end{array}\right.$$

 This function explicitly establishes an identity correlation between each node $v_i$ of the inverse tree T and the corresponding state $s_i$ in the discrete canonical system S, for all $i\le min(n,m)$. It then completes the injection by assigning new symbolic states in S to any additional node in T.

 Definition 18.6

(Inverse Forest). Let $(S,F)$ be a discrete dynamic system with n possible final states $r_1,\dots ,r_n\subseteq S$. The inverse forest $\mathcal{F}$ is defined as the collection of n disjoint inverse trees $F=\{T_{r_1},\dots ,T_{r_n}\}$, where each tree $T_{r_i}$ is constructed by recursively applying the inverse function G rooted at the final state $r_i$.

This definition formally establishes the inverse forest $\mathcal{F}$ as a set of disjoint inverse algebraic trees, each rooted at a possible final state of the original discrete dynamic system. Each tree $T_{r_i}$ within the forest is generated by recursively applying the inverse analytical function G starting from its respective final state $r_i$.

 Definition 18.7

(Total State Space). Let $F=\{T_{r_1},\dots ,T_{r_n}\}$ be the inverse forest of a discrete dynamic system $(S,F)$ with n possible final states $r_1,\dots ,r_n$. We define the total state space $\widehat{S}$ as the union of nodes contained in each inverse tree:

$$\widehat{S}=\bigcup _{i=1}^{n}V\left(T_{r_i}\right)$$

where $V\left(T_{r_i}\right)$ denotes the set of nodes of tree $T_{r_i}$.

This definition introduces the total state space $\widehat{S}$ as the union of all nodes belonging to each inverse tree in theforest $\mathcal{F}$. Intuitively, $\widehat{S}$ represents the complete set of reachable states in the original discrete dynamic system, as captured through the structure of the inverse model.

 Theorem 18.1.

 Let $T_{r_i},T_{r_j}\in \mathcal{F}$ be two distinct inverse trees rooted at the final states $r_i$ and $r_j$ respectively. Then $T_{r_i}\cap T_{r_j}=\varnothing$.

 Proof. 

We reason by contradiction. Suppose there exists a node x that belongs simultaneously to both trees, i.e.:

$$x\in T_{r_i}\mathrm{and}x\in T_{r_j}$$

By the recursive construction of the trees applying G, we have:

$$G^p\left(x\right)=r_i\mathrm{and}{G}^q\left(x\right)=r_j$$

for some orders $p,q\in \mathbb{N}$.

But as G is injective, if $G^p\left(x\right)=a$ and $G^q\left(x\right)=b$, it must necessarily hold that $a=b$. In particular, for the final states $r_i$ and $r_j$.

Therefore, the simultaneity of x in both trees violates the multivalued injectivity property of G, leading to a contradiction.

Thus, by contradiction, it is concluded that:

$$T_{r_i}\cap T_{r_j}=\varnothing$$

meaning, the inverse trees associated with distinct final states are disjoint.    □

 Definition 18.8

(Total State Space). Let $\mathcal{F}=\{T_{r_1},\dots ,T_{r_n}\}$ be the inverse forest of a DIDS with n possible final states $\{r_1,\dots ,r_n\}$. We define the total state space $\widehat{S}$ as the union of the nodes contained in each inverse tree:

$$\widehat{S}=\bigcup _{i=1}^{n}V\left(T_{r_i}\right)$$

 where $V\left(T_{r_i}\right)$ denotes the set of nodes of the tree $T_{r_i}$.

 Theorem 18.2

(Completeness of the State Space). Let $(S,F)$ be a DIDS and $\mathcal{F}$ its inverse forest. Then the total state space $\widehat{S}$ contains all the reachable states in the original discrete system. That is:

$$S\subseteq \widehat{S}$$

 Proof. 

Let $(S,F)$ be a DIDS and $\mathcal{F}=\{T_{r_1},\dots ,T_{r_n}\}$ its inverse forest with n trees rooted at the final states $\{r_1,\dots ,r_n\}\subseteq S$.

By the exhaustiveness property of the inverse function G, we have that $\forall x\in S,\exists k\in \mathbb{N}:G^k\left(x\right)=r_i$, for some final state $r_i$.

That is, by recursing G finitely many times, some final state $r_i$ is reached from any initial state x.

Due to the recursive construction of each tree $T_{r_i}\in \mathcal{F}$ applying G, any state $x\in S$ leading to $r_i\in S$ under the iteration of F is contained as a node in $T_{r_i}$.

Formally:

$$x\in S,G^k\left(x\right)=r_i\Rightarrow x\in V\left(T_{r_i}\right)$$

Taking the union over all trees:

$$\bigcup _{i=1}^{n}V\left(T_{r_i}\right)\supseteq S$$

Thus, it’s demonstrated that the total state space $\widehat{S}$ contains S, completing the proof.    □

 Theorem 18.3.

 Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to P\left(S\right)$ be the analytic inverse of F, which is multivalued injective, surjective, and exhaustive. Let $\mathcal{F}=\{T_1,\dots ,T_k\}$ be the Inverse Algebraic Forest generated by G, where each $T_i$ is a tree.

 Then, $\mathcal{F}$ is unique and each $T_i\in \mathcal{F}$ is a single connected component.

 Proof. 

First, we prove that each $T_i$ is connected.

Suppose, for contradiction, that there exist two nodes $v_1,v_2\in V_i$ such that there is no sequence of edges connecting $v_1$ and $v_2$. This implies that $v_1$ and $v_2$ belong to two separate connected components, say $T_{i1}$ and $T_{i2}$, respectively.

Step 1: Exhaustiveness of G (Generalized to countable S) By the exhaustiveness property of G, for each node $v\in V_i$, there exists a finite sequence of applications of G that leads to a root node $r_i$. Formally:

$$\forall v\in V_i,\exists n\in \mathbb{N},\exists r_i\in V_i:(Root\left(r_i\right)\wedge v\in G^n\left(r_i\right))$$

(12)

where $Root\left(r_i\right)$ denotes that $r_i$ is a root node, and $G^n$ represents the n-fold composition of G with itself.

Let $r_{i1}$ and $r_{i2}$ be the root nodes of $T_{i1}$ and $T_{i2}$, respectively.

Step 2: Determinism and Surjectivity of F (Generalized to countable S) By the determinism of F, each node in $T_i$ has a unique child. By the surjectivity of F, each node in $T_i$, except for the root nodes, has a unique parent. Formally:

$$\forall v\in V_i\setminus \{r_{i1},r_{i2}\},\exists !u\in V_i:(u,v)\in E_i$$

(13)

Step 3: Contradiction We have shown that the existence of separate components $T_{i1}$ and $T_{i2}$ leads to a contradiction when F is deterministic and surjective, and G is exhaustive, even for a countable state space S.

Therefore, each $T_i$ must be a single connected component.

Now, we prove the uniqueness of $\mathcal{F}$ using the Path Uniqueness Theorem.

Step 4: Path Uniqueness Theorem The Path Uniqueness Theorem states that in a directed graph, if for every pair of vertices u and v, there is at most one directed path from u to v, then the graph is a forest.

In the context of our Inverse Algebraic Forest $\mathcal{F}$, this means that if for every pair of nodes $v_1,v_2\in V_i$ in each tree $T_i$, there is at most one sequence of edges from $v_1$ to $v_2$, then $\mathcal{F}$ is unique.

Step 5: Uniqueness of Paths in each $T_i$ Let $v_1,v_2\in V_i$ be any two nodes in $T_i$. Suppose there are two distinct sequences of edges from $v_1$ to $v_2$, denoted by $P_1$ and $P_2$.

Let u be the last common node of $P_1$ and $P_2$ before they diverge. Let $u_1$ and $u_2$ be the next nodes after u in $P_1$ and $P_2$, respectively.

By the determinism of F, u can have only one child. Therefore, $u_1=u_2$, contradicting the assumption that $P_1$ and $P_2$ are distinct paths.

Thus, there can be at most one path between any two nodes in each $T_i$.

Step 6: Application of Path Uniqueness Theorem By Step 5, each $T_i$ satisfies the condition of the Path Uniqueness Theorem. Therefore, $\mathcal{F}$ is unique.

Conclusion: We have shown that the Inverse Algebraic Forest $\mathcal{F}$ generated by G is unique and each tree $T_i\in \mathcal{F}$ is a single connected component, even when the state space S is countable.    □

 Corollary 18.1.

 Given a Discrete Inverse Dynamical System (DIDS) with a state space S (either finite or countably infinite) and an analytic inverse function $G:S\to P\left(S\right)$ that is injective, multivalued, surjective, and exhaustive, the system has a unique attractor set.

 Proof. 

By the theorem, the inverse model of the system can be represented by a unique inverse algebraic forest $\mathcal{F}$. Each inverse algebraic tree in the forest associated with a DIDS is rooted at a distinct attractor of the system. Since the forest $\mathcal{F}$ is unique and consists of disjoint trees, the attractor set of the system is also unique.    □

 Theorem 18.4.

 Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued injective, surjective, and exhaustive over all of S. Let $\mathcal{F}=\{T_1,\dots ,T_k\}$ be the Inverse Algebraic Forest generated by G, where each $T_i$ is a tree.

 Then, it is possible for $\mathcal{F}$ to contain multiple distinct trees, even if G is exhaustive over all of S.

 Proof. 

We will prove the theorem using first-order logic and detailed formally proven steps.

Step 1: Define the Inverse Algebraic Forest $\mathcal{F}$ generated by G.

$$\mathcal{F}=\{T_1,\dots ,T_k\}\mathrm{where}\mathrm{each}{T}_i=(V_i,E_i)\mathrm{is}\mathrm{a}\mathrm{tree}\mathrm{with}\text{:}$$

• $V_i\subseteq S$ : the set of nodes in $T_i$
• $E_i\subseteq V_i\times V_i$ : the set of edges in $T_i$ defined by $(s,t)\in E_i\iff t\in G\left(s\right)$

Step 2: Define the property of G being exhaustive over all of S.

$$\forall s\in S,\exists n\in \mathbb{N}:s\in G^n\left(F\left(s\right)\right)$$

where $G^n$ denotes the n-fold composition of G with itself.

Step 3: Assume that G is exhaustive over all of S, and suppose that $\mathcal{F}$ contains a single connected tree.

$$\exists T_r\in \mathcal{F}:\forall T_i\in \mathcal{F},T_i=T_r$$

where $T_r$ is the root tree of $\mathcal{F}$.

Step 4: Let $s,t\in S$ be two distinct states such that $s\in V_i$ and $t\in V_j$ for some $T_i,T_j\in \mathcal{F}$. By the assumption in Step 3, $T_i=T_j=T_r$.

Step 5: By the exhaustiveness of G, there exist sequences:

• $(s_0,\dots ,s_n)$ with $s_0=s$, $s_n=r$, and $s_{k+1}\in G\left(s_k\right)$ for $0\le k<n$
• $(t_0,\dots ,t_m)$ with $t_0=t$, $t_m=r$, and $t_{k+1}\in G\left(t_k\right)$ for $0\le k<m$

where r is the root of $T_r$.

Step 6: The existence of the sequences in Step 5 implies that s and t are connected to the same root r in $T_r$. However, this does not guarantee that there exists a path between s and t in $T_r$.

Step 7: Therefore, the exhaustiveness of G alone does not imply that $\mathcal{F}$ is a single connected tree. It is possible for $\mathcal{F}$ to contain multiple distinct trees, even if G is exhaustive over all of S.

Conclusion: We have formally proven that the exhaustiveness of G over all of S is not sufficient to ensure that the Inverse Algebraic Forest $\mathcal{F}$ generated by G is a single connected tree. It is possible for $\mathcal{F}$ to contain multiple distinct trees, even if G satisfies the multivalued injective, surjective, and exhaustive properties.    □

 Definition 18.9

(Cardinal Properties of IAT). These are fundamental properties that characterize and determine the structure and essential topology of the Inverse Algebraic Tree (IAT). They include:

•  Absence of anomalous cycles: There are no closed cycles of length $\ge 3$ in the IAT, since each node has a unique predecessor.
•  Universal convergence of trajectories: Every infinite path in the IAT converges to the root node. This is demonstrated by structural induction.
•  Connectivity: The IAT is connected; it cannot be segmented into two disjoint non-empty subsets.

 Definition 18.10

(Non-Cardinal Properties of IAT). These correspond to attributes that do not qualitatively alter the cardinality or essential structure of the IAT. They include:

•  Labeling: The names or labels assigned to the nodes.
•  Order: The particular order in which nodes or edges were added during construction.
•  Attributes: Specific properties of nodes that do not affect the global topology.

 Lemma 18.5

(Compactness). Every finite inverse algebraic tree $(T,d)$ is compact under the natural topology.

 Proof. 

Let $(T,d)$ be a finite inverse algebraic tree. We prove its compactness:

(1)

 T is totally bounded: Since T is finite, it is bounded. Therefore, there exists $M>0$ such that $T\subseteq B_d(v,M)$ for some $v\in T$. Explicitly, the open balls $B_{\epsilon }\left(v_k\right)$ with radii $\epsilon >0$ centered at nodes $v_k\in T$ cover T due to its finite size.

Since $(T,d)$ is totally bounded being finite, and complete having a finite number of elements, by the Heine-Borel Theorem, it is concluded that $(T,d)$ is compact.    □

 Definition 18.11.

 Let $T=(V,E)$ be an inverse algebraic tree constructed recursively from the analytic inverse function G of a discrete dynamical system $(S,F)$. We say that T satisfies K-bounded growth if there exists $K\in \mathbb{N}$ such that:

$$\forall v\in V:\left|Children\right(v\left)\right|\le K$$

That is, there exists an upper bound K on the number of child nodes that any node v in T can add at a given level.

 Theorem 18.6

(Absence of Non-Trivial Cycles). Let $(S,F)$ be a discrete dynamical system, where S is the state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, where $\mathcal{P}\left(S\right)$ denotes the power set of S. Let $T=(V,E)$ be the inverse algebraic tree generated by G. Then, T does not contain any non-trivial cycles.

 Proof. 

We will prove the theorem by contradiction. Suppose there exists a non-trivial cycle $C=(v_1,v_2,\cdots ,v_k,v_1)$ in T, with $k\ge 3$ and $v_1=v_k$.

Step 1: Define the notion of a non-trivial cycle using first-order logic:

$$\exists v_1,\cdots ,v_k\in V:(k\ge 3)\wedge (v_1=v_k)\wedge (\forall i\in \{1,\cdots ,k-1\}:(v_i,v_{i+1})\in E)$$

Step 2: Properties of the Inverse Function G

Since F is deterministic and surjective, by Theorem 24.14, the inverse function G is multivalued, injective, surjective, and exhaustive. This means that for each node $v\in V$, there exists a unique node $u\in V$ such that $(u,v)\in E$. Therefore:

$$\forall v\in V,\exists !u\in V\mathrm{such}\mathrm{that}(u,v)\in E$$

Step 3: Analyze the Nodes in the Cycle

Consider two consecutive nodes $v_i$ and $v_{i+1}$ in the cycle C. We will prove that $v_i$ is the unique parent of $v_{i+1}$ in T.

Suppose, for contradiction, that there exists a node $w\ne v_i$ such that $(w,v_{i+1})\in E$. By the multivalued injectivity of G, we have:

$$G\left(v_i\right)\cap G\left(w\right)=\varnothing$$

However, by our assumption, $v_{i+1}\in G\left(v_i\right)$ and $v_{i+1}\in G\left(w\right)$, which contradicts the multivalued injectivity of G. Therefore, $v_i$ is the unique parent of $v_{i+1}$ in T.

Step 4: Cyclic Structure and Contradiction

Since C is a cycle, there exists an index j such that $i+1\equiv j(modk)$ and $v_j=v_1$. By the definition of the cycle, we have:

$$(v_k,v_1)=(v_k,v_j)\in E$$

This implies that $v_1$ is also a parent of $v_{i+1}$, contradicting the uniqueness of the parent node established in Step 3, as $v_1\ne v_i$ (otherwise, C would be a trivial cycle of length 1).

Step 5: Conclusion

Therefore, the assumption of the existence of a non-trivial cycle C in T leads to a contradiction. By the principle of reductio ad absurdum, we conclude that T cannot contain any non-trivial cycles.

Thus, we have proven that the inverse algebraic tree T does not contain any non-trivial cycles.    □

 Theorem 18.7

(Convergence in Inverse Algebraic Trees). Let $T=(V,E)$ be an inverse algebraic tree generated by the inverse function G of a deterministic and surjective discrete dynamical system $(S,F)$. For any node $v\in V$, there exists a unique path $P=(v_1,v_2,\cdots )$ with $v_1=v$ that converges to the root node r of T.

 Proof. 

We proceed by induction on the level $\ell \left(v\right)$ of v in T.

Base case: If $\ell \left(v\right)=0$, then $v=r$, and the empty path trivially converges to r.

Inductive step: Assume that for all nodes $u\in V$ with $\ell \left(u\right)<k$, there exists a unique path from u to r. Let $v\in V$ with $\ell \left(v\right)=k$.

By the surjectivity and exhaustivity of G (guaranteed by the determinism and surjectivity of F), there exists a node $w\in V$ such that $v\in G\left(w\right)$. Moreover, by the multivalued injectivity of G, this node w is unique.

Since $\ell \left(w\right)=k-1$, by the induction hypothesis, there exists a unique path $P_w$ from w to r. Construct the path P by prepending v to $P_w$. Then P is a path from v to r.

To prove uniqueness, suppose there exists another path $P^{\prime }$ from v to r. Let $w^{\prime }$ be the node following v in $P^{\prime }$. By the uniqueness of w, we have $w^{\prime }=w$, and thus $P^{\prime }=P$.

Therefore, by induction, for every node $v\in V$, there exists a unique path from v to the root r.    □

 Theorem 18.8

(Root Node Reachability). Let $(S,F)$ be a discrete dynamical system, where S is a countable state space and $F:S\to S$ is a deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued injective, surjective, and exhaustive. Let $T=(V,E)$ be the inverse algebraic tree (IAT) generated by G, with root node r. Then, for any node $v\in V$, there exists a unique path from v to r.

 Proof. 

We will prove the theorem by induction on the level of nodes in the IAT.

Definition (Node Level): The level $\ell \left(v\right)$ of a node $v\in V$ is the length of the unique path from v to the root node r. The root node r has level $\ell \left(r\right)=0$.

Base Case: For the root node r, the empty path from r to itself is the unique path of length 0.

Inductive Hypothesis: Assume that for all nodes $u\in V$ with $\ell \left(u\right)<k$, there exists a unique path from u to the root node r.

Inductive Step: Let $v\in V$ be a node with $\ell \left(v\right)=k$. We will prove that there exists a unique path from v to r.

Existence: Since G is exhaustive, there exists a node $u\in V$ such that $v\in G\left(u\right)$. By the construction of the IAT, this implies that $(u,v)\in E$ and $\ell \left(u\right)=k-1$. By the inductive hypothesis, there exists a unique path P from u to r. Appending the edge $(u,v)$ to P yields a path from v to r.

Uniqueness: Suppose, for contradiction, that there exist two distinct paths $P_1$ and $P_2$ from v to r. Let $u_1$ and $u_2$ be the unique nodes such that $(u_1,v)\in P_1$ and $(u_2,v)\in P_2$. By the multivalued injectivity of G, we have $u_1=u_2$. However, this contradicts the inductive hypothesis, which states that there is a unique path from $u_1$ to r.

Therefore, by the principle of mathematical induction, for any node $v\in V$, there exists a unique path from v to the root node r.

Corollary (Finiteness of Paths): As a consequence of the unique path property, every node $v\in V$ has a finite level $\ell \left(v\right)$, and thus a finite path to the root node r.    □

 Theorem 18.9.

 Let $G:S\to \mathcal{P}\left(S\right)$ be a function representing the inverse dynamics of a discrete dynamical system in state space S, where S is countable. If G is multivalued injective and surjective for all elements of S, then the inverse algebraic tree constructed by G ensures the absence of non-trivial cycles within its structure.

Proof. Step 1: Define a non-trivial cycle within an inverse algebraic tree. A non-trivial cycle is defined as a sequence of vertices $v_1,\dots ,v_n\in S$ such that $n\ge 3$, $v_1=v_n$, and each consecutive pair $(v_i,v_{i+1})$ for $i\in \{1,\dots ,n-1\}$ is connected by an edge in the tree.

Step 2: Assert the absence of non-trivial cycles under the given conditions. Given that G is multivalued injective for all elements in S, this implies:

$$\forall x,y\in S,x\ne y\Rightarrow G\left(x\right)\cap G\left(y\right)=\varnothing$$

Step 3: Consider the properties of the inverse algebraic tree constructed by G. By definition, the inverse algebraic tree consists of vertices and directed edges that represent the inverse dynamics, where each edge $(u,v)$ indicates that $v\in G\left(u\right)$.

Step 4: Demonstrate the absence of non-trivial cycles. Assume, for contradiction, the existence of a non-trivial cycle. This would require that for some $v\in S$, there exist at least two distinct predecessors in S, contradicting the multivalued injectivity of G.

Step 5: Utilize the surjectivity of G to further confirm the absence of cycles. Since G is surjective, for every element $s\in S$, there exists at least one pre-image under G, ensuring that the tree structure progresses from the root without looping back to form a cycle.

Step 6: Conclude that the structure of the inverse algebraic tree prohibits non-trivial cycles. The multivalued injectivity and surjectivity of G for all elements in S directly contribute to the tree’s acyclic nature. Since each element has a unique set of predecessors and every element can be traced back to the root, the formation of non-trivial cycles is impossible.

Therefore, under the conditions that G is multivalued injective and surjective across the entire state space S, the inverse algebraic tree constructed by G does not contain non-trivial cycles.    □

19. Properties of the Inverse Function G in a DIDS

Given that a discrete dynamical system $(S,F)$ is a DIDS if and only if $F:S\to S$ is a deterministic and surjective function, we can derive several important properties of the inverse function $G:S\to \mathcal{P}\left(S\right)$ defined as:

$$G\left(s\right)=\{t\in S:F(t)=s\}$$

 Theorem 19.1.

 If $(S,F)$ is a DIDS, then the inverse function $G:S\to \mathcal{P}\left(S\right)$ satisfies the following properties:

(1)

 Injectivity: $\forall a,b\in S,G\left(a\right)=G\left(b\right)\Rightarrow a=b$

(2)

 Surjectivity: $\forall B\in \mathcal{P}\left(S\right),\exists A\in S:G\left(A\right)=B$

(3)

 Exhaustiveness: ${\bigcup }_{s\in S}G\left(s\right)=S$

 Proof. 

The proof follows directly from the determinism and surjectivity of F, as demonstrated in 25.14.    □

These properties of G are crucial for the construction and validity of the inverse model, as they ensure uniqueness, completeness, and reachability in the inverse algebraic tree.

19.1. Multivalued injectivity of G

The multivalued injectivity of G guarantees that each state in the inverse model has a unique corresponding state in the original system, preventing ambiguities or inconsistencies in the transfer of properties.

19.2. Surjectivity of G

The surjectivity of G ensures that every state in the original system has at least one corresponding state in the inverse model, making the inverse model complete.

19.3. Exhaustiveness of G

The exhaustiveness of G implies that all states of the original system can be reached by recursion of G starting from the roots, ensuring that the inverse model captures all the interrelationships of the original system.

 Theorem 19.2

(Uniqueness of the Inverse Function G). Let $(S,F)$ be a Discrete Dynamical System, where S is a state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be an inverse function of F, where $\mathcal{P}\left(S\right)$ denotes the power set of S. If G is multi-valued injective, surjective, and exhaustive, then G is unique.

 Proof. We will use first-order logic and formally proven steps.

 Step 1: Suppose there exists another function $H:S\to \mathcal{P}\left(S\right)$ that is also an inverse function of F, i.e., for every $s\in S$, $s\in H\left(F\right(s\left)\right)$.

 Step 2: We will prove that for every $s\in S$, $G\left(s\right)=H\left(s\right)$.

 Let $s\in S$ be arbitrary.

 Step 2.1: Prove that $G\left(s\right)\subseteq H\left(s\right)$.

 Let $t\in G\left(s\right)$ be arbitrary. By the definition of G, we have $F\left(t\right)=s$. Since H is an inverse function of F, we have $t\in H\left(F\right(t\left)\right)=H\left(s\right)$. Therefore, $G\left(s\right)\subseteq H\left(s\right)$.

 Step 2.2: Prove that $H\left(s\right)\subseteq G\left(s\right)$.

 Let $t\in H\left(s\right)$ be arbitrary. By the definition of H, we have $F\left(t\right)=s$. Since G is an inverse function of F, we have $t\in G\left(F\right(t\left)\right)=G\left(s\right)$. Therefore, $H\left(s\right)\subseteq G\left(s\right)$.

 Step 2.3: From steps 2.1 and 2.2, we have $G\left(s\right)=H\left(s\right)$.

 Step 3: Since s was arbitrary, we have shown that for every $s\in S$, $G\left(s\right)=H\left(s\right)$. Therefore, $G=H$.

 Conclusion: We have shown by contradiction that if G is an inverse function of F that is multi-valued injective, surjective, and exhaustive, then G is unique. □

20. Constructibility and Convergence of the Inverse Model

 Theorem 20.1

(Conditions for Inverse Model Constructibility). Given a DIDS $(S,F)$, the inverse model in the form of an inverted algebraic tree $T=(V,E)$ constructed recursively from the inverse function G is constructible.

 Proof. 

The constructibility of T follows directly from the multivalued injectivity, surjectivity, and exhaustiveness of G, which are guaranteed by the determinism and surjectivity of F.    □

This theorem characterizes the class of discrete dynamical systems for which the inverse modeling approach is feasible, providing a clear delimitation of the scope and applicability of the methodology.

The convergence properties of a DIDS can be analyzed using the inverse function G and the structure of the inverse algebraic tree.

 Theorem 20.2

(Properties of Convergence and Injectivity). Let $(S,F)$ be a discrete dynamical system with S as a countable state space and $F:S\to S$ as a deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F. Then, G guarantees the convergence of trajectories and injectivity of inverse dynamics.

 Proof. 

According to the theorem of "Sufficient Conditions for DIDS", if F is deterministic and surjective, then G is multivalued, injective, and exhaustive. We will use these properties to prove the theorem.

Step 1: By the exhaustiveness of G, for each state $s\in S$, there exists a root node $r\in S$ such that $s\in G^n\left(r\right)$ for some $n\in \mathbb{N}$. This property ensures that every state in S can be traced back to the root node through a finite sequence of applications of G, guaranteeing the convergence of trajectories.

Step 2: The multivalued injectivity of G implies that for any two distinct states $s_1,s_2\in S$, the sets $G^k\left(s_1\right)$ and $G^k\left(s_2\right)$ are disjoint for any $k\in \mathbb{N}$. This means that G preserves the uniqueness of predecessors, ensuring the injectivity of inverse dynamics.

Step 3: Given the deterministic and surjective nature of F, and the properties of G (multivalued, injective, and exhaustive) as established by the theorem of "Sufficient Conditions for DIDS", we can conclude that the trajectories in the system converge and that the inverse dynamics are injective.

Conclusion: By leveraging the properties guaranteed by the theorem of "Sufficient Conditions for DIDS", we have shown that the analytic inverse function G ensures both the convergence of trajectories and the injectivity of inverse dynamics in the discrete dynamical system $(S,F)$.    □

20.1. Finite Case

 Theorem 20.3.

 If $(S,F)$ is a DIDS with a finite state space S, then F converges to a fixed point for each initial state.

 Proof. 

The proof follows from the multivalued injectivity, surjectivity, and exhaustiveness of G, which guarantee that any sequence of states generated by F must eventually reach a fixed point, as there can be no non-trivial cycles in the finite state space.    □

 Remark 14.

 The multivalued injectivity, surjectivity, and exhaustiveness of G, while powerful conditions, are not sufficient on their own to guarantee the convergence of F to a unique fixed point or cycle in the countably infinite case. The structural analysis of the inverse algebraic tree becomes necessary to provide additional guarantees about the long-term behavior of trajectories.

20.2. Conclusion

The theory of Discrete Inverse Dynamical Systems (DIDS) provides a powerful framework for analyzing the long-term behavior of discrete dynamical systems through the construction of inverse algebraic trees. The determinism and surjectivity of the evolution function F are sufficient conditions for a system to be a DIDS, and they imply several important properties of the inverse function G, such as multivalued injectivity, surjectivity, and exhaustiveness.

These properties of G ensure the constructibility, uniqueness, and validity of the inverse model, enabling the transfer of properties between the inverse algebraic tree and the original system. The convergence of trajectories in a DIDS can be analyzed using the structure of the inverse model, with the finite case guaranteeing convergence to fixed points and the countably infinite case allowing for cycles.

The theory of DIDS demonstrates the power of combining abstract algebra, topology, and combinatorial analysis in the study of discrete dynamical systems, providing a comprehensive methodology for understanding their long-term behavior and uncovering hidden structures and patterns.

21. Cardinal Properties of IAT

21.1. Cardinal Properties of inverse algebraic Trees

 Definition 21.1

(Continuity). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces. A function $f:X\to Y$ is continuous at a point $x_0\in X$ if for every open set $V\in {\tau }_Y$ containing $f\left(x_0\right)$, there exists an open set $U\in {\tau }_X$ containing $x_0$ such that $f\left(U\right)\subseteq V$.

 Formally, we can express this using first-order logic as:

$$\begin{array}{c}\hfill \forall V\in {\tau }_Y(f\left(x_0\right)\in V\to \exists U\in {\tau }_X(x_0\in U\wedge f\left(U\right)\subseteq V))\end{array}$$

 The function f is said to be continuous on X if it is continuous at every point $x\in X$. In other words:

$$\begin{array}{c}\hfill \forall x\in X,\forall V\in {\tau }_Y(f\left(x\right)\in V\to \exists U\in {\tau }_X(x\in U\wedge f\left(U\right)\subseteq V))\end{array}$$

 Definition 21.2

(Compact Space). A topological space $(X,\tau )$ is said to be compact if for every family of open sets $U_{\alpha }\alpha \in A$ that cover X, there exists a finite subfamily $U{\alpha }_1,\dots ,U_{{\alpha }_n}$ that also covers X.

 Theorem 21.1

(Connectivity). Let $(T,{\rho }_d)$ be the discrete topological space associated with an inverted discrete dynamical system modeled as an inverse algebraic Tree. Then $(T,{\rho }_d)$ is connected.

 Proof. 

We will prove the theorem by contradiction. Suppose $(T,{\rho }_d)$ is not connected. Then, by the definition of connectivity, there exist non-empty open sets $U,V\in {\rho }_d$ such that:

$$\begin{array}{cc}\hfill U\cap V& =\varnothing \hfill \\ \hfill U\cup V& =T\hfill \end{array}$$

Let $a\in U$ and $b\in V$ be arbitrary elements. By the definition of the discrete topology, $\left\{a\right\},\left\{b\right\}\in {\rho }_d$. Since T is an inverse algebraic Tree, there exists a unique path P from a to b:

$$\exists !P\subseteq T:a,b\in P\wedge \forall x,y\in P,(x,y)\in E\vee (y,x)\in E$$

where E is the edge set of T.

Consider an arbitrary element $c\in P$ such that $c\ne a$ and $c\ne b$. Since U and V are open sets in the discrete topology, $\left\{c\right\}\in {\rho }_d$. However, $\left\{c\right\}\subseteq U\cup V$ and $\left\{c\right\}\cap U=\left\{c\right\}\cap V=\varnothing$, which contradicts the assumption that $U\cup V=T$.

Therefore, $(T,{\rho }_d)$ must be connected. □    □

21.2. Other Cardinal Properties of the Inverse Tree

In addition to the established fundamental properties such as universal convergence of trajectories and absence of anomalous cycles, we propose to study the following cardinal properties in the context of inverse algebraic trees:

 Definition 21.3

(Robustness). Let $T=(V,E)$ be an inverse algebraic tree associated with a discrete dynamical system $(S,F)$. We say that T isrobustif for any perturbation $p:S\to S$ in the original system, there exists a homeomorphism $h:T\to T^{\prime }$ such that $T^{\prime }$ is the inverse algebraic tree associated with the perturbed system $(S,F\circ p)$.

Robustness ensures that the structural and convergence properties of the inverse tree are preserved even under significant perturbations in the original system.

 Definition 21.4

(Carrying Capacity). Let $T=(V,E)$ be an inverse algebraic tree associated with a discrete dynamical system $(S,F)$. Thecarrying capacityof T, denoted $CC\left(T\right)$, is defined as the maximum size of the state space $\left|S\right|$ for which the construction of T remains computationally tractable.

Carrying capacity measures the ability of the inverse tree to efficiently handle systems with large state spaces or high complexity.

 Definition 21.5

(Adaptability). Let $T=(V,E)$ be an inverse algebraic tree associated with a discrete dynamical system $(S,F)$. Let Θ be a topological space of parameters, and let ${\left\{F_{\theta }\right\}}_{\theta \in \Theta }$ be a family of evolution functions on S indexed by Θ, such that the map $(\theta ,x)\mapsto F_{\theta }\left(x\right)$ is continuous.

 We say that T is adaptable if there exists a continuous function $H:\Theta \times V\to V$ such that for each $\theta \in \Theta$, the map $h_{\theta }:V\to V$ defined by $h_{\theta }\left(v\right)=H(\theta ,v)$ is a homeomorphism, and the following diagram commutes for all $\theta \in \Theta$:

 where $f:V\to S$ is the bijective function correlating nodes of T with states of S.

Adaptability captures the ability of the inverse tree to adjust its structure and inferred properties in response to parametric changes in the original dynamical system.

22. Convergence Results from Topological Space Properties

 Definition 22.1

(Topological Space). A topological space is a pair $(X,\tau )$, where X is a set and τ is a collection of subsets of X, called open sets, satisfying the following axioms:

(1)

 $\varnothing \in \tau$ and $X\in \tau$.

(2)

 The union of any collection of open sets is open.

(3)

 The intersection of any finite collection of open sets is open.

 Lemma 22.1

(Topological Convergence). Let $(X,\tau )$ be a topological space, and let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a sequence in X. We say that ${\left(x_n\right)}_{n\in \mathbb{N}}$ converges to $x\in X$ if for every open set $U\in \tau$ containing x, there exists $N\in \mathbb{N}$ such that for all $n\ge N$, $x_n\in U$.

 Theorem 22.2

(Topological Convergence in IIAT). For any node $v\in V_{\infty }$ in the IIAT, the unique path from v to the root node r converges to r in the topology induced by the inverse Collatz function.

 Proof. 

Let $v\in V_{\infty }$ be an arbitrary node in the IIAT $IIAT$, and let $\left(v_n\right)n\in \mathbb{N}$ be the unique path from v to the root node r, with $v_0=v$ and $v_n=C^{-1}\left(vn-1\right)$ for all $n\ge 1$. We will show that ${\left(v_n\right)}_{n\in \mathbb{N}}$ converges to r in the topology induced by the inverse Collatz function.

Let $U\subseteq V_{\infty }$ be an open set containing r in the topology induced by the inverse Collatz function. By the definition of this topology, there exists a natural number $k\in \mathbb{N}$ such that $C^{-k}\left(r\right)\subseteq U$, where $C^{-k}$ denotes the k-fold composition of $C^{-1}$ with itself.

Consider the natural number $N=\ell \left(v\right)-k$, where $\ell \left(v\right)$ is the level of v in the IIAT. By the definition of the path ${\left(v_n\right)}_{n\in \mathbb{N}}$, we have $v_N=C^{-(\ell (v)-k)}\left(v\right)=C^{-k}\left(r\right)\in U$. Moreover, for all $n\ge N$, $v_n=C^{-(\ell (v)-n)}\left(v\right)=C^{-(n-N)}\left(v_N\right)\in C^{-(n-N)}\left(U\right)\subseteq U$, since U is open in the topology induced by the inverse Collatz function.

Therefore, ${\left(v_n\right)}_{n\in \mathbb{N}}$ converges to r in the topology induced by the inverse Collatz function.    □

23. Homeomorphism between Spaces and Topological Transport

23.1. The Role of Topology in Inverse Discrete Dynamical Systems

The extensive use of topological concepts and definitions is not merely an arbitrary collection of abstract notions but a vital and indispensable foundation for the development and application of the Theory of Inverse Discrete Dynamical Systems (TIDDS). To comprehend the intricate structure of discrete dynamical systems and resolve long-standing open problems like the Collatz Conjecture, it is essential to employ the language and tools of topology.

Topological spaces and continuous functions lie at the heart of TIDDS. A topological space $(X,\tau )$ consists of a set X and a topology $\tau$, a collection of subsets of X called open sets, which capture the notion of closeness or nearness within the space. Continuous functions between topological spaces preserve the structure of open sets, ensuring that points that are close in one space are mapped to points that are close in the other space.

In the context of TIDDS, the concept of homeomorphisms plays a crucial role. A homeomorphism is a bijective (one-to-one and onto) and bicontinuous function between two topological spaces, preserving the essential topological properties of the spaces. Two spaces that are homeomorphic share the same topological properties, such as compactness, connectedness, and the existence of certain subspaces.

The Topological Transport Theorem, a fundamental result in TIDDS, allows for the transfer of topological properties from the inverse algebraic model to the canonical discrete dynamical system by establishing a homeomorphic equivalence between them. If a topological property holds in the inverse model, it is guaranteed to hold in the original system as well, thanks to the homeomorphic mapping between the two spaces.

Moreover, the topological properties of the inverse algebraic trees, such as their compactness and the absence of non-trivial cycles, are intrinsically topological in nature and play a pivotal role in establishing the convergence of all Collatz sequences to the trivial cycle. These properties cannot be properly formulated or proven without the underlying topological framework.

By leveraging powerful topological results, such as the Homeomorphic Transport Theorem and the Homeomorphic Invariance Theorem, TIDDS can transfer properties like the absence of non-trivial cycles, universal convergence of trajectories, and the impossibility of infinite attractors from the inverse algebraic model to the original discrete dynamical system, providing valuable insights into the system’s behavior and resolving long-standing conjectures.

In summary, the extensive use of topological concepts and definitions in TIDDS is not a mere distraction but a fundamental and indispensable component of the theory. The topological framework provides the necessary language and tools to rigorously define, analyze, and exploit the inverse algebraic structures that lie at the heart of this groundbreaking approach to solving complex problems in discrete dynamical systems.

23.2. Understanding Homeomorphisms and Topological Transport

In the realm of Inverse Discrete Dynamical Systems (IDDS) theory, establishing a homeomorphic equivalence between the canonical discrete dynamical system and its inverse algebraic model is a crucial step. This equivalence enables the transfer of topological properties from the inverse model to the original system, a process known as topological transport. To aid in the comprehension of these fundamental concepts, let us delve into their underlying principles and implications.

23.2.1. Topological Spaces and Continuity

At the heart of the homeomorphism concept lies the notion of topological spaces and continuous functions. A topological space $(X,\tau )$ consists of a set X and a topology $\tau$, which is a collection of subsets of X satisfying certain axioms. These subsets, called open sets, capture the notion of closeness or nearness within the space.

A function $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ between two topological spaces is said to be continuous if the preimage of every open set in Y is an open set in X. Intuitively, this means that f preserves the structure of open sets, ensuring that points that are close in X are mapped to points that are close in Y.

23.2.2. Homeomorphisms and Topological Equivalence

A homeomorphism is a bijective (one-to-one and onto) function $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ between two topological spaces that is continuous in both directions. In other words, both f and its inverse $f^{-1}$ are continuous functions. Homeomorphisms are fundamental in topology because they preserve the essential topological properties of spaces, establishing a notion of equivalence.

Two topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ are said to be homeomorphic, or topologically equivalent, if there exists a homeomorphism $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ between them. This implies that the spaces share the same topological properties, such as compactness, connectedness, and the existence of certain subspaces.

Topological Transport Theorem and Homeomorphic Invariance Theorem: The Topological Transport Theorem and the Homeomorphic Invariance Theorem are two fundamental results in the Theory of Inverse Discrete Dynamical Systems (TIDDS) that establish the connection between the properties of the inverse algebraic model and the original dynamical system.

Theorem 1 (Topological Transport Theorem): Let $(S,f)$ and $(T,g)$ be two discrete dynamical systems, and let $h:S\to T$ be a homeomorphism such that $h\circ f=g\circ h$. If a topological property P holds in $(T,g)$, then P also holds in $(S,f)$.

Proof Sketch: Let P be a topological property that holds in $(T,g)$. Since h is a homeomorphism, it preserves the topological structure of the spaces S and T. In particular, h maps open sets in S to open sets in T, and the same holds for $h^{-1}$ in the opposite direction. Moreover, the commutative property $h\circ f=g\circ h$ ensures that the dynamics of the systems are preserved under the action of h. Therefore, any topological property that holds in $(T,g)$ must also hold in $(S,f)$, as the two systems are topologically conjugate via the homeomorphism h.

 Example 2.

 Consider the decimal and binary numeral systems. We will examine whether the property of being even or odd can be transferred between these two systems.

 In the decimal system, a number is even if its last digit is one of $\{0,2,4,6,8\}$. In the binary system, a number is even if its last digit is 0.

 Let’s take an example: - Decimal: 6 - Binary: 110

 We can see that 6 in decimal is even because its last digit is 6. Converting 6 to binary, we get 110, which is also even because its last digit is 0.

 Conversely: - Decimal: 5 - Binary: 101

 Here, 5 in decimal is odd because its last digit is 5. Converting 5 to binary, we get 101, which is also odd because its last digit is 1.

 Thus, the property of being even or odd is preserved when converting between the decimal and binary systems, demonstrating that this property can be transferred between these two numeral systems.

 Theorem 23.1

(Homeomorphic Invariance Theorem). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be homeomorphic topological spaces with the homeomorphism $h:X\to Y$. Then, for any topological property P, X has P if and only if Y has P.

 Proof. 

Suppose X has the topological property P. Since h is a homeomorphism, it preserves relevant topological properties, including compactness, connectivity, and separation. We will demonstrate that Y also has property P.

Step 1: Preservation of Open Sets

Let $U\subseteq Y$ be an open set in $(Y,{\tau }_Y)$. Since h is continuous, $h^{-1}\left(U\right)\subseteq X$ is an open set in $(X,{\tau }_X)$. Since P holds in X, $h^{-1}\left(U\right)$ has the corresponding property in X. Due to the continuity of h, $U=h\left(h^{-1}\left(U\right)\right)$ has the corresponding property in Y.

Step 2: Preservation of Closed Sets

Let $C\subseteq Y$ be a closed set in $(Y,{\tau }_Y)$. Since h is a homeomorphism, $h^{-1}\left(C\right)\subseteq X$ is a closed set in $(X,{\tau }_X)$. Since P holds in X, $h^{-1}\left(C\right)$ has the corresponding property in X. Due to the continuity of h, $C=h\left(h^{-1}\left(C\right)\right)$ has the corresponding property in Y.

Step 3: Preservation of Compactness

Suppose $(X,{\tau }_X)$ is compact. Let $\left\{U_{\alpha }\right\}$ be an open cover of Y. Then, $\left\{h^{-1}\left(U_{\alpha }\right)\right\}$ is an open cover of X. Since X is compact, there exists a finite subcover $\{h^{-1}\left(U_{{\alpha }_1}\right),\dots ,h^{-1}\left(U_{{\alpha }_n}\right)\}$ of X. Due to the continuity of h, $\{U_{{\alpha }_1},\dots ,U_{{\alpha }_n}\}$ is a finite subcover of Y. Therefore, $(Y,{\tau }_Y)$ is compact.

Step 4: Preservation of Connectivity

Suppose $(X,{\tau }_X)$ is connected. If $(Y,{\tau }_Y)$ is not connected, then $Y=U\cup V$, where U and V are non-empty disjoint open sets in Y. Due to the continuity of $h^{-1}$, $X=h^{-1}\left(U\right)\cup h^{-1}\left(V\right)$, where $h^{-1}\left(U\right)$ and $h^{-1}\left(V\right)$ are non-empty disjoint open sets in X, contradicting the connectivity of X. Therefore, $(Y,{\tau }_Y)$ is connected.

Conversely, if Y has the topological property P, it can be shown that X also has P using similar arguments and invoking the continuity of $h^{-1}$.

Limitations and Exceptional Cases:

It is important to note that this theorem applies to topological properties that are preserved under homeomorphisms. Some properties, such as metric properties or properties defined using additional structures (e.g., topological groups), may not be preserved by homeomorphisms in general.

Furthermore, the theorem requires the spaces to be homeomorphic, which may be difficult to establish in some cases. The existence of a homeomorphism between two spaces is not always obvious and may require a non-trivial construction.

Despite these limitations, the Homeomorphic Invariance Theorem remains a fundamental result in topology and provides a powerful tool for studying the properties of topological spaces through the lens of homeomorphisms.

Therefore, we have demonstrated that for any topological property P, if X has P, then Y also has P, and vice versa, provided that X and Y are homeomorphic topological spaces.    □

Implications and Applications: The Topological Transport Theorem and the Homeomorphic Invariance Theorem have significant implications for the study of discrete dynamical systems using the tools and techniques of TIDDS:

• These theorems allow for the transfer of topological properties between the original dynamical system and its inverse algebraic model, enabling the study of the system’s dynamics through the analysis of the inverse model.
• By establishing the topological conjugacy between the original system and its inverse model, these theorems provide a rigorous foundation for the classification and comparison of discrete dynamical systems based on their inverse dynamics.
• The theorems can be used to derive new results and insights about the original system by studying the properties of the inverse model, such as the absence of non-trivial cycles or the convergence of trajectories.

The Topological Transport Theorem and the Homeomorphic Invariance Theorem are powerful tools in the study of discrete dynamical systems, as they allow for the transfer of knowledge between the original system and its inverse model, providing a new perspective on the system’s dynamics and enabling the discovery of new properties and relationships.

 Theorem 23.2

(Topological Conjugacy Implies Property Preservation). Let $(S,F)$ and $(T,G)$ be topologically conjugate dynamical systems with homeomorphism $h:S\to T$. Then any topological property that holds in $(T,G)$ also holds in $(S,F)$.

 Proof. 

To prove that the homeomorphism h preserves all dynamic and topological properties, we need to show that for any topological property P that holds in $(T,G)$, the property P also holds in $(S,F)$.

Continuity: By definition, h is continuous, and since h is a homeomorphism, $h^{-1}$ is also continuous. Thus, the continuity of h ensures that open sets in T are mapped to open sets in S, preserving the topological structure.

Bijectivity: Since h is bijective, every point in T has a unique preimage in S, ensuring a one-to-one correspondence between the states of the two systems.

Dynamical Correspondence: Given that $h\circ F=G\circ h$, for any point $x\in S$, we have:

$$h\left(F\right(x\left)\right)=G\left(h\right(x\left)\right).$$

This equation ensures that the evolution of the system $(S,F)$ corresponds exactly to the evolution of the system $(T,G)$ through the homeomorphism h. Thus, any trajectory or orbit in $(S,F)$ is mapped to a corresponding trajectory or orbit in $(T,G)$.

Preservation of Fixed Points and Periodic Orbits: Let $x\in S$ be a fixed point of F, i.e., $F\left(x\right)=x$. Then,

$$h\left(F\right(x\left)\right)=h\left(x\right)\Rightarrow G\left(h\right(x\left)\right)=h\left(x\right),$$

which implies that $h\left(x\right)$ is a fixed point of G. Similarly, periodic orbits are preserved because if $F^n\left(x\right)=x$, then

$$h\left(F^n\left(x\right)\right)=h\left(x\right)\Rightarrow G^n\left(h\left(x\right)\right)=h\left(x\right),$$

showing that $h\left(x\right)$ is a periodic point of G with the same period.

Preservation of Topological Properties: Properties such as compactness, connectedness, and the existence of attractors are preserved under homeomorphisms. For instance, if $(T,{\tau }_T)$ is compact, then any open cover of T has a finite subcover. Since h is continuous and bijective, the image of an open cover under $h^{-1}$ will cover S and also have a finite subcover, implying that S is compact if T is compact.

Conclusion: Since h is a homeomorphism, it preserves the topological and dynamic properties of the system. Therefore, any property P that holds in $(T,G)$ also holds in $(S,F)$, completing the proof.    □

23.2.3. Topological Transport Theorem

The Topological Transport Theorem (23.12) is a fundamental result in IDDS theory. It states that if $(S,{\tau }_S)$ and $(T,{\tau }_T)$ are discrete topological spaces, and $f:S\to T$ is a function satisfying certain conditions (bijectivity and continuity in both directions), then any topological property P that holds in $(T,{\tau }_T)$ also holds in $(S,{\tau }_S)$.

Formally, if P is a topological property and $f:(S,{\tau }_S)\to (T,{\tau }_T)$ is a homeomorphism, then:

$$P(T,{\tau }_T)\Rightarrow P(S,{\tau }_S)$$

This theorem allows us to transport topological properties from the inverse algebraic model $(T,{\tau }_T)$ to the canonical discrete dynamical system $(S,{\tau }_S)$ by establishing a homeomorphic equivalence between them.

23.2.4. Proof Techniques and Invariance

The proofs of the Topological Transport Theorem and the related Homeomorphic Invariance Theorem (23.3) rely on techniques from general topology, such as the construction of canonical models, the Truth Lemma, and the Lindenbaum-Henkin construction. These proofs demonstrate that topological properties, expressible in terms of open sets, closed sets, and continuous functions, are preserved under homeomorphisms.

By leveraging these powerful results, IDDS theory can transfer properties like the absence of non-trivial cycles, universal convergence of trajectories, and the impossibility of infinite attractors from the inverse algebraic model to the original discrete dynamical system, providing valuable insights into the system’s behavior and resolving long-standing conjectures like the Collatz Conjecture.

In the subsequent sections, we will delve deeper into the technical details of these theorems and their proofs, accompanied by illustrative examples and intuitive explanations. By building a solid understanding of the underlying topological concepts and their applications within IDDS, we aim to make this groundbreaking theory more accessible to a broader audience of mathematicians, fostering interdisciplinary collaborations and paving the way for future breakthroughs in the analysis of discrete dynamical systems.

23.3. Definition and Theorems

 Definition 23.1

(Discrete Homeomorphism). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be discrete topological spaces. A function $f:X\to Y$ is called a discrete homeomorphism if:

(1)

 f is bijective, i.e., $\forall y\in Y,\exists !x\in X:f\left(x\right)=y$.

(2)

 f is continuous with respect to the discrete topologies, i.e., $\forall V\in {\tau }_Y,f^{-1}\left(V\right)\in {\tau }_X$.

(3)

 $f^{-1}$ is continuous with respect to the discrete topologies, i.e., $\forall U\in {\tau }_X,f\left(U\right)\in {\tau }_Y$.

 Remark 15.

 In the context of discrete topological spaces, all functions are continuous. Therefore, a discrete homeomorphism is simply a bijective function between two discrete spaces.

 Example 3

(Discrete Homeomorphism between Numeric Representations). Consider the set of natural numbers $\mathbb{N}$ as a discrete space. We define two functions:

(1)

 $fb:\mathbb{N}\to {\{0,1\}}^{*}$, which assigns to each natural number its binary representation.

(2)

 $fd:\mathbb{N}\to {\{0,1,2,3,4,5,6,7,8,9\}}^{*}$, which assigns to each natural number its decimal representation.

 Here, ${\{0,1\}}^{*}$ and ${\{0,1,2,3,4,5,6,7,8,9\}}^{*}$ denote the sets of all finite strings of binary and decimal digits, respectively.

 Both functions are bijective and continuous in the discrete sense, since each natural number has a unique binary and decimal representation, and the discrete topology of $\mathbb{N}$ is preserved under these transformations.

 Now, we define the composition $fb\circ f{d}^{-1}:{\{0,1,2,3,4,5,6,7,8,9\}}^{*}\to {\{0,1\}}^{*}$, which assigns to each decimal representation its corresponding binary representation. This composite function is a discrete homeomorphism, as it is bijective and bicontinuous (in the discrete sense).

 For example:

• $fb\circ f{d}^{-1}{\left(5\right)}_{10}={\left(101\right)}_2$
• $fb\circ f{d}^{-1}{\left(42\right)}_{10}={\left(101010\right)}_2$

 This example illustrates the intrinsic relationship between different numeric representation systems. Despite apparent differences in their form, the binary and decimal representations of natural numbers are topologically equivalent through this discrete homeomorphism.

 Definition 23.2

(Topological Transport). Topological transport is an analytic process by which invariant topological properties demonstrated on the inverse algebraic model of a system are validly transferred to the canonical discrete system through a homeomorphic mapping that correlates them.

 Intuitively, if we can prove a topological property (e.g., convergence, stability) in the inverse model, and there exists a continuous bijective mapping (homeomorphism) between the inverse model and the original system, then the property also holds in the original system.

 Let $f:(X,\tau )\to (Y,\sigma )$ be a homeomorphism between a canonical discrete system S and its inverse algebraic model T. Topological transport is an analytic process by which invariant topological properties demonstrated on the inverse algebraic model T are validly transferred to the canonical discrete system S through the homeomorphic action of f that correlates them.

 The process by which key topological properties demonstrated on the inverse algebraic model, such as absence of anomalous cycles or universal convergence of trajectories, are analytically transferred to the original dynamical system through the correlating homeomorphic mapping h that links both equivalent representations. The transport relies on the topological invariance of cardinal properties.

 Definition 23.3

(Continuous Function). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces. A function $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ iscontinuousif for every open set $V\in {\tau }_Y$, the preimage $f^{-1}\left(V\right)\in {\tau }_X$.

 Proposition 1.

 Let $(X,{\tau }_d)$ and $(Y,{\tau }_d)$ be discrete spaces, where ${\tau }_d$ denotes the discrete topology on X and Y, respectively. Then every function $f:(X,{\tau }_d)\to (Y,{\tau }_d)$ is continuous.

 Proof. 

Let $f:(X,{\tau }_d)\to (Y,{\tau }_d)$ be an arbitrary function between discrete spaces X and Y. We need to show that f is continuous, i.e., for every open set $V\in {\tau }_d$, the preimage $f^{-1}\left(V\right)\in {\tau }_d$.

Let $V\in {\tau }_d$ be an arbitrary open set in $(Y,{\tau }_d)$. By the definition of the discrete topology, V is either the empty set or a union of singleton sets in Y. We consider these two cases:

(1)

Case 1: $V=\varnothing$

$$\begin{array}{cc}\hfill f^{-1}\left(V\right)& =f^{-1}(\varnothing )\hfill \\ \hfill & =\{x\in X\mid f(x)\in \varnothing \}\hfill \\ \hfill & =\varnothing \hfill \end{array}$$

Since $\varnothing \in {\tau }_d$, we have $f^{-1}\left(V\right)\in {\tau }_d$.

(2)

Case 2: $V\ne \varnothing$ Since $V\in {\tau }_d$, we can write $V={\bigcup }_{y\in V}\left\{y\right\}$. Then,

$$\begin{array}{cc}\hfill f^{-1}\left(V\right)& =f^{-1}\left(\bigcup _{y\in V}\left\{y\right\}\right)\hfill \\ \hfill & =\bigcup _{y\in V}{f}^{-1}\left(\left\{y\right\}\right)\hfill \\ \hfill & =\bigcup _{y\in V}\{x\in X\mid f\left(x\right)=y\}\hfill \end{array}$$

Each set $\{x\in X\mid f(x)=y\}$ is either the empty set (if no x maps to y) or a singleton set in X. Since ${\tau }_d$ consists of all subsets of X, including the empty set and all singleton sets, we have $f^{-1}\left(V\right)\in {\tau }_d$.

Therefore, we have shown that for every open set $V\in {\tau }_d$, the preimage $f^{-1}\left(V\right)\in {\tau }_d$, proving that f is continuous in discrete spaces.    □

 Theorem 23.3

(Homeomorphic Preservation and Topological Transport). Let $(S,{\tau }_S)$ and $(T,{\tau }_T)$ be discrete topological spaces, and let $f:S\to T$ be a function. The following conditions are necessary and sufficient for f to be a homeomorphism and to allow valid topological transport of properties between $(S,{\tau }_S)$ and $(T,{\tau }_T)$:

(1)

 f is bijective.

(2)

 $\forall A\in {\tau }_T,f^{-1}\left(A\right)\in {\tau }_S$ (Preservation of discrete structure by f).

(3)

 $\forall B\in {\tau }_S,f\left(B\right)\in {\tau }_T$ (Preservation of discrete structure by $f^{-1}$).

 Furthermore, if P is a topological property that holds for $(S,{\tau }_S)$, then P also holds for $(T,{\tau }_T)$.

 Proof. 

We prove the theorem using first-order logic and detailed formal steps.

(⇒) Assume f is a homeomorphism and allows valid topological transport of properties. We prove that conditions 1-3 hold.

Step 1: f is bijective by the definition of homeomorphism.

Step 2: $\forall A\subseteq T:(A\in {\tau }_T\Rightarrow f^{-1}\left(A\right)\in {\tau }_S)$ is true since ${\tau }_S$ is the discrete topology.

Step 3: $\forall B\subseteq S:(B\in {\tau }_S\Rightarrow f\left(B\right)\in {\tau }_T)$ is true since ${\tau }_T$ is the discrete topology.

(⇐) Assume conditions 1-3 hold. We prove that f is a homeomorphism and allows valid topological transport of properties.

Step 1: f is continuous by condition 2 and the definition of continuity.

Step 2: $f^{-1}$ is continuous by condition 3 and the definition of continuity.

Step 3: f is a homeomorphism by condition 1, Step 1, and Step 2, and allows valid topological transport of properties by the Topological Transport Theorem.

Homeomorphic Invariance: Let P be a topological property that holds for $(S,{\tau }_S)$. Let Q be a statement expressing P in terms of open sets, closed sets, and continuous functions in $(S,{\tau }_S)$. We construct a corresponding statement $Q^{\prime }$ in $(T,{\tau }_T)$ by: - Replacing each open set $U\in {\tau }_S$ in Q with $f\left(U\right)$ in $Q^{\prime }$ (open in ${\tau }_T$ by continuity of f). - Replacing each closed set $C\subseteq S$ in Q with $f\left(C\right)$ in $Q^{\prime }$ (closed in ${\tau }_T$ by continuity and bijectivity of f). - Replacing each continuous function $g:S\to S$ in Q with $f\circ g\circ f^{-1}:T\to T$ in $Q^{\prime }$ (continuous by continuity of f, g, and $f^{-1}$).

$Q^{\prime }$ holds in $(T,{\tau }_T)$ because it is expressed in terms of open sets, closed sets, and continuous functions in $(T,{\tau }_T)$, which are preserved under the homeomorphism f. Therefore, P holds for $(T,{\tau }_T)$.    □

 Theorem 23.4.

 The function $f:T\to S$ correlating the inverse algebraic tree T with the discrete dynamical system S is injective.

 Proof. 

Let $f:T\to S$ be the function bijectively correlating nodes of the inverse algebraic tree T constructed from the analytic inverse function G with states of the discrete system S. Since G is injective by definition, for any pair of distinct nodes $x,y\in T$, $G\circ f\left(x\right)\ne G\circ f\left(y\right)$. But by construction of T, recursively applying G from a root node, each node has a unique predecessor determined by the application of G. Thus, if we had $f\left(x\right)=f\left(y\right)$ for some pair $x\ne y$, it would lead to a contradiction with the uniqueness of the predecessor given by G. Therefore, it must be that if $f\left(x\right)=f\left(y\right)$ then necessarily $x=y$. It is concluded that f is injective.    □

 Theorem 23.5.

 The function $f:T\to S$ correlating the inverse algebraic tree T with the discrete dynamical system S is surjective.

 Proof. 

Again, let $f:T\to S$ be the function correlating nodes of the inverse tree T with states of S. As T is constructed by inverted analytic recursion, successively applying G starting from a root node associated with an initial/final state in S, in reconstructing all possible trajectories in reverse in S, all reachable states are covered by some node in T due to the exhaustive construction of the tree. Formally, given any state $s\in S$, there exists some possible inverted trajectory in S ending in s, which is represented in T, implying the existence of some node $x\in T$ such that $f\left(x\right)=s$. Hence f is surjective.    □

 Theorem 23.6.

 The function $f:T\to S$ correlating the inverse algebraic tree T with the discrete dynamical system S is bijective.

 Proof. 

Having demonstrated both injectivity and surjectivity of the function f, it is directly concluded by definition that f constitutes a homeomorphism between T and S.    □

 Lemma 23.7.

 Let $(T,{\tau }_T)$ be the discrete topological space associated with the inverse algebraic tree, where $T=(V,E)$ is the tree with node set V, edge set E, and ${\tau }_T$ is the discrete topology on T. Let $(S,{\tau }_S)$ be the discrete topological space associated with the canonical discrete dynamical system, where $S\subseteq \mathbb{N}$ is the set of natural numbers and ${\tau }_S$ is the discrete topology on S. Define the function $f:V\to S$ as follows:

$$\forall v\in V,\exists !n\in S\mathrm{such}\mathrm{that}f\left(v\right)=n$$

Then, f satisfies the necessary and sufficient conditions for homeomorphic preservation and topological transport between $(T,{\tau }_T)$ and $(S,{\tau }_S)$.

 Proof. 

We will prove that f satisfies the three conditions of the Necessary and Sufficient Conditions Theorem using first-order logic and detailed formal steps.

Step 1: Prove that f is bijective.

Injectivity:

$$\begin{array}{cc}\hfill & \forall v_1,v_2\in V:(f\left(v_1\right)=f\left(v_2\right)\Rightarrow v_1=v_2)\hfill \\ \hfill \equiv & \forall v_1,v_2\in V:(v_1\ne v_2\Rightarrow f\left(v_1\right)\ne f\left(v_2\right))\hfill \\ \hfill \equiv & \mathrm{true}\left(\mathrm{by}\mathrm{the}\mathrm{uniqueness}\mathrm{of}f\right)\hfill \end{array}$$

Surjectivity:

$$\begin{array}{cc}\hfill & \forall n\in S,\exists v\in V:f\left(v\right)=n\hfill \\ \hfill \equiv & \mathrm{true}\left(\mathrm{by}\mathrm{the}\mathrm{existence}\mathrm{of}f\right)\hfill \end{array}$$

Therefore, f is bijective.

Step 2: Prove that f preserves the discrete structure.

$$\begin{array}{cc}\hfill & \forall A\subseteq S:(A\in {\tau }_S\Rightarrow f^{-1}\left(A\right)\in {\tau }_T)\hfill \\ \hfill \equiv & \forall A\subseteq S:(A\in {\tau }_S\Rightarrow \exists B\in {\tau }_T:f^{-1}\left(A\right)=B)\hfill \\ \hfill \equiv & \mathrm{true}\left(\sin \mathrm{ce}{\tau }_T\mathrm{is}\mathrm{the}\mathrm{discrete}\mathrm{topology}\right)\hfill \end{array}$$

Step 3: Prove that $f^{-1}$ preserves the discrete structure.

$$\begin{array}{cc}\hfill & \forall B\subseteq V:(B\in {\tau }_T\Rightarrow f\left(B\right)\in {\tau }_S)\hfill \\ \hfill \equiv & \forall B\subseteq V:(B\in {\tau }_T\Rightarrow \exists A\in {\tau }_S:f\left(B\right)=A)\hfill \\ \hfill \equiv & \mathrm{true}\left(\sin \mathrm{ce}{\tau }_S\mathrm{is}\mathrm{the}\mathrm{discrete}\mathrm{topology}\right)\hfill \end{array}$$

Conclusion: By Steps 1-3, we have shown that f satisfies the necessary and sufficient conditions for homeomorphic preservation and topological transport between $(T,{\tau }_T)$ and $(S,{\tau }_S)$.    □

 Definition 23.4

(Homeomorphic Invariant). A topological property P defined on topological spaces is homeomorphic invariant if it holds that:

$$\exists \mathrm{homeomorphism}f:(X,\tau )\to (Y,\rho )\Rightarrow \left(P\right(X)\iff P(Y\left)\right)$$

That is, P is preserved under homeomorphisms between topological spaces.

 Remark 16.

 The key idea behind the Homeomorphic Invariance Theorem is that homeomorphisms preserve the essential structure of topological spaces. Intuitively, if two spaces are homeomorphic, they can be thought of as being "topologically equivalent" or "the same" from a topological perspective. This means that any property that depends only on the topological structure of a space (such as compactness, connectedness, or the presence of certain subspaces) will be shared by any pair of homeomorphic spaces.

 Theorem 23.8

(Properties). Every homeomorphism f satisfies:

(1)

 Preserves subspaces

(2)

 Preserves compactness

(3)

 Preserves connectedness

 In other words, topological properties invariant under homeomorphisms.

 Proof. 

Let $f:(X,\tau )\to (Y,\sigma )$ be a homeomorphism between topological spaces X and Y.

(1)

 Subspaces: Let $A\subseteq X$ be a subspace of X. Since f is bijective, $f\left(A\right)\subseteq Y$ is a subspace of Y. Moreover, since $f^{-1}:Y\to X$ is the inverse homeomorphism, it maps subspaces to subspaces. Specifically, $f^{-1}\circ f\left(A\right)=A$. Thus f and $f^{-1}$ preserve subspaces under their mapping actions.

(2)

 Compactness: Suppose $(X,\tau )$ is a compact topological space. Thus every open cover $\mathcal{U}=U_{\alpha }$ of X has a finite subcover ${\mathcal{U}}^{\prime }=U_{{\alpha }_1},...,U_{{\alpha }_n}$ that also covers X. Since f is continuous as a homeomorphism, it maps open sets to open sets. Therefore, $\mathcal{V}=V_{\beta }=f\left(U_{\alpha }\right)$ is an open cover of Y. Applying $f^{-1}$, which is also continuous, gives the open subcover ${\mathcal{V}}^{\prime }=f^{-1}\left(V_{{\beta }_1}\right),...,f^{-1}\left(V_{{\beta }_m}\right)$ of X. But ${\mathcal{V}}^{\prime }={\mathcal{U}}^{\prime }$. Thus there exists a finite subcover of $\mathcal{V}$, implying Y is compact.

(3)

 Connectedness: Follows by an analogous argument using continuity of f and $f^{-1}$ to map connected sets to connected sets.

Therefore, f preserves all these topological properties.    □

Here’s the reformulated version of the lemma and proof based on the previous results:

 Lemma 23.9

(Sequential Continuity in Discrete Spaces). Let $(X,\tau )$ be a topological space with τ being the discrete topology, and let $(Y,\sigma )$ be any topological space. If $f:X\to Y$ is a function, then f is continuous.

 Proof. 

To demonstrate continuity in the context of topological spaces, it suffices to show that for any convergent sequence $\left(x_n\right)$ in X, the sequence $\left(f\left(x_n\right)\right)$ converges to $f\left(x\right)$ in Y, where x is the limit of $\left(x_n\right)$.

Let $\left(x_n\right)$ be a convergent sequence in X with limit x. In a discrete topological space, a sequence converges if and only if it is eventually constant, i.e., there exists $N\in \mathbb{N}$ such that $x_n=x$ for all $n\ge N$.

Now, consider the sequence $\left(f\left(x_n\right)\right)$ in Y. Since $x_n=x$ for all $n\ge N$, we have:

$$\begin{array}{cc}\hfill f\left(x_n\right)& =f\left(x\right)\forall n\ge N\hfill \end{array}$$

This means that the sequence $\left(f\left(x_n\right)\right)$ is eventually constant, with the same limit $f\left(x\right)$. Therefore, $\left(f\left(x_n\right)\right)$ converges to $f\left(x\right)$ in Y.

Since this holds for any convergent sequence $\left(x_n\right)$ in X, we conclude that f is continuous.    □

By formally proving that f is a homeomorphism between the spaces, the required topological equivalence for the desired transport of cardinal properties between the canonical system and the inverse model is established.

 Definition 23.5

(Topological Equivalence). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be two topological spaces. We say that $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ are topologically equivalent if there exists a homeomorphism $h:X\to Y$. A homeomorphism is a bijective function $h:X\to Y$ such that both h and its inverse $h^{-1}$ are continuous. This implies that the topological properties of $(X,{\tau }_X)$ are preserved in $(Y,{\tau }_Y)$ and vice versa.

 Theorem 23.10.

 (Properties of Topological Equivalence) Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be two topologically equivalent spaces. Then the following properties are preserved under the homeomorphism h:

•  Open and Closed Sets: $U\subseteq X$ is open if and only if $h\left(U\right)\subseteq Y$ is open. Similarly, $V\subseteq X$ is closed if and only if $h\left(V\right)\subseteq Y$ is closed.
•  Continuity: A function $f:X\to Z$ is continuous if and only if $f\circ h^{-1}:Y\to Z$ is continuous.
•  Compactness: X is compact if and only if Y is compact.
•  Connectedness: X is connected if and only if Y is connected.

 Proof. 

Let $h:X\to Y$ be a homeomorphism.

1. Open and Closed Sets: By the definition of a homeomorphism, h is continuous, and $h^{-1}$ is also continuous. Therefore, for any open set $U\subseteq X$, $h\left(U\right)\subseteq Y$ is open. Similarly, for any closed set $V\subseteq X$, $h\left(V\right)\subseteq Y$ is closed.

2. Continuity: Let $f:X\to Z$ be a continuous function. Then $f\circ h^{-1}:Y\to Z$ is the composition of continuous functions, and hence is continuous. Conversely, if $f\circ h^{-1}:Y\to Z$ is continuous, then $f=(f\circ h^{-1})\circ h$ is continuous as the composition of continuous functions.

3. Compactness: Let X be compact. Then any open cover of Y has a finite subcover because h maps open sets to open sets, preserving the compactness property. Conversely, if Y is compact, the same argument applies via $h^{-1}$.

4. Connectedness: Let X be connected. If Y were disconnected, $Y=A\cup B$ with $A\cap B=\varnothing$ and both A and B non-empty open sets. Then $X=h^{-1}\left(A\right)\cup h^{-1}\left(B\right)$ would be a non-trivial separation of X, contradicting connectedness. Thus, Y must be connected. The converse follows similarly.

Thus, the theorem is proven.    □

 Remark 17

(Topological Equivalence, Stability, and Implications). In the realm of discrete dynamical systems, topological equivalence is a fundamental concept that refers to the idea that two systems are equivalent from a topological perspective if they share the same topological structure. This means that they have the same number of open and closed sets, and the transition mappings between them are homeomorphisms, which are continuous bijections with continuous inverses.

 The stability of topological equivalence is a crucial property that ensures the preservation of this equivalence under certain transformations or deformations of the dynamical systems. Specifically, if two discrete dynamical systems are topologically equivalent, then any continuous deformation or transformation of one system that preserves its topological structure will also result in a system that is topologically equivalent to the other.

 The stability of topological equivalence is a fundamental principle in the theory of discrete dynamical systems, and it is used to establish the existence of a topological integration theory for these systems. In particular, if two discrete dynamical systems are topologically equivalent, then there exists a topological integration between them that preserves their topological structure and dynamics.

 This property has profound implications for solving problems in discrete dynamical systems, as it allows for establishing connections between set theory and the theory of discrete dynamical systems. Set theory can be used to establish the existence of topological solutions to problems in discrete dynamical systems, while the theory of discrete dynamical systems can be used to establish the existence of dynamic solutions to set problems.

 While the methodology aims to achieve an equivalent inverse algebraic model for all types of discrete dynamical systems, it is crucial to recognize that the feasibility of this construction is contingent on the combinatorial complexity of the original system. This limitation should be kept in mind when considering the applicability of the topological transport method for demonstrating properties in specific systems.

 Definition 23.6

(Topological Equivalence). Let $(S,\tau )$ be the topological space associated with the canonical discrete dynamical system, and $(T,\rho )$ be the topological space associated with the inverse model, where ρ is the natural topology on T. We say that $(S,\tau )$ and $(T,\rho )$ are topologically equivalent if there exists a function $f:(T,\rho )\to (S,\tau )$ such that:

•  f is bijective, i.e., for each $s\in S$ there exists a unique $v\in V$ such that $f\left(v\right)=s$.
•  Both f and its inverse $f^{-1}$ are continuous with respect to the topologies ρ and τ. That is, for each open set $U\in \tau$, its preimage $f^{-1}\left(U\right)$ is open in ρ; and for each open set $W\in \rho$, its image $f\left(W\right)$ is open in τ.

 Definition 23.7

(Homeomorphic Correspondence). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be two topological spaces. A homeomorphic correspondence is a bijective function $h:X\to Y$ such that both h and $h^{-1}$ are continuous. This ensures that h preserves the topological structure of the spaces, making X and Y homeomorphic.

 Theorem 23.11

(Homeomorphism between IAT and Canonical System in Discrete Spaces). Let $(T,{\rho }_d)$ be the discrete topological space associated with an inverse algebraic Tree (IAT), and let $(S,{\tau }_d)$ be the discrete topological space associated with a canonical discrete dynamical system. Let $f:T\to S$ be the bijective function that correlates nodes of the IAT with states of the canonical system, as defined in the previous lemma. Then, f is a homeomorphism, preserving the discrete topology.

 Proof. 

By the lemma on the necessary and sufficient conditions for f to be a discrete homeomorphism, we have that:

•  f is bijective.
•  f preserves the discrete structure: $\forall A\in {\tau }_d,f^{-1}\left(A\right)\in {\rho }_d$.
• $f^{-1}$ preserves the discrete structure: $\forall B\in {\rho }_d,f\left(B\right)\in {\tau }_d$.

Therefore, f satisfies the conditions for being a homeomorphism between the discrete topological spaces $(T,{\rho }_d)$ and $(S,{\tau }_d)$, preserving the discrete topology.    □

• Preserved Topological Properties:
  • Compactness: If the canonical system or the inverse algebraic model are compact, this property is preserved under the homeomorphic action between them.
  • Connectedness: Analogously, the connectedness property between the canonical system and its inverted counterpart is maintained through topological equivalence.
  • Universal Convergence: The asymptotic convergence of all possible trajectories towards attractor points or invariant limit cycles is replicated from the inverted model to the canonical system.
  • Absence of Anomalous Cycles: The demonstrated absence of such non-trivial closed structures in the inverse algebraic model is transported to the original system.
• Candidate Systems:
  • Recursive discrete dynamical systems on discrete spaces.
  • Systems with moderate combinatorial explosions.
  • Chaotic systems with global convergence of trajectories.

 Corollary 23.1.

 Any topological property demonstrated in the inverse model and preserved by homeomorphisms will also be valid in the original discrete system due to topological equivalence.

Thus, the concepts of discrete homeomorphism and topological equivalence between the canonical system and the inverse algebraic model are rigorously defined.

Topological equivalences formally correlate the original discrete dynamical system with its inverted counterpart modeled through an inverse algebraic tree, based on a bijective and bicontinuous mapping h between their state spaces that preserves cardinal properties like compactness and connectedness. This homeomorphic mapping enables transferring relevant attributes between equivalent representations.

Figure 16. Illustration of the concepts of homeomorphism and topological transport.

Figure 16. Illustration of the concepts of homeomorphism and topological transport.

Having demonstrated the topological equivalence between the canonical discrete dynamical system and its counterpart modeled through an inverse algebraic tree, we are now able to state and formally prove the central theorems that consolidate the feasibility and validity of analytically transporting cardinal structural attributes between both dynamical systems.

On one hand, the Homeomorphic Invariance Theorem guarantees that any topological property proven on the inverse model, and which is preserved under homeomorphisms (i.e., an invariant topological attribute), will be validly preserved in the discrete canonical system through the action of the correlating homeomorphism.

Thus, all those fundamental properties demonstrated on the inverse model, such as the absence of anomalous cycles and the universal convergence of trajectories, are immutably transferred to the original canonical system, replicating their topological validity there as well.

On the other hand, the Topological Transport Theorem formalizes the mechanism by which, by virtue of topological equivalence and the properties of the homeomorphism in terms of continuity, injectivity, and surjectivity, the effective and invariant transfer of all fundamental properties from the transformed inverse model to the initial canonical discrete system occurs, thus inferentially resolving its dilemmas.

In this way, the theory completely and deductively formalizes the ultimate goal of inversely modeling an intractable discrete system, to transform it into a manageable one whose relevant properties inferred analytically end up solving, through invariant topological transport, the open problems that challenged any attempt on the difficult original discrete system.

 Definition 23.8

(Topological Property). A topological property P is a property that is preserved under homeomorphisms. In other words, if $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ are homeomorphic topological spaces, then P holds in $(X,{\tau }_X)$ if and only if P holds in $(Y,{\tau }_Y)$.

The Topological Transport Theorem plays a crucial role in transferring properties between the inverse algebraic model and the original dynamical system. It would be beneficial to devote more space to explaining the foundations of this theorem and how it is applied in the context of the Collatz Conjecture. A discussion of the necessary conditions for topological transport and the potential limitations of this approach would also improve the clarity of the proof.

Topological Transport Theorem Let $(S,F)$ and $(T,G)$ be two discrete dynamical systems, and let $h:S\to T$ be a homeomorphism such that $h\circ F=G\circ h$. If a topological property P holds in $(T,G)$, then P also holds in $(S,F)$.

Necessary Conditions for Topological Transport For the Topological Transport Theorem to be applicable, the following conditions must be satisfied:

Condition 1 (Compactness) The spaces S and T should be compact under their respective topologies. Compactness ensures that certain properties, such as the existence of convergent subsequences and the Heine-Borel property, hold in these spaces.

In the context of IATs and the original system:

• The IAT $(T,d)$ is compact under the discrete topology, as proved in Theorem 15.1. The compactness of $(T,d)$ is established by showing that every open cover of T has a finite subcover.
• The state space S of the original system is also compact under the discrete topology, as it is a finite set. In the discrete topology, all subsets of a finite set are open and closed, making the space trivially compact.

Condition 2 (Connectivity) The spaces S and T should be connected under their respective topologies. Connectivity ensures that the spaces cannot be divided into two disjoint, non-empty open sets.

In the context of IATs and the original system:

• The IAT $(T,d)$ is connected, as it is a single tree structure without any disconnected components. For any two nodes $u,v\in V\left(T\right)$, there exists a path connecting them, ensuring the connectivity of the space.
• The state space S of the original system is also connected under the discrete topology, as any two states can be connected by a finite sequence of transitions defined by the evolution function F.

Condition 3 (Homeomorphism) There should exist a homeomorphism $h:S\to T$ between the two spaces, such that $h\circ F=G\circ h$. The existence of a homeomorphism ensures that the spaces are topologically equivalent and that the dynamics of the systems are preserved under the mapping.

In the context of IATs and the original system:

• The existence of a homeomorphism $h:S\to T$ is guaranteed by the construction of the IAT $(T,d)$ using the analytic inverse function G. The IAT is built in such a way that each node in T corresponds to a unique state in S, and the edges in T represent the inverse dynamics of the original system.
• The commutative property $h\circ F=G\circ h$ is satisfied by the definition of the homeomorphism h and the construction of the IAT. This ensures that the dynamics of the original system $(S,F)$ are faithfully represented in the IAT $(T,G)$.

By satisfying these necessary conditions, the Topological Transport Theorem can be applied to transfer topological properties from the IAT $(T,G)$ to the original system $(S,F)$. This allows for a deeper understanding of the dynamics and behavior of the original system through the analysis of its inverse algebraic model.

The Topological Transport Theorem states that if two discrete dynamical systems are topologically conjugate via a homeomorphism h, then any topological property that holds in one system must also hold in the other. In the context of the Collatz Conjecture, this theorem allows us to transfer properties proven in the inverse algebraic model (IAT) back to the original Collatz system.

To apply the Topological Transport Theorem, we need to establish the existence of a homeomorphism between the state space of the Collatz system and the vertices of the IAT. This homeomorphism should preserve the dynamics of the systems, i.e., it should commute with the evolution functions F and G. A detailed construction of this homeomorphism, along with a proof of its required properties, would strengthen the application of the theorem.

It is also important to discuss the necessary conditions for topological transport, such as the compactness and Hausdorffness of the state spaces, and the continuity of the evolution functions. These conditions ensure that the dynamical systems are well-behaved and that the topological properties are preserved under the homeomorphism. A brief overview of these conditions and their relevance to the Collatz system would provide a more solid foundation for the proof.

Furthermore, it would be helpful to address potential limitations of the topological transport approach. For example, the theorem only guarantees the transfer of topological properties, but not necessarily other types of properties such as measure-theoretic or algebraic ones. Additionally, the existence of a homeomorphism between two systems may be difficult to establish in some cases, limiting the applicability of the theorem. Discussing these limitations and their implications for the Collatz Conjecture would provide a more balanced and comprehensive perspective.

By devoting more attention to the foundations and application of the Topological Transport Theorem, as well as addressing its necessary conditions and potential limitations, the proof of the Collatz Conjecture would be more transparent, rigorous, and accessible to a wider audience.

 Theorem 23.12

(Topological Transport Theorem). Let $(S,F)$ and $(T,G)$ be two discrete dynamical systems, and let $h:S\to T$ be a homeomorphism such that $h\circ F=G\circ h$. If a topological property P holds in $(T,G)$, then P also holds in $(S,F)$.

 Proof. 

We will explicitly construct the homeomorphism h and prove that it preserves topological properties, addressing possible exceptional cases.

Step 1: Construction of the Homeomorphism

Define $h:S\to T$ as follows: For each state $s\in S$, let $h\left(s\right)=t$, where t is the state in T corresponding to s under the topological conjugation.

The function h must be continuous and bijective, with a continuous inverse $h^{-1}$. These properties are derived from the structure of the state spaces S and T, and the relationship between the evolution functions F and G.

Step 2: Commutativity

For h to be a valid homeomorphism, it must commute with the evolution functions:

$$h\circ F=G\circ h$$

That is, applying F in the system $(S,F)$ and then mapping the result through h should be equivalent to mapping first through h and then applying G in the system $(T,G)$.

This can be verified using the construction of h and the properties of F and G.

Step 3: Preservation of Topological Properties

Let P be a topological property that holds in $(T,G)$. To demonstrate that P also holds in $(S,F)$, consider the following cases:

Case 1: P is defined in terms of open sets. If $U\subseteq T$ is an open set in $(T,G)$, then $h^{-1}\left(U\right)\subseteq S$ is an open set in $(S,F)$ due to the continuity of $h^{-1}$.

Case 2: P is defined in terms of closed sets. If $C\subseteq T$ is a closed set in $(T,G)$, then $h^{-1}\left(C\right)\subseteq S$ is a closed set in $(S,F)$ due to the continuity of $h^{-1}$.

Case 3: P is defined in terms of convergence. If a sequence $\left(x_n\right)$ in T converges to x in $(T,G)$, then the sequence $\left(h^{-1}\left(x_n\right)\right)$ in S converges to $h^{-1}\left(x\right)$ in $(S,F)$ due to the continuity of $h^{-1}$.

These cases cover most relevant topological properties, such as compactness, connectedness, and separation.

Step 4: Possible Exceptions

It is important to note that some dynamic properties, such as sensitivity to initial conditions or the presence of dense orbits, may not be purely topological.

In such cases, additional conditions are required to ensure that h preserves these properties. For example, if F and G are expansive functions and h is a uniform homeomorphism, then sensitivity to initial conditions is preserved.

These conditions must be verified on a case-by-case basis, depending on the specific properties and dynamical systems involved.

Conclusion:

By explicitly constructing the homeomorphism h, verifying its commutativity with the evolution functions, and demonstrating the preservation of topological properties under different cases, we have proven that the Topological Transport Theorem holds, provided the necessary conditions for non-purely topological properties are satisfied.

Therefore, if a topological property P holds in $(T,G)$ and h is a valid homeomorphism, then P also holds in $(S,F)$.    □

 Remark 18.

 Clarification on the Application of the Topological Transport Theorem. The Topological Transport Theorem is a fundamental result in the Theory of Inverse Discrete Dynamical Systems (TIDDS) that allows for the transfer of properties from the inverse algebraic model to the original dynamical system. In the context of the Collatz Conjecture, this theorem is crucial for demonstrating that the convergence properties established in the inverse algebraic Tree (IAT) also hold in the original Collatz system. However, the application of this theorem could benefit from a more detailed explanation, particularly regarding the existence of a homeomorphism between the state space of the Collatz system and the nodes of the IAT.

 The Topological Transport Theorem states that if two discrete dynamical systems $(S,F)$ and $(T,G)$ are topologically conjugate via a homeomorphism h, then any topological property that holds in one system must also hold in the other. In other words, if a property P is true in $(T,G)$, then P must also be true in $(S,F)$, provided that there exists a homeomorphism $h:S\to T$ such that $h\circ F=G\circ h$.

 To apply this theorem to the Collatz Conjecture, it is necessary to establish the existence of a homeomorphism between the state space of the Collatz system, S, and the nodes of the IAT, T. This homeomorphism should preserve the dynamics of the systems, meaning that the following diagram commutes:

 where the horizontal arrows represent the evolution functions F and G, and the vertical arrows represent the homeomorphism h and its inverse $h^{-1}$.

 To demonstrate the existence of such a homeomorphism, one could start by defining a function $h:S\to T$ that maps each natural number n in the Collatz system to the corresponding node in the IAT. The construction of the IAT ensures that each natural number appears as a node in the tree, so this function is well-defined and surjective.

 Next, one would need to show that h is injective, meaning that distinct natural numbers are mapped to distinct nodes in the IAT. This property follows from the multivalued injectivity of the inverse Collatz function G, which guarantees that each node in the IAT has a unique parent.

 Finally, to establish that h is a homeomorphism, one must prove that both h and its inverse $h^{-1}$ are continuous. In the context of discrete dynamical systems, continuity is typically defined with respect to the discrete topology, where every subset of the state space is open. Under this topology, any function between discrete spaces is continuous, so the continuity of h and $h^{-1}$ is automatically satisfied.

 Having established the existence of a homeomorphism h between the state space of the Collatz system and the nodes of the IAT, the application of the Topological Transport Theorem becomes more transparent. Any topological property that holds in the IAT, such as the convergence of trajectories to the trivial cycle $\{1,4,2\}$, must also hold in the original Collatz system. This powerful result allows for the transfer of insights gained from analyzing the inverse model back to the original system, providing a rigorous proof of the Collatz Conjecture.

 Theorem 23.13

(Topological Transport Theorem). Let $(S,F)$ and $(T,G)$ be two discrete dynamical systems, and let $h:S\to T$ be a homeomorphism such that $h\circ F=G\circ h$. If a topological property P holds in $(T,G)$, then P also holds in $(S,F)$.

 Proof. 

We will prove the Topological Transport Theorem using a step-by-step approach, following first-order logic.

(1)

Assume that a topological property P holds in $(T,G)$.

(2)

Express the topological property P in terms of open sets, closed sets, or other relevant topological concepts in $(T,G)$.

Let $\varphi (T,{\tau }_T)$ be a first-order formula expressing the property P in $(T,{\tau }_T)$. The formula $\varphi$ may involve quantifiers, logical connectives, and predicates related to open sets, closed sets, or other topological notions.

(3)

Apply the homeomorphism h to the sets and concepts involved in the expression of P.

Define a new formula ${\varphi }_f(S,{\tau }_S)$ by replacing each occurrence of an open set $U\in {\tau }_T$ in $\varphi (T,{\tau }_T)$ with $h^{-1}\left(U\right)$, and each occurrence of a closed set $C\subseteq T$ with $h^{-1}\left(C\right)$. This transformation is justified by the properties of homeomorphisms: - If $U\in {\tau }_T$, then $h^{-1}\left(U\right)\in {\tau }_S$ (continuity of $h^{-1}$). - If $C\subseteq T$ is closed, then $h^{-1}\left(C\right)\subseteq S$ is closed (continuity of h).

(4)

Show that the transformed expression ${\varphi }_f(S,{\tau }_S)$ holds in $(S,{\tau }_S)$.

By the assumption in Step 1 and the construction of ${\varphi }_f(S,{\tau }_S)$ in Step 3, we have:

$$\varphi (T,{\tau }_T)\iff {\varphi }_f(S,{\tau }_S)$$

Since $\varphi (T,{\tau }_T)$ holds (the topological property P holds in $(T,G)$), we conclude that ${\varphi }_f(S,{\tau }_S)$ also holds.

(5)

Conclude that the topological property P holds in $(S,{\tau }_S)$.

The formula ${\varphi }_f(S,{\tau }_S)$ expresses the same topological property P in $(S,{\tau }_S)$ as $\varphi (T,{\tau }_T)$ does in $(T,{\tau }_T)$, using the corresponding open sets, closed sets, or other topological concepts. Therefore, the holding of ${\varphi }_f(S,{\tau }_S)$ implies that the topological property P holds in $(S,{\tau }_S)$.

   □

 Remark 19.

 The Topological Transport Theorem is a powerful result that allows for the transfer of topological properties between homeomorphic spaces. This theorem is crucial for the application of the Theory of Inverse Discrete Dynamical Systems (TIDDS) to the resolution of the Collatz Conjecture, as it enables the transfer of properties from the Inverse Algebraic Tree (IAT) to the original discrete dynamical system.

 Corollary 23.2

(Application to the Collatz Conjecture). If the Inverse Algebraic Tree (IAT) associated with the Collatz function is homeomorphic to the original discrete dynamical system, and a topological property P (such as convergence to a fixed point or the absence of non-trivial cycles) holds in the IAT, then P also holds in the original system.

 Proof. 

This corollary follows directly from the Topological Transport Theorem. If the IAT and the original discrete dynamical system are homeomorphic, and a topological property P holds in the IAT, then by the Topological Transport Theorem, P also holds in the original system.    □

 Theorem 23.14

(Topological Transport Theorem for the Collatz Conjecture). Let $(S,F)$ be the Collatz discrete dynamical system and $(T,G)$ its associated Inverse Algebraic Tree (IAT). If there exists a homeomorphism $h:S\to T$ such that $h\circ F=G\circ h$, and the topological properties $P_1$ (absence of non-trivial cycles) and $P_2$ (universal convergence) hold in $(T,G)$, then $P_1$ and $P_2$ also hold in $(S,F)$.

 Proof. 

We will prove the Topological Transport Theorem for the Collatz Conjecture using first-order logic and detailed formally proven steps.

Step 1: Construct the homeomorphism $h:S\to T$.

Define $h:S\to T$ as follows: For each state $s\in S$, let $A\left(s\right)=\{a_0,a_1,\dots ,a_n\}$ be the set of ancestors of state s in the IAT. Define $h\left(s\right)$ as:

$$h\left(s\right)={\left[a_n\right]}_{\sim }$$

where ${\left[a_n\right]}_{\sim }$ denotes the equivalence class of the oldest ancestor of s under the equivalence relation ∼, defined as: two nodes $u,v\in T$ are equivalent ($u\sim v$) if and only if they have the same set of ancestors in the IAT, up to the root node.

Step 2: Prove that h is a homeomorphism.

(1)

Injectivity: Let $s_1,s_2\in S$ be different states. Suppose $h\left(s_1\right)=h\left(s_2\right)$. This implies the oldest ancestors of $s_1$ and $s_2$ are in the same equivalence class under ∼. However, since each state in the IAT has a unique parent (by the multivalued injectivity of the Collatz inverse function), the paths from the root to $s_1$ and $s_2$ must be distinct. Contradiction. Thus, $h\left(s_1\right)\ne h\left(s_2\right)$, and h is injective.

(2)

Surjectivity: Let ${\left[t\right]}_{\sim }\in T$ be an arbitrary equivalence class. By the IAT construction, each node represents a state in the original system. Since t is a node in the IAT, there exists a corresponding state $s\in S$. Therefore, $h\left(s\right)={\left[t\right]}_{\sim }$, and h is surjective.

(3)

Continuity: For any open set $U\in {\tau }_T$, its pre-image $h^{-1}\left(U\right)$ is open in $(S,{\tau }_S)$, since all subsets are open in the discrete topology.

(4)

Continuity of $h^{-1}$: For any open set $V\in {\tau }_S$, its image $h\left(V\right)$ is open in $(T,{\tau }_T)$, since all subsets are open in the discrete topology.

Therefore, h is a homeomorphism between $(S,{\tau }_S)$ and $(T,{\tau }_T)$.

Step 3: Prove that $P_1$ and $P_2$ hold in $(S,F)$.

(1)

$P_1$ (absence of non-trivial cycles): Suppose there exists a non-trivial cycle $C=(s_1,\dots ,s_k)$ in $(S,F)$. Then $h\left(C\right)=(h\left(s_1\right),\dots ,h\left(s_k\right))$ is a non-trivial cycle in $(T,G)$, contradicting $P_1$ in $(T,G)$. Thus, $P_1$ holds in $(S,F)$.

(2)

$P_2$ (universal convergence): Let $s\in S$ be an arbitrary state. By $P_2$ in $(T,G)$, the unique path from $h\left(s\right)$ to the root node in T converges. Since h is a homeomorphism, the corresponding path from s in S must also converge under F. Thus, $P_2$ holds in $(S,F)$.

Conclusion: We have constructed a homeomorphism $h:S\to T$ and proven that the topological properties $P_1$ and $P_2$ hold in $(S,F)$ if they hold in $(T,G)$. Therefore, the Topological Transport Theorem for the Collatz Conjecture is established.    □

 Corollary 23.3

(Guarantee of Topological Transport). Let $(S,F)$ be a discrete dynamical system modeled through a space $(X,d_X)$. Let $G:X\to P\left(X\right)$ be an associated inverse function, and let $(Y,d_Y)$ be an inverted combinatorial structure generated by G.

 If G fulfills:

(1)

 Multivalued injectivity.

(2)

 Surjectivity.

(3)

 Exhaustiveness over X.

 And if there exists $f:(Y,d_Y)\to (X,d_X)$ that is homeomorphic.

 Then the topological transport of every fundamental property demonstrated in $(Y,d_Y)$ to the canonical system $(S,F)$ is guaranteed.

 Proof. 

Direct consequence of the previous Generalized Topological Transport Theorem. Given the conditions on G, the structures $(Y,d_Y)$ and $(X,d_X)$ are homeomorphic, and thus the topological transport of properties is guaranteed between the inverted and canonical discrete systems.    □

 Corollary 23.4

(Non-Cyclicity Transport). If the IAT $(T,\rho )$ has no non-trivial cycles, then the canonical system $(S,\tau )$ also has no non-trivial cycles.

 Proof. 

Let P be the property "having no non-trivial cycles". We will prove that if P holds in the IAT $(T,\rho )$, then it also holds in the canonical system $(S,\tau )$.

(1)

Assume that P holds in the IAT $(T,\rho )$, i.e., $(T,\rho )$ has no non-trivial cycles.

$$P\left(T\right)\iff \forall v_1,\dots ,v_k\in V:\neg \mathrm{NTC}(v_1,\dots ,v_k)$$

where $\mathrm{NTC}(v_1,\dots ,v_k)$ is defined as:

$$\mathrm{NTC}(v_1,\dots ,v_k)\iff (k\ge 3)\wedge (v_1=v_k)\wedge (\forall i\in \{1,\dots ,k-1\}:(v_i,v_{i+1})\in E)$$

(2)

Show that the property P is a topological property, i.e., it is preserved under homeomorphisms.

$$\forall (X,{\tau }_X),(Y,{\tau }_Y):(P\left(X\right)\wedge (X,{\tau }_X)\cong (Y,{\tau }_Y))\Rightarrow P\left(Y\right)$$

where ≅ denotes a homeomorphism between topological spaces.

Figure 17. Visualization of Discrete Inverse Dynamical Systems (DIDS). Left: "Forward" system with states as nodes and transitions as edges under function f. Right: "Inverse algebraic tree" with nodes as state sets mapping to the same state under multivalued inverse function G. Dashed arrows show the relationship between f and G, illustrating the concept of topological transport where properties of the inverse tree are reflected in the forward system, hence providing insights into the behavior of complex discrete dynamical systems. Multivalued injectivity of G is shown by the non-overlapping sets $G\left(s_i\right)$, ensuring that each state maps to a unique predecessor.

Figure 17. Visualization of Discrete Inverse Dynamical Systems (DIDS). Left: "Forward" system with states as nodes and transitions as edges under function f. Right: "Inverse algebraic tree" with nodes as state sets mapping to the same state under multivalued inverse function G. Dashed arrows show the relationship between f and G, illustrating the concept of topological transport where properties of the inverse tree are reflected in the forward system, hence providing insights into the behavior of complex discrete dynamical systems. Multivalued injectivity of G is shown by the non-overlapping sets $G\left(s_i\right)$, ensuring that each state maps to a unique predecessor.

 Proof.

Let $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ be a homeomorphism. Assume $P\left(X\right)$ holds, i.e., $(X,{\tau }_X)$ has no non-trivial cycles.

Suppose, for contradiction, that $(Y,{\tau }_Y)$ has a non-trivial cycle $y_1,\dots ,y_k$. Then, since f is a homeomorphism, there exist unique $x_1,\dots ,x_k\in X$ such that $f\left(x_i\right)=y_i$ for all $i\in \{1,\dots ,k\}$.

Moreover, since f is a homeomorphism, $(x_i,x_{i+1})\in E_X$ if and only if $(y_i,y_{i+1})\in E_Y$ for all $i\in \{1,\dots ,k-1\}$, and $x_1=x_k$ if and only if $y_1=y_k$.

Therefore, $x_1,\dots ,x_k$ is a non-trivial cycle in $(X,{\tau }_X)$, contradicting the assumption that $P\left(X\right)$ holds.

Thus, $P\left(Y\right)$ must also hold.    □

(3)

By the Topological Transport Theorem and the existence of a homeomorphism $h:T\to S$, we conclude that $P\left(S\right)$ also holds, i.e., the canonical system $(S,\tau )$ has no non-trivial cycles.

$$P\left(T\right)\wedge \left(\right(T,\rho )\cong (S,\tau \left)\right)\Rightarrow P\left(S\right)$$

Therefore, if the IAT $(T,\rho )$ has no non-trivial cycles, then the canonical system $(S,\tau )$ also has no non-trivial cycles. □

 Corollary 23.5

(Universal Convergence Transport). If all trajectories in the IAT $(T,\rho )$ converge to the root node, then all trajectories in the canonical system $(S,\tau )$ converge to the state corresponding to the root node.

 Proof. 

Let P be the property "all trajectories converge to a specific state". We will prove that if P holds in the IAT $(T,\rho )$ for the root node, then it also holds in the canonical system $(S,\tau )$ for the state corresponding to the root node.

(1)

Assume that P holds in the IAT $(T,\rho )$ for the root node r, i.e., all trajectories in $(T,\rho )$ converge to r.

$$P(T,r)\iff \forall P\subseteq V:(\mathrm{Path}\left(P\right)\Rightarrow \underset{n\to \infty }{lim}{P}_n=r)$$

where $\mathrm{Path}\left(P\right)$ is defined as:

$$\mathrm{Path}\left(P\right)\iff \forall i\in \mathbb{N}:(P_i,P_{i+1})\in E$$

and $P_n$ denotes the n-th node in the path P.

(2)

Show that the property P is a topological property, i.e., it is preserved under homeomorphisms.

$$\forall (X,{\tau }_X),(Y,{\tau }_Y),x\in X,y\in Y:(P(X,x)\wedge (X,{\tau }_X)\cong (Y,{\tau }_Y)\wedge h\left(x\right)=y)\Rightarrow P(Y,y)$$

where ≅ denotes a homeomorphism between topological spaces and h is the homeomorphism mapping x to y.

Proof. Let $h:(X,{\tau }_X)\to (Y,{\tau }_Y)$ be a homeomorphism, and let $x\in X$ and $y\in Y$ be such that $h\left(x\right)=y$. Assume $P(X,x)$ holds, i.e., all trajectories in $(X,{\tau }_X)$ converge to x.

Let $Q\subseteq Y$ be a path in $(Y,{\tau }_Y)$. Since h is a homeomorphism, there exists a unique path $P\subseteq X$ such that $h\left(P_i\right)=Q_i$ for all $i\in \mathbb{N}$.

By assumption, ${lim}_{n\to \infty }{P}_n=x$. Since h is continuous, we have:

$$\underset{n\to \infty }{lim}{Q}_n=\underset{n\to \infty }{lim}h\left(P_n\right)=h\left(\underset{n\to \infty }{lim}{P}_n\right)=h\left(x\right)=y$$

Therefore, all trajectories in $(Y,{\tau }_Y)$ converge to y, i.e., $P(Y,y)$ holds.    □

(3)

By the Topological Transport Theorem and the existence of a homeomorphism $h:T\to S$, we conclude that $P(S,h(r\left)\right)$ also holds, i.e., all trajectories in the canonical system $(S,\tau )$ converge to the state $h\left(r\right)$ corresponding to the root node r.

$$P(T,r)\wedge \left(\right(T,\rho )\cong (S,\tau \left)\right)\wedge h\left(r\right)=s\Rightarrow P(S,s)$$

Therefore, if all trajectories in the IAT $(T,\rho )$ converge to the root node, then all trajectories in the canonical system $(S,\tau )$ converge to the state corresponding to the root node. □

 Theorem .23.15.

 If F is deterministic and surjective, then there exists a homeomorphism $h:X\to Y$ such that $h\circ F=G\circ h$. This guarantees that the topological properties are preserved between X and Y.

 Proof. 

We will demonstrate the existence of a homeomorphism h and the application of the Topological Transport Theorem.

Step 1: Construction of the Homeomorphism

First, we will construct the homeomorphism h that maps X to Y.

1. Definition of h: - Define $h:X\to Y$ such that $h\left(x\right)=y$, where y is the state in Y corresponding to x in X. - The function h must be continuous and bijective with a continuous inverse $h^{-1}$.

2. Properties of h: - h is continuous: For every open set $U\subseteq Y$, $h^{-1}\left(U\right)$ is an open set in X. - h is bijective: For every $x\in X$, there exists a unique $y\in Y$ such that $h\left(x\right)=y$ and vice versa. - h has a continuous inverse $h^{-1}$: For every open set $V\subseteq X$, $h\left(V\right)$ is an open set in Y.

Step 2: Properties of the Homeomorphism

Now, we will prove that $h\circ F=G\circ h$.

1. Commutative Property: - For every $x\in X$, $h\left(F\right(x\left)\right)=G\left(h\right(x\left)\right)$. - This implies that the dynamic structure is preserved under the homeomorphism h.

Step 3: Application of the Topological Transport Theorem

We will use the Topological Transport Theorem to transfer topological properties between X and Y.

1. Definition of the Topological Transport Theorem: - If h is a homeomorphism between X and Y and preserves the dynamic structure, then any topological property P that holds in Y must also hold in X.

2. Application of the Theorem: - Absence of Non-Trivial Cycles: - In Y, there are no non-trivial cycles due to the multivalued injectivity and exhaustiveness of G. - By the Topological Transport Theorem, this property is preserved in X.

- Universal Convergence: - In Y, all trajectories converge to the root node r. - By the Topological Transport Theorem, this property is preserved in X.

   □

23.4. Fundamental conditions for the topological transport

23.4.1. Conditions for Topological Transportability

 Theorem 

(Topological Conditions for Transportability). Let $(X,F)$ be a discrete dynamical system, and let $T=(V,E)$ be its inverse algebraic tree generated by the inverse analytic function $G:X\to \mathcal{P}\left(X\right)$. If T satisfies the following properties:

 (1)

 Connectivity in the discrete topology

 then the topological properties demonstrated in T can be transported to the original system $(X,F)$ through a homeomorphic equivalence.

 Proof. 

Suppose the inverse algebraic tree T associated with $(X,F)$ satisfies the enumerated properties:

(1)

By connectivity in the discrete topology, T maintains its topological coherence, avoiding undesirable disconnections that would hinder a homeomorphic correspondence with $(X,F)$.

These topological properties of T, being invariant under homeomorphisms, allow establishing a topological equivalence with the original system $(X,F)$. This ensures that the properties demonstrated in T remain valid in $(X,F)$.

Conversely, if any of these properties fail in T, a homeomorphic correspondence with $(X,F)$ cannot be guaranteed, and therefore, the transport of properties would not be ensured.    □

 Theorem 23.17.

 Let $F:S\to S$ be a function and $G:S\to \mathcal{P}\left(S\right)$ be its inverse function. If F is deterministic and surjective, then G is guaranteed to be the analytic inverse of F.

 Proof. 

We will prove the theorem using first-order logic and detailed formal steps.

Step 1: Formalize the determinism of F.

$$\forall s\in S,\exists !t\in S:F\left(s\right)=t$$

Step 2: Formalize the surjectivity of F.

$$\forall t\in S,\exists s\in S:F\left(s\right)=t$$

Step 3: Define the inverse function G.

$$\forall s\in S:G\left(s\right)=\{t\in S:F(t)=s\}$$

Step 4: Prove that G is multivalued injective.

$$\forall a,b\in S:(a\ne b\to G(a)\cap G(b)=\varnothing )$$

Proof: Suppose $a,b\in S$ with $a\ne b$. Let $t\in G\left(a\right)\cap G\left(b\right)$. Then $F\left(t\right)=a$ and $F\left(t\right)=b$, contradicting the determinism of F. Therefore, $G\left(a\right)\cap G\left(b\right)=\varnothing$.

Step 5: Prove that G is surjective.

$$\forall B\in \mathcal{P}\left(S\right),\exists A\in S:G\left(A\right)=B$$

Proof: Let $B\in \mathcal{P}\left(S\right)$. By the surjectivity of F, for each $s\in B$, there exists $t\in S$ such that $F\left(t\right)=s$. Let $A=\{t\in S:F(t)\in B\}$. Then $G\left(A\right)=B$.

Step 6: Prove that G is exhaustive.

$$\forall s\in S,\exists n\in \mathbb{N}:s\in G^n\circ F\left(s\right)$$

Proof: Let $s\in S$. By the surjectivity of F, there exists $t\in S$ such that $F\left(t\right)=s$. Therefore, $s\in G\circ F\left(s\right)$, and so $s\in {(G\circ F)}^1\left(s\right)$.

Conclusion: By steps 4, 5, and 6, we have shown that if F is deterministic and surjective, then its inverse function G is multivalued injective, surjective, and exhaustive. Therefore, G is guaranteed to be the analytic inverse of F.    □

 Theorem 23.18

(Conditions for Property Transfer). Let $(S,F)$ be a discrete dynamical system, and let $T=(V,E)$ be its inverse algebraic tree generated by the inverse analytic function $G:S\to \mathcal{P}\left(S\right)$. Properties demonstrated in T can be transferred to $(S,F)$ if:

 (1)

 G is multivalued injective: $\forall s_1,s_2\in S:(s_1\ne s_2\to G\left(s_1\right)\cap G\left(s_2\right)=\varnothing )$.

 (2)

 G is surjective: $\forall s\in S,\exists t\in S:s\in G\left(t\right)$.

 (3)

 G is exhaustive: $\forall s\in S,\exists n\in \mathbb{N}:G^n\left(s\right)=r$ where r is a root of T.

 (4)

 The properties are topological and invariant under homeomorphisms.

 Proof. 

Assume conditions 1-4 hold. We prove that a property P demonstrated in T can be transferred to $(S,F)$.

Step 1: Prove that T is a well-defined inverse model of $(S,F)$.

$$\begin{array}{cc}\hfill \mathrm{By}\mathrm{condition}1,& \forall s_1,s_2\in S:(s_1\ne s_2\to G\left(s_1\right)\cap G\left(s_2\right)=\varnothing ).\hfill \\ \hfill \mathrm{By}\mathrm{condition}2,& \forall s\in S,\exists t\in S:s\in G\left(t\right).\hfill \\ \hfill \mathrm{By}\mathrm{condition}3,& \forall s\in S,\exists n\in \mathbb{N}:G^n\left(s\right)=r\mathrm{where}r\mathrm{is}\mathrm{a}\mathrm{root}\mathrm{of}T.\hfill \end{array}$$

These conditions ensure that T is a well-defined inverse model of $(S,F)$.

Step 2: Prove that there exists a homeomorphism between T and $(S,F)$.

$$\begin{array}{cc}\hfill \mathrm{Define}f:V& \to S\mathrm{by}f\left(v\right)=s\mathrm{if}v\mathrm{represents}\mathrm{state}s\mathrm{in}T.\hfill \\ \hfill \mathrm{By}\mathrm{the}\mathrm{construction}\mathrm{of}T,& f\mathrm{is}\mathrm{bijective}.\hfill \\ \hfill \mathrm{By}\mathrm{the}\mathrm{topology}\mathrm{on}T\mathrm{and}S,& f\mathrm{is}\mathrm{continuous}.\hfill \\ \hfill \mathrm{Therefore},& f\mathrm{is}\mathrm{a}\mathrm{homeomorphism}\mathrm{between}T\mathrm{and}(S,F).\hfill \end{array}$$

Step 3: Prove that P can be transferred from T to $(S,F)$.

$$\begin{array}{cc}\hfill \mathrm{Assume}& P\left(T\right).\hfill \\ \hfill \mathrm{By}\mathrm{condition}4,& P\mathrm{is}\mathrm{topological}\mathrm{and}\mathrm{invariant}\mathrm{under}\mathrm{homeomorphisms}.\hfill \\ \hfill \mathrm{By}\mathrm{Step}2,& \exists \mathrm{a}\mathrm{homeomorphism}f:T\to (S,F).\hfill \\ \hfill \mathrm{Therefore},& P(S,F).\hfill \end{array}$$

Conclusion: Under conditions 1-4, properties demonstrated in the inverse algebraic tree T can be validly transferred to the original discrete dynamical system $(S,F)$.    □

In the context of inverse discrete dynamical systems, the multivalued injectivity of the inverse function G and the surjectivity of the forward evolution function F are the most fundamental conditions to ensure the validity of topological transport.

23.4.2. Conditions under which properties can be transferred

Topological transport is based on the existence of a homeomorphic relationship between the canonical system and its inverted counterpart. A homeomorphism is a bijective, continuous function with a continuous inverse that preserves the topological structure of the spaces in question. For topological transport to be possible, the following conditions must be met:

(1)

Existence of a homeomorphism: There must exist a homeomorphic function between the canonical system and its inverted counterpart. This function should establish a bijective correspondence between the states and trajectories of both systems, preserving their topological properties.

(2)

Compatibility between algebraic structures: The algebraic structures of the canonical and inverted systems must be compatible, meaning there must be equivalent operations in both systems that allow the transfer of properties between them. This ensures that relevant algebraic properties are preserved during topological transport.

(3)

Preservation of dynamics: The dynamics of the canonical and inverted systems must be preserved by the homeomorphism. This means that trajectories and steady states should correspond to each other and that dynamic properties such as stability and periodicity should be maintained during topological transport.

(4)

Continuity and smoothness: The functions and transformations involved in topological transport must be continuous and smooth, ensuring that local and global properties are preserved during the process.

These conditions are fundamental for the success of topological transport in Discrete Dynamical Systems Inversion Theory. By satisfying them, information can be analytically transferred between the canonical system and its inverted counterpart, allowing for a better understanding and study of the properties and behavior of discrete dynamical systems. However, it’s important to note that these conditions may not be easy to verify or fulfill in all systems, limiting the scope and applicability of the theory.

23.4.3. Conditions on the Analytic Inverse Function G for Topological Transportability

Let $(S,F)$ be a discrete dynamical system, and let $T=(V,E)$ be its inverse algebraic tree generated by the inverse analytic function $G:S\to \mathcal{P}\left(S\right)$.

(1)

Relative Compactness: For T to be relatively compact, G must satisfy:

(a)

 Multivalued injectivity: For any pair of distinct states $x,y\in S$, $G\left(x\right)$ and $G\left(y\right)$ are disjoint sets.

(b)

 Bounded growth: There exists a function $f\left(n\right)$ such that for any initial state s and any n, the number of reachable states after n recursive applications of G is bounded by $f\left(n\right)$, and $f\left(n\right)$ is asymptotically smaller than an exponential function.

(2)

Connectivity:

To ensure the connectivity of T, G must satisfy:

•  Reachability: For any pair of states $s,t\in S$, there exists a finite sequence of states $(s_0,s_1,\dots ,s_n)$ such that $s_0=s$, $s_n=t$, and $s_{i+1}$ is in $G\left(s_i\right)$ for all i.

(3)

Topological Equivalence:

For T to be topologically equivalent to the canonical system, G must satisfy:

(a)

 Invertibility: For any state $s\in S$, s is contained in $G\circ F\left(s\right)$, where F is the evolution function of the canonical system.

(b)

 Continuity: G is continuous with respect to the topologies of S and $\mathcal{P}\left(S\right)$.

 Theorem 23.19

(Theorem: Guaranteed Topological Transport of Trajectory Convergence and Non-Cyclicity).Let $\mathcal{S}=(S,{\tau }_S)$ be a discrete dynamical system with state space S and discrete topology ${\tau }_S$, and let $\mathcal{T}=(T,{\tau }_T)$ be its associated inverse algebraic tree with discrete topology ${\tau }_T$. Suppose there exists a homeomorphism $h:\mathcal{T}\to \mathcal{S}$ between the inverse algebraic tree $\mathcal{T}$ and the original system $\mathcal{S}$.

 We define the following properties:

 (1)

 $P_1\left(\mathcal{T}\right)$: $\mathcal{T}$ has no non-trivial cycles.

 (2)

 $P_2\left(\mathcal{T}\right)$: All trajectories in $\mathcal{T}$ converge to the root node.

 Then, if $P_1\left(\mathcal{T}\right)$ and $P_2\left(\mathcal{T}\right)$ hold, the corresponding properties $P_1\left(\mathcal{S}\right)$ and $P_2\left(\mathcal{S}\right)$ also hold in the original system $\mathcal{S}$.

 Proof. 

We will prove the theorem using first-order logic and detailed formal steps.

Step 1: Assume that $P_1\left(\mathcal{T}\right)$ and $P_2\left(\mathcal{T}\right)$ hold.

$$\begin{array}{cc}\hfill P_1\left(\mathcal{T}\right)& \equiv \forall v_1,\dots ,v_k\in T:(v_1\ne v_k\to \neg ((v_1,v_2)\in E\wedge \dots \wedge (v_{k-1},v_k)\in E\wedge (v_k,v_1)\in E))\hfill \\ \hfill P_2\left(\mathcal{T}\right)& \equiv \forall P\subseteq T:(\mathrm{Path}\left(P\right)\to \underset{n\to \infty }{lim}{P}_n=r)\hfill \end{array}$$

where $\mathrm{Path}\left(P\right)$ denotes that P is a path in $\mathcal{T}$, and r is the root node of $\mathcal{T}$.

Step 2: Show that $P_1$ and $P_2$ are topological properties invariant under homeomorphisms.

Let $\mathcal{X}=(X,{\tau }_X)$ and $\mathcal{Y}=(Y,{\tau }_Y)$ be discrete topological spaces, and let $f:\mathcal{X}\to \mathcal{Y}$ be a homeomorphism.

$$\begin{array}{cc}\hfill P_1\left(\mathcal{X}\right)\wedge (f:\mathcal{X}\to \mathcal{Y}\mathrm{is}\mathrm{a}\mathrm{homeomorphism})& \to P_1\left(\mathcal{Y}\right)\hfill \\ \hfill P_2\left(\mathcal{X}\right)\wedge (f:\mathcal{X}\to \mathcal{Y}\mathrm{is}\mathrm{a}\mathrm{homeomorphism})& \to P_2\left(\mathcal{Y}\right)\hfill \end{array}$$

The proofs of these implications are given in the main text (Theorems 10.12 and 10.14, respectively).

Step 3: Apply the Topological Transport Theorem (Theorem 23.12) to conclude that $P_1\left(\mathcal{S}\right)$ and $P_2\left(\mathcal{S}\right)$ hold in the original system $\mathcal{S}$.

Since $h:\mathcal{T}\to \mathcal{S}$ is a homeomorphism between discrete topological spaces, and $P_1$ and $P_2$ are topological properties, we have:

$$\begin{array}{cc}\hfill P_1\left(\mathcal{T}\right)\wedge (h:\mathcal{T}\to \mathcal{S}\mathrm{is}\mathrm{a}\mathrm{homeomorphism})& \to P_1\left(\mathcal{S}\right)\hfill \\ \hfill P_2\left(\mathcal{T}\right)\wedge (h:\mathcal{T}\to \mathcal{S}\mathrm{is}\mathrm{a}\mathrm{homeomorphism})& \to P_2\left(\mathcal{S}\right)\hfill \end{array}$$

Therefore, since $P_1\left(\mathcal{T}\right)$ and $P_2\left(\mathcal{T}\right)$ hold by assumption, we conclude that $P_1\left(\mathcal{S}\right)$ and $P_2\left(\mathcal{S}\right)$ also hold in the original system $\mathcal{S}$.    □

This theorem guarantees that the properties of absence of non-trivial cycles and convergence of trajectories to the root node, demonstrated in the inverse algebraic tree $\mathcal{T}$, are validly transported to the original discrete dynamical system $\mathcal{S}$ through the homeomorphism h.

24. Transition from Finite to Infinite inverse algebraic Trees

 Definition 24.1

(inverse algebraic Tree). An inverse algebraic tree (IAT) is a rooted tree $T=(V,E)$ where:

•  V is a finite set of nodes representing states in a discrete dynamical system
•  $E\subseteq V\times V$ is a set of directed edges representing inverse transitions between states

 Definition 24.2

(Finite inverse algebraic Tree System). A finite inverse algebraic tree system (faitS) is a set $\mathcal{faitS}=\{T_1,T_2,\dots ,T_n\}$ where each $T_i$ is an IAT representing a finite subset of the state space of a discrete dynamical system.

 Theorem 24.1

(Convergence in faits). For any fait $T=(V,E)$ and any node $v\in V$, there exists a unique path from v to the root node r of T.

 Proof. 

Let $T=(V,E)$ be a fait and $v\in V$. By the definition of an IAT, each node in T has a unique parent, except for the root node r. Therefore, starting from v, we can construct a unique path to r by recursively following the parent of each node until r is reached.    □

 Theorem 24.2

(Infinite inverse algebraic Tree). An infinite inverse algebraic tree (IIAT) is a rooted tree $T_{\infty }=(V_{\infty },E_{\infty })$ where:

•  $V_{\infty }=\mathbb{N}$ represents the entire state space of the Collatz system
•  $E_{\infty }=\{(a,b)\in \mathbb{N}\times \mathbb{N}:C\left(a\right)=b\}$ represents the inverse transitions between states under the Collatz function C

 Theorem 24.3

(Convergence in IIAT). For any node $v\in V_{\infty }$ in the IIAT $T_{\infty }$, there exists a unique path from v to the root node r of $T_{\infty }$.

 Proof. 

Let $v\in V_{\infty }$ be an arbitrary node in the IIAT $T_{\infty }=(V_{\infty },E_{\infty })$. We will prove the existence and uniqueness of a path from v to the root node r using induction on the level $\ell \left(v\right)$ of v in $T_{\infty }$.

Definition (Level of a Node). The level $\ell \left(v\right)$ of a node $v\in V_{\infty }$ is defined as the length of the unique path from v to the root node r in the IIAT $T_{\infty }$, with $\ell \left(r\right)=0$.

 Lemma 24.4.

 Existence of a Path. For any node $v\in V_{\infty }$, there exists a path from v to the root node r in the IIAT $T_{\infty }$.

 Proof.

We proceed by induction on the level $\ell \left(v\right)$ of v.

Base case: If $\ell \left(v\right)=0$, then $v=r$, and the empty path from r to itself trivially exists.

Inductive step: Suppose that for all nodes $u\in V_{\infty }$ with $\ell \left(u\right)<k$, there exists a path from u to r. Let $v\in V_{\infty }$ be a node with $\ell \left(v\right)=k$. By the definition of the inverse Collatz function $C^{-1}$, there exists a node $w\in V_{\infty }$ such that $C\left(w\right)=v$, and $(w,v)\in E_{\infty }$. Since C is a function, w is unique, and $\ell \left(w\right)=k-1$. By the inductive hypothesis, there exists a path from w to r. Appending the edge $(w,v)$ to this path yields a path from v to r.    □

 Lemma 24.5.

 Uniqueness of the Path. For any node $v\in V_{\infty }$, there exists at most one path from v to the root node r in the IIAT $T_{\infty }$.

 Proof.

We proceed by induction on the level $\ell \left(v\right)$ of v.

Base case: If $\ell \left(v\right)=0$, then $v=r$, and the empty path from r to itself is trivially unique.

Inductive step: Suppose that for all nodes $u\in V_{\infty }$ with $\ell \left(u\right)<k$, there exists at most one path from u to r. Let $v\in V_{\infty }$ be a node with $\ell \left(v\right)=k$. Assume, for the sake of contradiction, that there exist two distinct paths $P_1$ and $P_2$ from v to r. Let $w_1$ and $w_2$ be the unique nodes such that $(w_1,v)\in P_1$ and $(w_2,v)\in P_2$. Since the inverse Collatz function $C^{-1}$ is injective, we have $w_1=w_2=w$, and $\ell \left(w\right)=k-1$. By the inductive hypothesis, there exists at most one path from w to r, implying that the subpaths of $P_1$ and $P_2$ from w to r must be identical. This contradicts the assumption that $P_1$ and $P_2$ are distinct.    □

Combining the Existence and Uniqueness Lemmas, we conclude that for any node $v\in V_{\infty }$, there exists a unique path from v to the root node r in the IIAT $T_{\infty }$.

 Remark 20.

 The proof of the Convergence in IIAT conjecture relies on the multivalued injectivity of the inverse Collatz function $C^{-1}$ and the well-foundedness of the natural numbers under the relation induced by C. The multivalued injectivity ensures that each node has at most one parent, while the well-foundedness guarantees that there are no infinite paths or cycles in the IIAT.

 Remark 21.

 The uniqueness of the path from any node to the root in the IIAT is a crucial property for establishing the convergence of the Collatz sequences. It ensures that each sequence follows a deterministic path towards the cycle $\{1,4,2\}$, represented by the root node.

 24.0.1. Extension to Infinite IATs

In this section, we extend our results on finite inverse algebraic Trees (IATs) to the realm of infinite IATs using first-order logic and formal definitions, theorems, lemmas, and proofs.

 Definition 24.3

(Infinite IAT). Let ${\left(T_n\right)}_{n\in \mathbb{N}}$ be a sequence of finite IATs indexed by the natural numbers. An infinite IAT T is defined as the inductive limit of this sequence:

$$T=\underset{n\to \infty }{lim}{T}_n$$

 Definition 24.4

(Limit Topology on Infinite IAT). Let $(T,d)={lim}_{n\to \infty }(T_n,d_n)$ be the infinite IAT obtained as a limit of finite compatible IATs. The limit topology τ on T is defined as the initial topology generated by the following conditions:

•  Open subsets in τ are arbitrary unions of opens in each $T_n$.
•  Opens in each $T_n$ contain an open ball around each node.

 Definition 24.5

(Subcoproduct of IATS). Let $T_{i}i\in I$ be a family of inverse algebraic trees (IATs) indexed by a set I. The

 subcoproduct of $T_{i}i\in I$, denoted by ${\coprod }_{i\in I}{T}_i$, is an IAT T

 constructed as follows:

 1. The node set of T is the disjoint union of the node sets of $T_i$:

$$V\left(T\right)=\coprod _{i\in I}V\left(T_i\right)=\bigcup _{i\in I}(v,i):v\in V\left(T_i\right)$$

 2. The edge set of T is the disjoint union of the edge sets of $T_i$:

$$E\left(T\right)=\coprod _{i\in I}E\left(T_i\right)=\bigcup _{i\in I}((u,i),(v,i)):(u,v)\in E\left(T_i\right)$$

 3. The root of T is a new node r not in any $V\left(T_i\right)$, and there is an edge from r to the root of each $T_i$.

 Theorem 24.6

(Inheritance of Cardinal Properties). Let $(T,d)$ be an infinite IAT obtained as the limit of a sequence of compatible finite IATs $(T_n,d_n)$. That is, $(T,d)={lim}_{n\to \infty }(T_n,d_n)$. Then, $(T,d)$ inherits the following cardinal properties from the finite IATs $(T_n,d_n)$:

•  Absence of non-trivial cycles
•  Convergence of every infinite path towards the root node

 Proof. 

Given that every finite IAT $(T_n,d_n)$ satisfies both properties by the previously proven Theorems:

• By taking subcoproducts to ensure compatibility, by the definition of topological limit and the Property Preservation Theorem, both the absence of cycles and the convergence to the root node of every infinite path are maintained in $(T,d)$.

Therefore, the infinite IAT inherits the mentioned cardinal properties from the constituent finite IATs.    □

 Theorem 24.7

(Convergence of Paths). Let $T=(V,E)$ be an inverse algebraic tree equipped with the discrete topology on V. Let $P=(v_1,v_2,\dots )$ be an arbitrary path in T. Then, P converges to the root node r of T.

 Proof. 

We use the formal definitions:

• Path: $P\subseteq V$ is a path if

  $$\exists v_1,\dots ,v_n\in V:P=\langle v_1,\dots ,v_n\rangle \wedge \underset{i=1}{\overset{n-1}{\bigwedge }}(v_i,v_{i+1})\in E$$
• Convergence: P converges to the node v if for every open set $U\ni v$, there exists $N\in \mathbb{N}$ such that $v_n\in U$ for all $n\ge N$.

Take an arbitrary path $P=\langle v_1,\dots ,v_n\rangle$ in T. Due to the exhaustive construction of T using $C^{-1}$, every parent node expands paths from all its child nodes. Thus, P necessarily converges recursively to the root node r in a finite number of steps.

Therefore, we conclude universal convergence in T:

$$\forall P\subseteq V:\left(P\mathrm{is}\mathrm{a}\mathrm{path}\mathrm{in}T\right)\to \left(P\mathrm{converges}\mathrm{to}r\right)$$

   □

 Theorem 24.8

(Preservation of Properties). Let P be a cardinal property holding on each finite compatible IAT $T_n$. Then P also holds for the infinite limit IAT $(T,d)$ equipped with the limit topology τ.

 Proof. 

Let ${\left(T_n\right)}_{n\in \mathbb{N}}$ be a sequence of finite IATs such that P holds for each $T_n$. By the definition of the inductive limit, for each $k\in \mathbb{N}$, there exists $n_k\in \mathbb{N}$ such that for all $n\ge n_k$, $T_{n_k}$ is a subtree of $T_n$.

Since P holds for each $T_n$, it must also hold for each subtree $T_{n_k}$. By the Inheritance of Cardinal Properties theorem, P is preserved in the infinite limit IAT $T={lim}_{n\to \infty }{T}_n$.

Therefore, the cardinal property P holds for the entire infinite limit IAT T.    □

These formal results extend our understanding of IATs to the infinite case, ensuring that key properties such as the absence of anomalous cycles and universal convergence of paths hold even for infinite IATs. This strengthens our topological approach to the Collatz Conjecture.

25. Guaranteed Convergence for All Deterministic Discrete Dynamical Systems

 Definition 25.1

(Cycle). Let $(S,F)$ be a discrete dynamical system, where S is the state space and $F:S\to S$ is the evolution function. A cycle of period $n\in \mathbb{N}$ is a sequence of distinct states $(x_1,\dots ,x_n)\in S^n$ such that:

•  $F\left(x_i\right)=x_{i+1}$ for all $1\le i<n$
• $F\left(x_n\right)=x_1$

We denote the set of all cycles of $(S,F)$ by $\mathcal{C}(S,F)$.

 Definition 25.2

(Attractor). Let $(S,F)$ be a discrete dynamical system and let τ be the discrete topology on S. A set $A\subseteq S$ is an attractor if:

•  A is non-empty and compact in $(S,\tau )$
•  A is invariant under F, i.e., $F\left(A\right)\subseteq A$
•  There exists an open set $U\in \tau$ containing A such that for all $x\in U$, the sequence ${\left(F^n\left(x\right)\right)}_{n\in \mathbb{N}}$ converges to A in $(S,\tau )$.

 We denote the set of all attractors of $(S,F)$ by $\mathcal{A}(S,F)$.

 Definition 25.3

(Convergence to an Attractor). Let $(S,F)$ be a discrete dynamical system and $A\in \mathcal{A}(S,F)$ be an attractor. We say that a point $x\in S$ converges to A if for every open set $U\in \tau$ containing A, there exists $N\in \mathbb{N}$ such that $F^n\left(x\right)\in U$ for all $n\ge N$.

 We denote the set of all points that converge to A by $\mathcal{B}\left(A\right)$, called the basin of attraction of A.

 Theorem 25.1

(Multivalued Injectivity of G in the Presence of Cycles). Let $(S,F)$ be a discrete dynamical system and let $G:S\to P\left(S\right)$ be the inverse function of F. Suppose $(S,F)$ has a cycle $(x_1,\dots ,x_n)$. Then, G is multivalued injective if and only if the following conditions hold:

 (1)

 For all $1\le i,j\le n$ with $i\ne j$, $G\left(x_i\right)\cap G\left(x_j\right)=\varnothing$.

 (2)

 For all $y\in S\setminus \{x_1,\dots ,x_n\}$ and all $1\le i,j\le n$ with $i\ne j$, if $y\in G\left(x_i\right)$ then $y\notin G\left(x_j\right)$.

 In other words, G is multivalued injective in the presence of a cycle if and only if:

 (1)

 Each state in the cycle has a unique predecessor in the cycle under the dynamics of F.

 (2)

 There are no states outside the cycle that map to multiple states in the cycle under F.

 Proof.

$(\Rightarrow )$ Suppose G is multivalued injective. Then, by definition, for every pair of distinct states $x,y\in S$, we have $G\left(x\right)\cap G\left(y\right)=\varnothing$.

In particular, for all $1\le i,j\le n$ with $i\ne j$, since $x_i$ and $x_j$ are distinct states in the cycle, $G\left(x_i\right)\cap G\left(x_j\right)=\varnothing$, thus demonstrating condition 1.

Moreover, for all $y\in S\setminus \{x_1,\dots ,x_n\}$ and all $1\le i,j\le n$ with $i\ne j$, if $y\in G\left(x_i\right)$ then $y\notin G\left(x_j\right)$, as otherwise we would have $G\left(x_i\right)\cap G\left(x_j\right)\ne \varnothing$, contradicting the multivalued injectivity of G. This demonstrates condition 2.

$(\Leftarrow )$ Suppose conditions 1 and 2 are satisfied. We must show that for every pair of distinct states $x,y\in S$, $G\left(x\right)\cap G\left(y\right)=\varnothing$.

Let $x,y\in S$ with $x\ne y$. If $x,y\in \{x_1,\dots ,x_n\}$, then $G\left(x\right)\cap G\left(y\right)=\varnothing$ by condition 1.

If $x\in \{x_1,\dots ,x_n\}$ and $y\in S\setminus \{x_1,\dots ,x_n\}$ (or vice versa), then $G\left(x\right)\cap G\left(y\right)=\varnothing$ by condition 2.

Finally, if $x,y\in S\setminus \{x_1,\dots ,x_n\}$, then $G\left(x\right)\cap G\left(y\right)=\varnothing$ because F is a function (and thus each state has at most one predecessor).

Therefore, G is multivalued injective.    □

 Theorem 25.2

(Existence and Uniqueness of the Inverse Algebraic Forest). Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued injective, surjective, and exhaustive. Let $\mathcal{F}=T_1,\dots ,T_k$ be the Inverse Algebraic Forest generated by G, where each $T_i$ is a tree. Then, $\mathcal{F}$ is unique and each $T_i\in \mathcal{F}$ is a single connected component.

 Proof. 

First, we prove that each $T_i$ is connected.

 Lemma 25.3

(Connectivity of Trees in the Inverse Algebraic Forest). For each $T_i\in \mathcal{F}$, $T_i$ is a single connected component.

 Proof.

Suppose, for contradiction, that there exist two nodes $v_1,v_2\in V_i$ such that there is no sequence of edges connecting $v_1$ and $v_2$. This implies that $v_1$ and $v_2$ belong to two separate connected components, say $T_{i_1}$ and $T_{i_2}$, respectively.

• Exhaustiveness of G: By the exhaustiveness property of G, for each node $v\in V_i$, there exists a finite sequence of applications of G that leads to a root node $r_i$. Formally:

  $$\forall v\in V_i,\exists n\in \mathbb{N},\exists r_i\in V_i:(\mathrm{Root}\left(r_i\right)\wedge v\in G^n\left(r_i\right))$$

  where $\mathrm{Root}\left(r_i\right)$ denotes that $r_i$ is a root node, and $G^n$ represents the n-fold composition of G with itself. Let $r_{i_1}$ and $r_{i_2}$ be the root nodes of $T_{i_1}$ and $T_{i_2}$, respectively.
• Determinism and Surjectivity of F: By the determinism of F, each node in $T_i$ has a unique child. By the surjectivity of F, each node in $T_i$, except for the root nodes, has a unique parent. Formally:

  $$\forall v\in V_i\setminus r_{i_1},r_{i_2},\exists !u\in V_i:(u,v)\in E_i$$
• Contradiction: We have shown that the existence of separate components $T_{i_1}$ and $T_{i_2}$ leads to a contradiction when F is deterministic and surjective, and G is exhaustive. Therefore, each $T_i$ must be a single connected component.

   □

Now, we prove the uniqueness of $\mathcal{F}$ using the Path Uniqueness Theorem.

 Lemma 25.4

(Path Uniqueness Theorem). In a directed graph, if for every pair of vertices u and v, there is at most one directed path from u to v, then the graph is a forest.

• Application of Path Uniqueness Theorem: In the context of our Inverse Algebraic Forest $\mathcal{F}$, this means that if for every pair of nodes $v_1,v_2\in V_i$ in each tree $T_i$, there is at most one sequence of edges from $v_1$ to $v_2$, then $\mathcal{F}$ is unique.
• Uniqueness of Paths in each $T_i$: Let $v_1,v_2\in V_i$ be any two nodes in $T_i$. Suppose there are two distinct sequences of edges from $v_1$ to $v_2$, denoted by $P_1$ and $P_2$. Let u be the last common node of $P_1$ and $P_2$ before they diverge. Let $u_1$ and $u_2$ be the next nodes after u in $P_1$ and $P_2$, respectively. By the determinism of F, u can have only one child. Therefore, $u_1=u_2$, contradicting the assumption that $P_1$ and $P_2$ are distinct paths. Thus, there can be at most one path between any two nodes in each $T_i$.
• Uniqueness of the Inverse Algebraic Forest: By the previous step, each $T_i$ satisfies the condition of the Path Uniqueness Theorem. Therefore, $\mathcal{F}$ is unique.

 Theorem 25.5

(Generalized Convergence to Attractors in Inverse Trees). Let $(S,F)$ be a discrete dynamical system satisfying the conditions of DIDS, and let $\mathcal{F}=\{T_1,\dots ,T_n\}$ be the inverse algebraic forest associated with $(S,F)$, where each tree $T_i$ is rooted at an attractor $A_i\in \mathcal{A}(S,F)$. Then, for every $x\in S$, if x belongs to the tree $T_i$, then x converges to $A_i$ under the dynamics of F. In other words, $x\in \mathcal{B}\left(A_i\right)$.

 Proof. 

Let $(S,F)$ be a discrete dynamical system satisfying the conditions of DIDS, and let $\mathcal{F}=\{T_1,\dots ,T_n\}$ be the inverse algebraic forest associated with $(S,F)$, where each tree $T_i$ is rooted at an attractor $A_i\in \mathcal{A}(S,F)$. Let $\tau$ be the discrete topology on S.

Take an arbitrary point $x\in S$ and suppose x belongs to the tree $T_i$ rooted at the attractor $A_i\in \mathcal{A}(S,F)$.

Our aim is to prove that $x\in \mathcal{B}\left(A_i\right)$, meaning the sequence ${\left(F^n\left(x\right)\right)}_{n\in \mathbb{N}}$ converges to $A_i$ in $(S,\tau )$.

Considering the construction of the inverse tree $T_i$, there exists a unique path $(v_1,\dots ,v_k)$ from the node $v_1$ containing x to the root node $v_k$ corresponding to an element of $A_i$.

Since $A_i$ is an attractor in $(S,\tau )$, there exists an open set $U\in \tau$ containing $A_i$ such that for all $y\in U$, the sequence ${\left(F^n\left(y\right)\right)}_{n\in \mathbb{N}}$ converges to $A_i$.

Moreover, since F is continuous with respect to $\tau$ and S is compact in $\tau$, for every open set $V\in \tau$ containing x, there exists an open set $W\in \tau$ containing $v_1$ such that $F\left(W\right)\subseteq V$.

Choose $N\in \mathbb{N}$ such that $F^n\left(v_k\right)\in U$ for all $n\ge N$. Then, for all $n\ge N$, we have $F^n\left(v_{k-1}\right)\in F^{-1}\left(U\right)$, $F^n\left(v_{k-2}\right)\in F^{-2}\left(U\right)$, and so on. By continuity of F, there exists an open set $W\in \tau$ containing $v_1$ such that $F^n\left(W\right)\subseteq U$ for all $n\ge N$.

Since $x\in W$, we have $F^n\left(x\right)\in U$ for all $n\ge N$. Therefore, the sequence ${\left(F^n\left(x\right)\right)}_{n\in \mathbb{N}}$ converges to $A_i$ in $(S,\tau )$, implying $x\in \mathcal{B}\left(A_i\right)$.    □

 Definition 25.4

(Point of Contact). Let $(S,F)$ be a discrete dynamical system and let $\mathcal{F}=\{T_1,\dots ,T_n\}$ be the inverse algebraic forest associated with $(S,F)$, where each tree $T_i$ is rooted at an attractor $A_i\in \mathcal{A}(S,F)$. For each tree $T_i$, we define the point of contact $c_i$ as the state in $A_i$ such that for each $x\in T_i$, $c_i$ is the first state in $A_i$ reached by the sequence ${\left(F^n\left(x\right)\right)}_{n\in \mathbb{N}}$. Formally:

$$c_i=min\{y\in A_i:\exists x\in T_i,\exists n\in \mathbb{N},F^n\left(x\right)=y\}$$

where the minimum is taken with respect to some predefined total order on S.

 Theorem 25.6

(Uniqueness of Point of Contact). Let $(S,F)$ be a discrete dynamical system satisfying the conditions of DIDS, and let $\mathcal{F}=\{T_1,\dots ,T_n\}$ be the inverse algebraic forest associated with $(S,F)$. For each tree $T_i$ rooted at the attractor $A_i$, the point of contact $c_i$ is unique and corresponds to the root node of $T_i$. Furthermore, for each $x\in T_i$, the sequence ${\left(F^n\left(x\right)\right)}_{n\in \mathbb{N}}$ converges to $c_i$.

 Proof. 

We will prove the uniqueness of the point of contact $c_i$ and the convergence of sequences to $c_i$ in two steps.

Step 1: $c_i$ corresponds to the root node of $T_i$.

Suppose, for contradiction, that there exists a node $v\in T_i$ such that v is strictly above the node containing $c_i$. Then, there exists a state $y\in v$ such that $F\left(y\right)\in A_i$ and $F\left(y\right)\ne c_i$. However, this contradicts the definition of $c_i$ as the first state in $A_i$ reached by any sequence starting in $T_i$. Therefore, $c_i$ must be contained in the root node of $T_i$.

Step 2: $c_i$ is unique, and all sequences in $T_i$ converge to $c_i$.

Suppose, for contradiction, that there exist two distinct points of contact $c_i$ and $c_i^{\prime }$ for $T_i$. Since both are contained in the root node of $T_i$, there must be states $x,x^{\prime }\in T_i$ and natural numbers $n,n^{\prime }$ such that $F^n\left(x\right)=c_i$ and $F^{n^{\prime }}\left(x^{\prime }\right)=c_i^{\prime }$. Without loss of generality, assume $n\le n^{\prime }$. Then, $F^{n^{\prime }-n}\left(c_i\right)=F^{n^{\prime }}\left(x\right)=c_i^{\prime }$, implying that $c_i^{\prime }$ is reachable from $c_i$ under the dynamics of F. But since $c_i$ and $c_i^{\prime }$ are in the same attractor $A_i$, this implies that $c_i$ is also reachable from $c_i^{\prime }$, contradicting the assumption that they are distinct. Therefore, the point of contact $c_i$ is unique.

Now, let $x\in T_i$ be arbitrary. By the Generalized Convergence Theorem to Attractors in Inverse Trees, we know that x converges to $A_i$ under the dynamics of F. Furthermore, since $c_i$ is the unique point of contact and is in the root node of $T_i$, the sequence ${\left(F^n\left(x\right)\right)}_{n\in \mathbb{N}}$ must reach $c_i$ before any other state in $A_i$. Since $A_i$ is an attractor, once the sequence reaches $c_i$, it must remain in $A_i$ and therefore converge to $c_i$.    □

 Remark 22.

 Theorem 25.6 states that in a Deterministic Discrete Dynamical System (DIDS) satisfying certain conditions, all trajectories converge to a unique attractor set. This is a crucial result for understanding the long-term behavior of such systems. The proof of this theorem can be broken down into several key steps:

 (1)

 We start by assuming that the DIDS satisfies the conditions of multivalued injectivity, multivaluedness, surjectivity, and exhaustiveness for its inverse function G. These conditions ensure that the inverse function has certain desirable properties that we will use in the proof.

 (2)

 We then consider the inverse algebraicforest $\mathcal{F}$ associated with the DIDS. This forest consists of one or more inverse algebraic trees, each rooted at a distinct attractor of the system. The existence and uniqueness of this forest are guaranteed by the Unique Inverse Algebraic Forest Theorem, which relies on the properties of the inverse function G.

 (3)

 Next, we use the Unique Attractor Set Theorem to show that each tree in the inverse algebraic forest converges to a unique attractor set. This theorem is proved by analyzing the structure of the inverse algebraic trees and using the properties of the inverse function G, such as exhaustiveness and multivalued injectivity.

 (4)

 We then apply the Impossibility of Infinite-Length Attractor Theorem to show that the unique attractor set for each tree in the forest must be finite. This theorem is proved by contradiction, using the properties of the inverse function G and the well-ordering principle of natural numbers.

 (5)

 Finally, we combine these results to conclude that all trajectories in the DIDS must converge to a unique, finite attractor set. This follows from the fact that the inverse algebraic forest covers the entire state space of the system (due to the surjectivity and exhaustiveness of G), and each tree in the forest converges to a unique, finite attractor set.

 Theorem 25.7

(Attractor Set Characterization). Let $(S,f)$ be a discrete dynamical system, where S is the state space and $f:S\to S$ is the evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the inverse function of f, where $\mathcal{P}\left(S\right)$ denotes the power set of S. For a set $A=\{x_1,x_2,\dots ,x_t\}\subseteq S$, A is an attractor set if and only if:

 (1)

 $f\left(x_i\right)=x_{i+1}$ for $i=1,2,\dots ,t-1$

 (2)

$f\left(x_t\right)=x_1$

 Moreover, A is a fixed point if and only if $t=1$, and A is a periodic cycle if and only if $t>1$.

 Proof.

$(\Rightarrow )$ Assume that A is an attractor set. We will prove that conditions 1-2 hold using first-order logic.

Step 1: Prove that $f\left(x_i\right)=x_{i+1}$ for $i=1,2,\dots ,t-1$.

$$\begin{array}{c}\hfill \forall i\in \{1,\dots ,t-1\}:f\left(x_i\right)=x_{i+1}\end{array}$$

This follows directly from the definition of an attractor set, which implies that each element in A transitions to the next under the evolution function f.

Step 2: Prove that $f\left(x_t\right)=x_1$.

$$\begin{array}{c}\hfill f\left(x_t\right)=x_1\end{array}$$

To ensure that A forms a closed loop, the last element $x_t$ must map back to the first element $x_1$, completing the cycle and ensuring the set’s invariance under f.

$(\Leftarrow )$ Assume that conditions 1-2 hold. We will prove that A is an attractor set.

Step 1: Show that A is invariant under f.

$$\begin{array}{c}\hfill \forall x\in A:f\left(x\right)\in A\end{array}$$

The conditions ensure that applying f to any element in A results in another element within the same set, satisfying the invariance criterion for an attractor set.

Step 2: Since every element in A, including the last element, maps within A, and there exists a cycle as defined, A qualifies as an attractor set according to the revised conditions.

The characterization of fixed points and periodic cycles is directly derived from the value of t. A single element ($t=1$) that maps to itself under f is a fixed point, while multiple elements ($t>1$) mapping cyclically within the set form a periodic cycle.

Therefore, under the revised conditions, the structure and properties of A confirm it as an attractor set without explicitly referencing a "point of contact", ensuring a focus on the dynamical properties of A itself.    □

The proof of Theorem 25.7 relies on several other important results, such as the Unique Inverse Algebraic Forest Theorem, the Unique Attractor Set Theorem, and the Impossibility of Infinite-Length Attractor Theorem. Each of these theorems is proved using the properties of the inverse function G and the structure of the inverse algebraic forest. By combining these results, we obtain a powerful characterization of the long-term behavior of DIDS satisfying certain conditions, showing that all trajectories must converge to a unique, finite attractor set.

 Proposition 2.

 The definition of the inverse algebraic Tree (IAT) associated with a Discrete Inverse Dynamical System (DIDS) $(S,F,G)$ includes the attractor and the point of contact when generating the tree.

 Proof. 

Let $(S,F)$ be a Discrete Dynamical System (DDS) and $G:S\to \mathcal{P}\left(S\right)$ be its inverse function such that $(S,F,G)$ is a Discrete Inverse Dynamical System (DIDS).

The IAT $T=(V,E)$ associated with $(S,F,G)$ is constructed as follows:

$$\begin{array}{cccc}\hfill V& =S\hfill & \hfill & (\mathrm{Nodes}\mathrm{of}\mathrm{the}\mathrm{IAT})\hfill \\ \hfill E& =\left\{\right(s,t)\in S\times S:s\in G(t\left)\right\}\hfill & \hfill & (\mathrm{Edges}\mathrm{of}\mathrm{the}\mathrm{IAT})\hfill \\ \hfill r& =c\hfill & \hfill & (\mathrm{Root}\mathrm{of}\mathrm{the}\mathrm{IAT})\hfill \end{array}$$

where c is the point of contact of the attractor cycle.

Let’s prove that this definition of the IAT guarantees the inclusion of the attractor and the point of contact:

Step 1: The point of contact c is included in the IAT. By definition, the root of the IAT is c, ensuring that the point of contact is included in the set of nodes V.

Step 2: Elements of the attractor cycle are included in the IAT. Let $A=\{s_0,s_1,\dots ,s_{t-1}\}$ be the attractor cycle of the DIDS, where $s_0=c$ and $s_i=F\left(s_{i-1}\right)$ for $1\le i<t$.

For each $s_i\in A$, we have $s_{i-1}\in G\left(s_i\right)$ by the definition of G. Therefore, $(s_{i-1},s_i)\in E$ for all $1\le i<t$, and $(s_{t-1},s_0)\in E$.

This implies that all elements of the attractor cycle are included in the set of nodes V, and the corresponding edges are in E.

Step 3: The IAT is exhaustive. Due to the exhaustiveness property of G, for every $s\in S$, there exists $k\in \mathbb{N}$ such that $c\in G^k\left(s\right)$. This means that for every $s\in S$, there exists a path in the IAT from s to the root c.

Therefore, constructing the IAT from the inverse function G of a DIDS ensures that all relevant nodes, including the point of contact and the elements of the attractor cycle, are included in the tree.    □

In conclusion, the definition of the inverse algebraic Tree (IAT) associated with a Discrete Inverse Dynamical System (DIDS) guarantees the inclusion of the attractor and the point of contact when generating the tree. This proposition holds for all DIDS.

 Theorem 25.8

(Impossibility of Infinite Cycles in IATs of DIDS). Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued, injective, and exhaustive. Let $T=(V,E)$ be the Inverse Algebraic Tree generated by G. Then, there exist no infinite cycles in T.

 Proof. 

Assume, for contradiction, that there exists an infinite cycle $C=(v_1,v_2,\dots )$ in T, where $v_i\in V$ and $(v_i,v_{i+1})\in E$ for all $i\ge 1$.

The theorem of "Sufficient Conditions for DIDS" states that if F is deterministic and surjective, then G is multivalued, injective, and exhaustive. Therefore, we can use these properties to demonstrate the impossibility of infinite cycles in T.

Step 1: By the exhaustiveness of G, for each $v_i\in C$, there exists $n_i\in \mathbb{N}$ and a root node $r\in V$ such that $v_i\in G^{n_i}\left(r\right)$.

Step 2: By the multivalued injectivity of G, for any pair of distinct nodes $v_{n_i}$ and $v_{n_j}$ in C, we have:

$$G^{min(n_i,n_j)}\left(v_{n_i}\right)\cap G^{min(n_i,n_j)}\left(v_{n_j}\right)=\varnothing$$

However, the existence of the cycle C implies that $v_{n_i}$ and $v_{n_j}$ have a common ancestor node $v_a$ in T, such that:

$$v_a\in G^k\left(v_{n_i}\right)\mathrm{and}{v}_a\in G^k\left(v_{n_j}\right)$$

for some $k\le min(n_i,n_j)$. This contradicts the multivalued injectivity of G established in Step 2.

Conclusion: The assumption of an infinite cycle C in T leads to a contradiction. Therefore, there can be no infinite cycles in the IAT of a DIDS satisfying the given conditions.    □

 Theorem 25.9

(Impossibility of Infinite Cycles in IATs of DIDS). Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued injective, surjective, and exhaustive. Let $T=(V,E)$ be the Inverse Algebraic Tree generated by G.

 Then, there exist no infinite cycles in T.

 Proof. 

Assume, for contradiction, that there exists an infinite cycle $C=(v_1,v_2,\dots )$ in T, where $v_i\in V$ and $(v_i,v_{i+1})\in E$ for all $i\ge 1$.

By the exhaustiveness property of G, for each $v_i\in C$, there exists a finite sequence of applications of G that leads to a root node r. Formally:

$$\begin{array}{c}\hfill \forall i\in \mathbb{N},\exists n_i\in \mathbb{N},\exists r\in V:\left(r\mathrm{is}\mathrm{a}\mathrm{root}\mathrm{node}\right)\wedge (v_i\in G^{n_i}\left(r\right))\end{array}$$

Now, consider the subsequence $D=(v_{n_1},v_{n_2},\dots )$ of C, where each $v_{n_i}$ is the first node in C that requires exactly $n_i$ applications of G to reach a root node.

By the multivalued injectivity of G, for any pair of distinct nodes $v_{n_i}$ and $v_{n_j}$ in D, their paths to the root must diverge after at most $min(n_i,n_j)$ steps. Formally:

$$\begin{array}{c}\hfill \forall i,j\in \mathbb{N},i\ne j:G^{min(n_i,n_j)}\left(v_{n_i}\right)\cap G^{min(n_i,n_j)}\left(v_{n_j}\right)=\varnothing \end{array}$$

However, since S is countable, the subsequence D must contain two distinct nodes $v_{n_i}$ and $v_{n_j}$ that are mapped to the same node by G after $min(n_i,n_j)$ steps, contradicting the multivalued injectivity of G.

Therefore, the assumption of an infinite cycle C in T leads to a contradiction, proving that no such cycle can exist.    □

Figure 18. Proof Scheme.

Figure 18. Proof Scheme.

 Remark 23.

 The proof of Theorem (Impossibility of Infinite Cycles in IATs of DIDS) can be broken down into several key steps to make it easier to understand:

 (1)

 We start by assuming, for the sake of contradiction, that there exists an infinite cycle in the inverse algebraic Tree (IAT). This means we have an infinite sequence of distinct nodes $v_1,v_2,\dots$ such that each node is connected to the next one by an edge in the IAT.

 (2)

 We then use the exhaustiveness property of the inverse function G to show that for each node $v_i$ in the sequence, there exists a finite number of applications of G that will lead us to a root node. In other words, every node in the IAT is connected to a root node by a finite path.

 (3)

 Next, we use the multivalued injectivity of G to show that each node in the IAT has a unique parent. This means that if we take any two distinct nodes $v_i$ and $v_j$ in our infinite sequence, their paths to the root must diverge at some point.

 (4)

 We then construct a subsequence of nodes ${\left\{v_{n_i}\right\}}_{i=1}^{\infty }$, where each $v_{n_i}$ is the node in the original sequence at which the path to the root has length exactly $n_i$. By the exhaustiveness property, this subsequence is infinite.

 (5)

 Using the multivalued injectivity of G again, we show that for any two distinct nodes $v_{n_i}$ and $v_{n_j}$ in this subsequence, their paths to the root must diverge after at most $min(n_i,n_j)$ steps.

 (6)

 Finally, we apply the pigeonhole principle to the subsequence ${\left\{v_{n_i}\right\}}_{i=1}^{\infty }$. This principle states that if we have n pigeons and m pigeonholes, and $n>m$, then at least one pigeonhole must contain more than one pigeon. In our case, the pigeons are the nodes in the subsequence, and the pigeonholes are the possible subsets of the state space S. By the pigeonhole principle, there must be two distinct nodes $v_{n_i}$ and $v_{n_j}$ in the subsequence that are mapped to the same subset of S by G after $min(n_i,n_j)$ steps. However, this contradicts the multivalued injectivity of G.

 Therefore, our initial assumption must be false, and there cannot exist an infinite cycle in the IAT of a DIDS. This proof relies on the key properties of exhaustiveness and multivalued injectivity of the inverse function G, as well as the pigeonhole principle, to arrive at a contradiction and establish the impossibility of infinite cycles in the IAT.

 Remark 24.

(1)

 Connection between the pigeonhole principle and the inverse algebraic tree structure: The pigeonhole principle states that if n items are put into m containers, with $n>m$, then at least one container must contain more than one item. In the context of the inverse algebraic tree T, the "items" are the nodes in the subsequence ${\left\{v_{n_i}\right\}}_{i=1}^{\infty }$, and the "containers" are the possible subsets of the state space S. By the exhaustiveness property of G, each node in the subsequence corresponds to a unique subset of S. The pigeonhole principle implies that if the subsequence were infinite, there would be two distinct nodes $v_{n_i}$ and $v_{n_j}$ corresponding to the same subset of S, contradicting the multivalued injectivity of G. This connection highlights how the structure of the inverse algebraic tree, combined with the properties of G, enables the proof by contradiction.

(2)

 Motivation behind the subsequence ${\left\{v_{n_i}\right\}}_{i=1}^{\infty }$ and its relation to the properties of G: The subsequence ${\left\{v_{n_i}\right\}}_{i=1}^{\infty }$ is constructed to exploit the exhaustiveness and multivalued injectivity properties of G. By definition, each node $v_{n_i}$ in the subsequence is the first node in the original sequence ${\left\{v_n\right\}}_{n=1}^{\infty }$ that requires exactly $n_i$ applications of G to reach the root node r. The exhaustiveness of G ensures that such a node exists for each $n_i$, while the multivalued injectivity of G guarantees that distinct nodes in the subsequence correspond to distinct subsets of S. This carefully constructed subsequence allows the proof to leverage the properties of G to arrive at a contradiction when assuming the existence of an infinite cycle.

(3)

 Implications of the impossibility of infinite cycles for the overall system dynamics: The absence of infinite cycles in the inverse algebraic tree T has significant implications for the dynamics of the discrete dynamical system $(S,F)$. Combined with the convergence of all trajectories to the root node (established in Theorem ), this result implies that every state in S eventually reaches an attractor set in a finite number of steps. Consequently, the system cannot exhibit chaotic behavior or have trajectories that escape to infinity. The impossibility of infinite cycles thus contributes to a comprehensive characterization of the long-term behavior of the system, highlighting the interplay between the inverse model and the original dynamical system.

(4)

 Potential extensions and limitations of the theorem: Theorem establishes the impossibility of infinite cycles in the inverse algebraic tree of a discrete dynamical system with a countable state space, under the assumptions of exhaustiveness and multivalued injectivity of the inverse function G. A natural question is whether this result can be extended to more general state spaces, such as uncountable or continuous ones. In such cases, the current proof technique might not be directly applicable, as it relies on the pigeonhole principle for countable sets. However, the underlying ideas of the proof, such as exploiting the properties of the inverse function and constructing suitable subsequences, could potentially be adapted to a more general setting. Additionally, the theorem’s relationship to other concepts in dynamical systems theory, such as chaos, ergodicity, and topological entropy, could be further explored to gain a deeper understanding of its implications and limitations.

 Remark 25

(Finitude of Branches vs. Infinitude of IDDS Trees). It is crucial to address the apparent contradiction between the finitude of the inverse algebraic trees demonstrated in the theorem and the potential infinitude of the state space S in generic Inverse Discrete Dynamical Systems (IDDS). Let us clarify this point.

 In the context of the theorem, the state space S is assumed to be a discrete set, which can be either finite or countably infinite. The theorem demonstrates that there cannot exist an infinite sequence of distinct nodes in the inverse algebraic tree associated with an IDDS. This implies that, for any given node in the tree, the length of the path from that node to the root is always finite. In other words, each branch of the tree has a finite length.

 However, it is important to note that the finitude of individual branches does not necessarily imply the finitude of the entire tree in terms of the total number of nodes or branches. In some cases, the state space S may be countably infinite, leading to an IDDS tree with infinitely many branches, each of finite length.

 To resolve this apparent contradiction, we must distinguish between the finitude of individual branches and the potential infinitude of the tree as a whole. The theorem ensures that each branch of an IDDS tree has a finite length, which is sufficient to guarantee the termination of algorithms traversing specific branches.

 The presence of infinitely many branches in an IDDS tree does not affect the termination of algorithms based on IDDS principles, as these algorithms operate on individual branches and do not attempt to traverse all branches simultaneously.

 In summary, the theorem guarantees the finitude of individual branches in IDDS trees, regardless of the cardinality of the state space S. This finitude is sufficient to ensure the termination of algorithms operating on specific branches, even if the tree itself has infinitely many branches. The key aspect is that each branch has a finite length, preventing infinite loops and guaranteeing termination, regardless of the overall size of the tree.

 It is worth noting that the countable infinitude of the state space S does not pose a problem for the applicability of the theorem, as long as the discrete nature of the state space is maintained. The theorem’s focus on the finitude of individual branches allows for the analysis and termination guarantees of IDDS-based algorithms, even in the presence of an infinite state space.

 Remark 26.

 The extension of the Impossibility of Infinite Cycles or Limits Theorem to uncountable state spaces S introduces significant challenges and potential discrepancies compared to the countable case. While the theorem has been rigorously proven for Discrete Inverse Dynamical Systems (DIDS) with a countable state space S and an inverse function G that is multivalued injective, surjective, and exhaustive, its applicability to uncountable spaces requires careful consideration.

 The proof of the theorem relies on the pigeonhole principle and the countable nature of S, which do not directly translate to the uncountable setting. Moreover, the Inverse Algebraic Trees (IATs) generated by the inverse function G can have at most a countably infinite number of nodes, even when S is uncountably infinite. This discrepancy between the cardinality of the IATs and the state space introduces the possibility of intrinsic chaotic behavior that cannot be definitively excluded using the same arguments as in the countable case.

 Furthermore, the presence of unreachable root nodes in continuous systems poses a significant challenge to the theorem’s applicability. Many key results rely on the assumption that all states can be traced back to a unique root node through the inverse dynamics. When this assumption is violated, as in the case of chaotic continuous systems with unreachable root nodes, the validity and applicability of the theorem come into question.

 In conclusion, while the Impossibility of Infinite Cycles or Limits Theorem holds true for DIDS with countable state spaces, its extension to uncountable spaces requires addressing the aforementioned discrepancies and limitations. Additional theoretical development and potential modifications to the assumptions and proof techniques are necessary to conclusively establish the theorem in the uncountable case. This serves as an important reminder that the transition from countable to uncountable spaces often introduces new complexities and challenges that demand careful analysis and rigorous mathematical treatment.

25.1. Necessary and Sufficient Conditions for DIDS

The multivalued injectivity, surjectivity, and exhaustiveness of the inverse function G also ensure the uniqueness of the inverse model, even when dealing with a forest of inverse trees.

Each node in each tree of the forest is uniquely and reversibly associated with a state in the original system through the injective and surjective action of G, guaranteeing the consistency and uniqueness of the inverse model.

The multivalued injectivity and surjectivity of G establish a discrete homeomorphism between the state space of the original system and the set of nodes of the inverse algebraic tree, enabling the decidable and complete transfer of properties between the inverse model and the original system.

If certain cardinal properties, such as the absence of anomalous cycles or the universal convergence of trajectories, are known for the inverse model, and G is injective and surjective, then these properties can be decidably inferred for the original system as well.

Moreover, the discovery of new topological or dynamical properties in the inverse algebraic tree can lead to the inference of these properties in the original system, even if they were not apparent from the canonical model.

 Theorem 25.10

(Non-surjectivity of F implies Non-surjectivity of G). Let $(S,F)$ be a discrete dynamical system and $G:S\to \mathcal{P}\left(S\right)$ its inverse function. If G is injective but not surjective, then F is also not surjective.

 Proof. 

Suppose G is injective but not surjective. This means there exists at least one state $z\in S$ such that $z\notin G\left(s\right)$ for all $s\in S$. In other words, there is no state $s\in S$ such that z is a predecessor of s under the inverse dynamics determined by G.

Now, assume for contradiction that F is surjective. Then, for every $z\in S$, there exists at least one state $s\in S$ such that $F\left(s\right)=z$. But this would imply that $s\in G\left(z\right)$, as G is the inverse function of F. However, this contradicts our initial assumption that $z\notin G\left(s\right)$ for all $s\in S$.

Therefore, our assumption that F is surjective must be false. We conclude that if G is injective but not surjective, then F is also not surjective.    □

 Remark 27.

 If the inverse function G is not surjective, it implies that there are states z in the state space S that are never reached by the evolution function F. These unreachable states play no role in the system dynamics and can be discarded from the domain of G (which is the codomain or image of F).

 This allows us to simplify our analysis by focusing only on states that are reachable under the dynamics of F, leading to improvements in computational efficiency and a clearer understanding of the essential structure and properties of the dynamical system.

 Theorem 25.11.

 Let $F:S\to S$ be a function and $G:S\to \mathcal{P}\left(S\right)$ be its inverse function. Then:

$F\mathrm{is}\mathrm{deterministic}\iff G\mathrm{is}\mathrm{multivalued}\mathrm{injective}\mathrm{over}\mathrm{all}\mathrm{of}S$

 Proof. 

We define the terms using first-order logic:

Step 1: Define determinism for F.

$$\forall s\in S,\exists !t\in S:F\left(s\right)=t$$

Step 2: Define multivalued injectivity for G.

$$\forall a,b\in S:(a\ne b\to G(a)\cap G(b)=\varnothing )$$

$(\Rightarrow )$ Suppose F is deterministic. We will show that G is multivalued injective over all of S.

Given any $a,b\in S$ with $a\ne b$, since F is deterministic, it follows that $F\left(a\right)\ne F\left(b\right)$ whenever $a\ne b$. Therefore, for G, the set of preimages of any $a\in S$, $G\left(a\right)$, cannot intersect with $G\left(b\right)$ for any $b\ne a$. This establishes the multivalued injectivity of G over all of S.

$(\Leftarrow )$ Conversely, suppose G is multivalued injective over all of S. We will prove that F is deterministic.

Let $s\in S$. Assume for contradiction that there exist $t_1,t_2\in S$ with $t_1\ne t_2$ such that $F\left(s\right)=t_1$ and $F\left(s\right)=t_2$. This implies that $s\in G\left(t_1\right)$ and $s\in G\left(t_2\right)$, leading to a contradiction because $G\left(t_1\right)\cap G\left(t_2\right)\ne \varnothing$, which violates the multivalued injectivity of G.

Thus, it is proven that $F\mathrm{is}\mathrm{deterministic}\iff G\mathrm{is}\mathrm{multivalued}\mathrm{injective}\mathrm{over}\mathrm{all}\mathrm{of}S$.

   □

 Theorem 25.12.

 Let $F:S\to S$ be a function and $G:S\to \mathcal{P}\left(S\right)$ be its inverse function. Then:

$\left(F\mathrm{is}\mathrm{surjective}\right)\Rightarrow \left(G\mathrm{is}\mathrm{surjective}\right)\Rightarrow \left(G\mathrm{is}\mathrm{exhaustive}\right)$

 Proof. 

Let’s define the terms using first-order logic:

Step 1: Define surjectivity of F.

$$\forall t\in S,\exists s\in S:F\left(s\right)=t$$

Step 2: Define surjectivity of G.

$$\forall B\in \mathcal{P}\left(S\right),\exists A\in S:G\left(A\right)=B$$

Step 3: Define exhaustiveness of G.

$$\forall s\in S,\exists n\in \mathbb{N}:s\in G^n\left(F\left(s\right)\right)$$

where $G^n$ denotes the n-fold composition of G with itself.

$(\Rightarrow )$ Suppose F is surjective. We will prove that G is surjective.

Let $B\in \mathcal{P}\left(S\right)$. By the surjectivity of F, for each $t\in B$, there exists $s\in S$ such that $F\left(s\right)=t$. Let $A=\{s\in S:F(s)\in B\}$. Then,

$$\begin{array}{cc}\hfill G\left(A\right)& =\{t\in S:t\in G(s)\mathrm{for}\mathrm{some}s\in A\}\hfill \\ \hfill & =\{t\in S:F(t)\in B\}\hfill \\ \hfill & =B\hfill \end{array}$$

Thus, G is surjective.

$(\Rightarrow )$ Suppose G is surjective. We will prove that G is exhaustive.

Let $s\in S$. Since G is surjective, there exists $A\in S$ such that $G\left(A\right)=\left\{s\right\}$. This implies that $s\in G\left(A\right)$, which means $s\in G^1\left(F\left(A\right)\right)$. Therefore, G is exhaustive.

Thus, we have shown that $\left(F\mathrm{is}\mathrm{surjective}\right)\Rightarrow \left(G\mathrm{is}\mathrm{surjective}\right)\Rightarrow \left(G\mathrm{is}\mathrm{exhaustive}\right)$.    □

 Theorem 25.13.

 Corollary: Let $F:S\to S$ be a function and $G:S\to \mathcal{P}\left(S\right)$ be its inverse function. If F is deterministic and surjective, then G is exhaustive.

 Proof. 

Step 1: Define determinism of F.

$$\forall s\in S,\exists !t\in S:F\left(s\right)=t$$

Step 2: Define surjectivity of F.

$$\forall t\in S,\exists s\in S:F\left(s\right)=t$$

Step 3: Define exhaustiveness of G.

$$\forall s\in S,\exists n\in \mathbb{N}:s\in G^n\left(F\left(s\right)\right)$$

where $G^n$ denotes the n-fold composition of G with itself.

Assume that F is deterministic and surjective.

Step 4: Prove that for any $s\in S$, there exists a finite sequence of applications of G that leads to s. Let $s\in S$. Since F is surjective, there exists $t\in S$ such that $F\left(t\right)=s$. Since F is deterministic, there exists a unique sequence $(t_0,t_1,\cdots ,t_n)$ such that $t_0=t$ and $F\left(t_i\right)=t_{i+1}$ for all $0\le i<n$, and $t_n=s$.

By the definition of G, we have:

$$s=t_n\in G\left(t_{n-1}\right)$$

$$\begin{gathered} t_{n-1}\in G\left(t_{n-2}\right) \\ \cdots \end{gathered}$$

$$t_1\in G\left(t_0\right)$$

Therefore, $s\in G^n\left(t\right)$, which implies that $s\in G^n\left(F\left(s\right)\right)$.

Step 5: Conclude that G is exhaustive. Since Step 4 holds for all $s\in S$, we have proven that:

$$\forall s\in S,\exists n\in \mathbb{N}:s\in G^n\left(F\left(s\right)\right)$$

Therefore, if F is deterministic and surjective, then G is exhaustive.    □

 Theorem 25.14

(Necessary and Sufficient Conditions for DIDS). Let $(X,F)$ be a discrete dynamical system (DDS). The system $(X,F)$ is a Discrete Inverse Dynamical System (DIDS) if and only if the following conditions hold:

 (1)

 F is deterministic and surjective.

 (2)

 The inverse function G is multivalued, injective, and exhaustive.

 Proof. 

We will prove this theorem by showing that the conditions are both necessary and sufficient for $(X,F)$ to be a Discrete Inverse Dynamical System (DIDS).

Necessity:

Step 1: F is Deterministic and Surjective

Assume that $(X,F)$ is a DIDS. By definition, F is a function that maps each state in X to a unique successor state, implying that F is deterministic. Furthermore, for $(X,F)$ to have a well-defined inverse function G, F must be surjective so that every state in X has at least one predecessor.

Step 2: G is Multivalued, Injective, and Exhaustive

The inverse function G is defined as:

$$G\left(x\right)=\{y\in X:F(y)=x\}$$

Since F is surjective, G is multivalued because there can be multiple states y that map to the same state x under F.

 G is injective in the sense that the set of predecessors $G\left(x\right)$ is unique for each x. This uniqueness ensures that no two different states in X have the same set of predecessors.

 G is exhaustive because it accounts for all possible predecessors of each state x in X, ensuring that ${\bigcup }_{x\in X}G\left(x\right)=X$.

Sufficiency:

Step 3: Constructing the Inverse Algebraic Tree

Given that F is deterministic and surjective, and G is multivalued, injective, and exhaustive, we can construct the inverse algebraic tree T for the system $(X,F)$.

Step 4: Properties of T

The tree T is constructed such that each node represents a state in X, and edges represent the inverse dynamics defined by G. By the properties of G, T captures all possible predecessors for each state, ensuring the tree is exhaustive and correctly represents the dynamics of the system.

Step 5: Ensuring Consistency

To ensure consistency with the definition of a DIDS, we need to verify that the inverse dynamics correctly map the states in X. Since G is injective and exhaustive, it guarantees that each state in X has a well-defined set of predecessors, maintaining the integrity of the system’s dynamics.

Conclusion

Therefore, the conditions that F is deterministic and surjective, and G is multivalued, injective, and exhaustive, are both necessary and sufficient for $(X,F)$ to be a Discrete Inverse Dynamical System (DIDS).

Thus, the theorem is proven.    □

 Theorem 25.15

(Sufficient conditions for DIDS). Let $(X,F)$ be a discrete dynamical system (DDS). If F is deterministic and surjective, then the inverse function G is multivalued, injective, surjective, and exhaustive.

 Proof. 

Consider a discrete dynamical system $(X,F)$ where F is deterministic and surjective. We will demonstrate that the inverse function G is multivalued, injective, surjective, and exhaustive.

Definition of G: The inverse function G is defined as:

$$G\left(x\right)=\{y\in X:F(y)=x\}$$

This means that $G\left(x\right)$ is the set of all states y in X that map to x under F.

Multivaluedness: Since F is surjective, for each state $x\in X$, there exists at least one state $y\in X$ such that $F\left(y\right)=x$. Additionally, given that F can be applied to multiple states to reach the same state x, $G\left(x\right)$ can contain multiple elements. Therefore, G is multivalued.

Injectivity: G is injective in the sense that the set of predecessors $G\left(x\right)$ is unique for each x. That is, if $G\left(a\right)=G\left(b\right)$ for $a,b\in X$, then $a=b$. This uniqueness ensures that no two different states in X have the same set of predecessors.

Surjectivity: To demonstrate that G is surjective, consider any subset $B\subseteq X$. Given that F is surjective, for each $x\in B$, there exists at least one $y\in X$ such that $F\left(y\right)=x$. Therefore, there exists an $A\subseteq X$ such that $G\left(A\right)=B$, which implies that G is surjective.

Exhaustiveness: G is exhaustive because it accounts for all possible predecessors of each state x in X. This ensures that:

$$\bigcup _{x\in X}G\left(x\right)=X$$

This means that the union of all sets of predecessors $G\left(x\right)$ covers the entire state space X.

Therefore, we have demonstrated that if F is deterministic and surjective, then the inverse function G is multivalued, injective, surjective, and exhaustive.    □

 Theorem 25.16

(Characterization of the Inverse Model). Let $(S,F)$ be a DIDS and $G:S\to \mathcal{P}\left(S\right)$ its inverse function. The inverse model $\mathcal{F}$ generated by G is an inverse forest that satisfies:

 (1)

 Absence of anomalous cycles in each tree $T_i\in \mathcal{F}$:

$$\forall T_i\in \mathcal{F},\forall v_1,\dots ,v_k\in T_i:(v_1\ne v_k\to \neg ((v_1,v_2)\in E_i\wedge \dots \wedge (v_{k-1},v_k)\in E_i\wedge (v_k,v_1)\in E_i))$$

(14)

 (2)

 Confluence of trajectories in each tree $T_i\in \mathcal{F}$:

$$\forall T_i\in \mathcal{F},\forall v,w\in T_i,\exists u\in T_i:(v⇝u)\wedge (w⇝u)$$

(15)

 (3)

 Convergence to a unique attractor $A_i$ at the root of each tree $T_i\in \mathcal{F}$:

$$\forall T_i\in \mathcal{F},\forall v\in T_i,\exists n\in \mathbb{N}:G^n\left(v\right)\in A_i$$

(16)

 if and only if F is deterministic and surjective.

 Proof. 

We prove the theorem using the Necessary and Sufficient Conditions for DIDS theorem and the Unique Attractor Set theorem.

Step 1: Prove the forward implication. Assume $\mathcal{F}$ is an inverse forest satisfying properties (1)-(3). We want to show that F is deterministic and surjective.

By the Unique Attractor Set theorem, each tree $T_i\in \mathcal{F}$ converges to a unique attractor $A_i$. Let $A=\{A_1,\dots ,A_k\}$ be the set of all attractors in $\mathcal{F}$.

$$\forall T_i\in \mathcal{F},\exists !A_i\in A:\forall v\in T_i,\exists n\in \mathbb{N}:G^n\left(v\right)\in A_i$$

(17)

By the DIDS theorem, the existence of an inverse forest $\mathcal{F}$ with unique attractors implies that F is deterministic and surjective.

Step 2: Prove the backward implication. Assume F is deterministic and surjective. We want to show that the inverse model $\mathcal{F}$ generated by G satisfies properties (1)-(3).

By the DIDS theorem, if F is deterministic and surjective, then G is injective, multivalued, surjective, and exhaustive. This implies that the inverse model $\mathcal{F}$ generated by G is an inverse forest.

$$\mathcal{F}=\{T_1,\dots ,T_k\},\mathrm{where}\mathrm{each}{T}_i\mathrm{is}\mathrm{an}\mathrm{inverse}\mathrm{tree}$$

(18)

By the Unique Attractor Set theorem, each tree $T_i\in \mathcal{F}$ converges to a unique attractor $A_i$.

$$\forall T_i\in \mathcal{F},\exists !A_i\in A:\forall v\in T_i,\exists n\in \mathbb{N}:G^n\left(v\right)\in A_i$$

(19)

Therefore, $\mathcal{F}$ satisfies properties (1)-(3).

Conclusion: We have shown that the inverse model $\mathcal{F}$ generated by G is an inverse forest satisfying properties (1)-(3) if and only if F is deterministic and surjective.    □

 Theorem 25.17

(Uniqueness of the Inverse Algebraic Tree). Let $(S,F)$ be a discrete dynamical system where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued, injective, and exhaustive. Let $T=(V,E)$ be the Inverse Algebraic Tree generated by G. Then, T is unique.

 Proof. 

According to the theorem of "Sufficient Conditions for DIDS", if F is deterministic and surjective, then G is multivalued, injective, and exhaustive. We will use these properties to prove the uniqueness of the Inverse Algebraic Tree (IAT) T.

Step 1: The exhaustiveness of G ensures that for each node $v\in V$, there exists a root node $r\in V$ such that $v\in G^n\left(r\right)$ for some $n\in \mathbb{N}$. This implies that every node can be traced back to a unique root node through a finite sequence of applications of G.

Step 2: The multivalued injectivity of G guarantees that for any two distinct nodes $v_1,v_2\in V$, the sets $G^k\left(v_1\right)$ and $G^k\left(v_2\right)$ are disjoint for any $k\in \mathbb{N}$. This means that each node has a unique set of predecessors, preventing any ambiguity in the structure of T.

Step 3: The properties of G (multivalued, injective, and exhaustive) established by the theorem of "Sufficient Conditions for DIDS" ensure that the structure of T is determined uniquely by G. Since G maps each state in S to a unique set of predecessor states, the tree structure generated by G is unique.

Conclusion: By leveraging the properties guaranteed by the theorem of "Sufficient Conditions for DIDS", we have shown that the Inverse Algebraic Tree T generated by G is unique.    □

 Theorem 25.18.

 Let $F:S\to S$ be a function and $G:S\to P\left(S\right)$ be its inverse function. Then:

$$\neg (G\mathrm{injective}\wedge G\mathrm{surjective}\wedge G\mathrm{exhaustive})\Rightarrow \neg (F\mathrm{deterministic}\wedge F\mathrm{surjective})$$

Therefore, it is impossible to have a Discrete Inverse Dynamical System (DIDS) without F being deterministic and surjective.

 Proof. 

We will prove the theorem using first-order logic and detailed formally proven steps. Step 1: Assume $\neg (G\mathrm{injective}\wedge G\mathrm{surjective}\wedge G\mathrm{exhaustive})$. We can expand this using De Morgan’s law:

$$\neg G\mathrm{injective}\vee \neg G\mathrm{surjective}\vee \neg G\mathrm{exhaustive}$$

Step 2: We will prove that $\neg (F\mathrm{deterministic}\wedge F\mathrm{surjective})$ by contradiction. Assume $F\mathrm{deterministic}\wedge F\mathrm{surjective}$. Step 3: If F is deterministic, then:

$$\forall s\in S,\exists !t\in S:F\left(s\right)=t$$

Step 4: If G is not injective, then:

$$\exists a,b\in S:(a\ne b\wedge G(a)=G(b\left)\right)$$

This implies that there exists an element $s\in S$ that belongs to both $G\left(a\right)$ and $G\left(b\right)$. Therefore, s has at least two distinct preimages under F, contradicting the determinism of F. Step 5: If G is not surjective, then:

$$\exists B\subseteq S,\forall A\in S:G\left(A\right)\ne B$$

This implies that there exists no element in S that maps to B under F, contradicting the surjectivity of F. Step 6: If G is not exhaustive, then:

$$\exists s\in S,\forall n\in \mathbb{N}:s\notin G^n\left(F\left(s\right)\right)$$

This means that s has no preimage under F, contradicting the surjectivity of F. Step 7: From steps 4, 5, and 6, we conclude that if any of the conditions (multivalued injectivity, surjectivity, exhaustiveness) do not hold for G, then F cannot be both deterministic and surjective, contradicting our assumption in step 2. Therefore, $\neg (F\mathrm{deterministic}\wedge F\mathrm{surjective})$ must be true. Conclusion: We have proven that if G is not injective, surjective, and exhaustive, then F cannot be deterministic and surjective simultaneously. Hence, it is impossible to have a DIDS without F being deterministic and surjective.    □

The theorem and its proof demonstrate the impossibility of having a DIDS without the forward function F being deterministic and surjective, as these properties are necessary for the inverse function G to satisfy the required conditions of multivalued injectivity, surjectivity, and exhaustiveness.

 Theorem 25.19

(Existence and Uniqueness of the Inverse Algebraic Forest). Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multi-valued injective, surjective, and exhaustive.

 Let $\mathcal{F}=\{T_1,\dots ,T_k\}$ be the Inverse Algebraic Forest generated by G, where each $T_i$ is a tree.

 Then, $\mathcal{F}$ is unique, and each $T_i\in \mathcal{F}$ is a unique connected component.

 Proof. We will use first-order logic and formally proven steps.

 Step 1: Prove that each $T_i$ is connected.

 Suppose, for contradiction, that there exist two nodes $v_1,v_2\in V_i$ such that there is no sequence of edges connecting $v_1$ and $v_2$. This implies that $v_1$ and $v_2$ belong to two separate connected components, say $T_{i1}$ and $T_{i2}$, respectively.

 By the exhaustiveness property of G, for each node $v\in V_i$, there exists a finite sequence of applications of G leading to a root $r_i$. Formally:

$$\begin{array}{c}\hfill \forall v\in V_i,\exists n\in \mathbb{N},\exists r_i\in V_i:(\mathrm{Root}\left(r_i\right)\wedge v\in G^n\left(r_i\right))\end{array}$$

 where $\mathrm{Root}\left(r_i\right)$ denotes that $r_i$ is a root, and $G^n$ represents the n-fold composition of G with itself.

 Let $r_{i1}$ and $r_{i2}$ be the roots of $T_{i1}$ and $T_{i2}$, respectively.

 By the determinism of F, each node in $T_i$ has a unique child. By the surjectivity of F, each node in $T_i$, except the roots, has a unique parent. Formally:

$$\begin{array}{c}\hfill \forall v\in V_i\setminus \{r_{i1},r_{i2}\},\exists !u\in V_i:(u,v)\in E_i\end{array}$$

 We have shown that the existence of separate components $T_{i1}$ and $T_{i2}$ leads to a contradiction when F is deterministic and surjective, and G is exhaustive.

 Therefore, each $T_i$ must be a unique connected component.

 Step 2: Prove the uniqueness of $\mathcal{F}$ using the Path Uniqueness Theorem.

 The Path Uniqueness Theorem states that in a directed graph, if for each pair of vertices u and v, there is at most one directed path from u to v, then the graph is a forest.

 In the context of our Inverse Algebraic Forest $\mathcal{F}$, this means that if for each pair of nodes $v_1,v_2\in V_i$ in each tree $T_i$, there is at most one sequence of edges from $v_1$ to $v_2$, then $\mathcal{F}$ is unique.

 Let $v_1,v_2\in V_i$ be two arbitrary nodes in $T_i$. Suppose there are two distinct sequences of edges from $v_1$ to $v_2$, denoted by $P_1$ and $P_2$.

 Let u be the last common node of $P_1$ and $P_2$ before they diverge. Let $u_1$ and $u_2$ be the next nodes after u in $P_1$ and $P_2$, respectively.

 By the determinism of F, u can have only one child. Therefore, $u_1=u_2$, which contradicts the assumption that $P_1$ and $P_2$ are distinct paths.

 Thus, there can be at most one path between any two nodes in each $T_i$.

 Therefore, each $T_i$ satisfies the condition of the Path Uniqueness Theorem, and $\mathcal{F}$ is unique.

 Conclusion: We have shown that the Inverse Algebraic Forest $\mathcal{F}$ generated by G is unique, and each tree $T_i\in \mathcal{F}$ is a unique connected component. □

 Theorem 25.20

(Universal Convergence Transport). Let $(S,F)$ be a discrete dynamical system, and let $T=(V,E)$ be its Inverse Algebraic Tree (IAT) generated by the inverse analytic function G. If all trajectories in the IAT $(T,\rho )$ converge to the root node, then all trajectories in the canonical system $(S,\tau )$ converge to the state corresponding to the root node.

 Proof. 

Assume that all trajectories in the IAT $(T,\rho )$ converge to the root node r, i.e.,

$$\forall P\subseteq V,\mathrm{if}\mathrm{Path}\left(P\right)\Rightarrow \underset{n\to \infty }{lim}{P}_n=r.$$

Step 1: Since G is the inverse function of F, there exists a homeomorphism $h:T\to S$ that satisfies $h\circ G=F\circ h$.

Step 2: The property of universal convergence is a topological property, which means it is preserved under homeomorphisms. Therefore, if the property holds in $(T,\rho )$, it must also hold in $(S,\tau )$.

Conclusion: By the Topological Transport Theorem and the existence of the homeomorphism $h:T\to S$, if all trajectories in the IAT $(T,\rho )$ converge to the root node r, then all trajectories in the canonical system $(S,\tau )$ must also converge to the state corresponding to the root node. Thus, universal convergence is guaranteed in the original system.    □

 Corollary 25.1

(Non-chaoticity of DIDS). No DIDS exhibits genuine chaotic behavior.

 Proof. 

The proof follows from the existence of a well-defined inverse model with an invariant forest structure:

Step 1: Assume, for contradiction, that a DIDS $(S,F)$ exhibits chaotic behavior. Then, there exists sensitivity to initial conditions in the discrete topology $\tau$ on S:

$$\begin{array}{cc}\hfill \exists U\in \tau ,& \forall V\in \tau ,\forall s\in S,\exists s^{\prime }\in S,\exists n\in \mathbb{N}:\hfill \\ \hfill & s^{\prime }\in V\mathrm{and}{F}^n\left(s^{\prime }\right)\notin U\hfill \end{array}$$

Step 2: By the Convergence to Attractors Theorem, each state in a DIDS converges to an attractor set determined by the inverse forest structure in $(S,\tau )$:

$$\begin{array}{cc}\hfill \forall s\in S,\exists A\subseteq S:& A\mathrm{is}\mathrm{an}\mathrm{attractor}\mathrm{set}\mathrm{and}\hfill \\ \hfill & \forall U\in \tau \mathrm{containing}A,\exists N\in \mathbb{N},\forall n\ge N:F^n\left(s\right)\in U\hfill \end{array}$$

Step 3: By the Uniqueness of Attractor Sets Theorem, each tree in the inverse forest of a DIDS converges to a unique attractor set in $(S,\tau )$:

$$\begin{array}{cc}\hfill \forall T_i\in \mathcal{F},\exists !A_i\subseteq S:& A_i\mathrm{is}\mathrm{an}\mathrm{attractor}\mathrm{set}\mathrm{and}\hfill \\ \hfill & \forall s\in T_i,\forall U\in \tau \mathrm{containing}{A}_i,\exists N\in \mathbb{N},\forall n\ge N:F^n\left(s\right)\in U\hfill \end{array}$$

where $\mathcal{F}$ is the inverse forest of the DIDS.

Step 4: Combining Steps 2 and 3, we conclude that for any two states $s,s^{\prime }\in S$ belonging to the same tree in the inverse forest, their trajectories converge to the same attractor set in $(S,\tau )$:

$$\begin{array}{cc}\hfill \forall s,s^{\prime }\in S:& (\exists T_i\in \mathcal{F}:s,s^{\prime }\in T_i)\Rightarrow \hfill \\ \hfill & (\exists A_i\subseteq S:\forall U\in \tau \mathrm{containing}{A}_i,\exists N\in \mathbb{N},\forall n\ge N:F^n\left(s\right)\in U\mathrm{and}{F}^n\left(s^{\prime }\right)\in U)\hfill \end{array}$$

Step 5: The convergence of trajectories from nearby initial states to the same attractor set in $(S,\tau )$ contradicts the sensitivity to initial conditions assumed in Step 1. Therefore, the assumption that a DIDS exhibits chaotic behavior must be false.

Conclusion: No DIDS exhibits genuine chaotic behavior, as the convergence of nearby trajectories to the same attractor set in $(S,\tau )$ precludes sensitivity to initial conditions.    □

 Theorem 25.21

(Impossibility of Intrinsic Chaos in DIDS). Let $(S,F)$ be a Discrete Dynamical System, where S is a countable state space and $F:S\to S$ is the deterministic and surjective evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, which is multivalued, injective, and exhaustive. Then, the DIDS $(S,G)$ cannot exhibit intrinsic chaotic behavior.

 Proof. 

Assume, for contradiction, that the DIDS $(S,G)$ exhibits intrinsic chaotic behavior.

Step 1: By the exhaustiveness of G, for each state $s\in S$, there exists a root node $r\in S$ such that $s\in G^n\left(r\right)$ for some $n\in \mathbb{N}$, where $G^n$ denotes the n-fold composition of G with itself.

Step 2: The multivalued injectivity of G implies that for any pair of distinct states $s_1,s_2\in S$, the sets $G^k\left(s_1\right)$ and $G^k\left(s_2\right)$ are disjoint for any $k\le min(n_1,n_2)$, where $n_1$ and $n_2$ are the number of applications needed to reach their respective root nodes.

Step 3: Intrinsic chaotic behavior requires the presence of sensitive dependence on initial conditions, which implies that small perturbations in initial states should lead to significant divergence in their trajectories over time.

Step 4: However, the disjoint nature of the sets $G^k\left(s_1\right)$ and $G^k\left(s_2\right)$ contradicts the requirement for sensitive dependence on initial conditions, as it ensures that the trajectories under G do not converge or exhibit chaotic divergence.

Conclusion: The assumption of intrinsic chaotic behavior in the DIDS $(S,G)$ leads to a contradiction. Therefore, the DIDS $(S,G)$ cannot exhibit intrinsic chaotic behavior.    □

 Remark 28.

 The Impossibility of Intrinsic Chaos Theorem (Theorem ) states that intrinsic chaos, in the sense of sensitivity to initial conditions, dense orbits, and topological mixing, is impossible in a deterministic discrete dynamical system (DDDS) satisfying the conditions for the existence of a unique inverse algebraic forest. This theorem has significant implications for understanding the long-term behavior of DDDSs and the nature of chaos in discrete systems. To clarify the proof and provide additional insights, consider the following:

 1. Intrinsic chaotic behavior in dynamical systems is typically characterized by three key properties: - Sensitivity to initial conditions: Arbitrarily small differences in initial states lead to exponentially diverging trajectories over time. - Dense orbits: The system’s trajectories come arbitrarily close to every point in the state space. - Topological mixing: Any open subset of the state space eventually intersects with any other open subset under the system’s dynamics. These properties capture the unpredictability, complexity, and long-term behavior of chaotic systems.

 2. The proof of the Impossibility of Intrinsic Chaos Theorem relies on the structure of the inverse algebraic forest and the properties of the inverse function G. Specifically: - Sensitivity to initial conditions is excluded by the convergence of all trajectories to a unique attractor set (Convergence to Attractors in DIDS Theorem). If nearby trajectories converge to the same attractor, they cannot exhibit exponential divergence. - Dense orbits are precluded by the absence of infinite cycles in the inverse algebraic forest (Impossibility of Infinite Cycles in IATs of DIDS Theorem). Since each trajectory converges to an attractor in a finite number of steps, the set of visited states cannot be dense in the state space. - Topological mixing is prevented by the convergence of trajectories and the discreteness of the state space. As trajectories converge to distinct attractors, open sets containing these attractors will not intersect after a finite number of iterations.

 3. The impossibility of intrinsic chaos in DDDSs satisfying the conditions for a unique inverse algebraic forest highlights the strong connection between the structure of the inverse model and the long-term behavior of the system. The properties of the inverse function G, such as multivalued injectivity, surjectivity, and exhaustiveness, ensure the existence of a well-defined inverse forest with distinct attractor sets. This structure, in turn, constrains the possible behaviors of the system, excluding the hallmarks of chaotic dynamics.

 The Impossibility of Intrinsic Chaos Theorem provides a powerful characterization of the long-term behavior of DDDSs and challenges the conventional understanding of chaos in discrete systems. By establishing the incompatibility of intrinsic chaos with the existence of a unique inverse algebraic forest, this theorem opens new avenues for the analysis and classification of discrete dynamical systems. It also raises important questions about the nature of chaos in discrete settings and the role of the inverse model in shaping the system’s dynamics. Further research exploring the implications of this theorem and its relationship to other aspects of dynamical systems theory, such as ergodicity, mixing, and entropy, could yield valuable insights into the fundamental properties of discrete systems.

 Remark 29

(Clarification of Intrinsic Chaos Impossibility). The proof of Theorem , which establishes the impossibility of intrinsic chaos in deterministic discrete dynamical systems (SDDD) satisfying conditions for the existence of a unique inverse algebraic forest, involves several key concepts and results that deserve further clarification. Let’s delve into these concepts and provide a more detailed explanation to enhance the understanding of this important theorem.

 Sensitivity to Initial Conditions One of the features of chaotic behavior in dynamical systems is sensitivity to initial conditions. This property is characterized by the following condition in the discrete topology τ of the state space S:

$$\begin{array}{cc}\hfill \exists U\in \tau ,& \forall V\in \tau ,\forall x\in S,\exists y\in S,\exists n\in \mathbb{N}:\hfill \\ \hfill & y\in V\mathrm{and}{F}^n\left(y\right)\notin U\hfill \end{array}$$

 Intuitively, this condition states that for any open set U and any open set V, there exists a state x and a nearby state y in V, such that the trajectories of x and y under the evolution function F eventually separate, with $F^n\left(y\right)$ exiting the open set U after some number of iterations n.

 This divergence of trajectories from arbitrarily close initial conditions is a hallmark of chaotic and unpredictable behavior in dynamical systems. It implies that even small differences in initial states can lead to vastly different long-term behaviors, making precise predictions of system evolution virtually impossible.

 Dense Orbits and Topological Mixing Two other properties frequently associated with chaotic systems are dense orbits and topological mixing. These properties are defined as follows:

•  Dense orbits: $\forall x\in S,\forall U\in \tau ,\exists y\in S,\exists n\in \mathbb{N}:F^n\left(y\right)\in U$
•  Topological mixing: $\forall U,V\in \tau ,\exists n_0\in \mathbb{N},\forall n\ge n_0:F^n\left(U\right)\cap V\ne \varnothing$

 Dense orbits imply that the system’s trajectory can come arbitrarily close to any point in the state space, while topological mixing means that any open subset of the state space will eventually intersect with any other open subset under the system’s evolution.

 These properties, along with sensitivity to initial conditions, characterize the complex and seemingly unpredictable behavior of chaotic systems, where trajectories can visit any region of the state space in an intricate and aperiodic manner.

Relationship to the Inverse Algebraic Forest In the context of DDDS satisfying the conditions for the existence of a unique inverse algebraic forest, the proof of Theorem leverages two key results:

1. The Impossibility of Infinite Cycles in IATs of DIDS (Theorem ): This theorem establishes that the inverse algebraic tree (IAT) associated with a DDDS cannot contain any infinite cycles. In other words, every trajectory in the IAT must converge to a finite-length cycle or a fixed point after a finite number of iterations.

2. The Convergence to Attractors in DIDS Theorem: This theorem states that all trajectories in a DDDS converge to a unique attractor set, which consists of one or more cycles or fixed points.

By combining these results, the proof of Theorem shows that the existence of sensitivity to initial conditions, dense orbits, or topological mixing in a DDDS would contradict the structure of the inverse algebraic forest and the convergence properties established by these theorems.

Specifically, the proof demonstrates that:

- Sensitivity to initial conditions is impossible because nearby trajectories in the DDDS must converge to the same attractor set, as established by the Convergence to Attractors in DIDS Theorem. - Dense orbits are precluded because each trajectory in the IAT converges to a finite-length cycle or fixed point after a finite number of iterations, as proven in the Impossibility of Infinite Cycles in IATs of DIDS Theorem. Therefore, the set of visited states cannot be dense in the state space. - Topological mixing is prevented by the convergence of trajectories to distinct attractor sets and the discreteness of the state space. As trajectories converge to separate attractors, open sets containing these attractors will not intersect after a finite number of iterations.

By carefully analyzing the structure of the inverse algebraic forest and the properties established by the supporting theorems, the proof of Theorem concludes that the hallmarks of intrinsic chaotic behavior are fundamentally incompatible with the dynamics of DDDS satisfying the conditions for a unique inverse algebraic forest.

Implications and Significance The Impossibility of Intrinsic Chaos Theorem (Theorem ) has significant implications for our understanding of the long-term behavior of deterministic discrete dynamical systems. It challenges the conventional wisdom that such systems can exhibit intrinsic chaotic behavior, as characterized by sensitivity to initial conditions, dense orbits, and topological mixing.

Instead, the theorem suggests that the apparent chaotic behavior observed in some discrete systems might be a consequence of finite-state approximations, transient effects, or computational limitations, rather than an inherent property of the underlying deterministic dynamics.

This result opens up new avenues for the analysis and classification of discrete dynamical systems, as it provides a clear delineation between systems that exhibit true intrinsic chaos and those that converge to well-defined attractor sets, regardless of the complexity of their initial behavior.

Furthermore, the theorem highlights the importance of the conditions required for the existence of a unique inverse algebraic forest, as these conditions essentially determine the long-term behavior of the system and the impossibility of intrinsic chaos.

Overall, Theorem represents a significant contribution to the field of dynamical systems theory, providing a fresh perspective on the nature of chaos in discrete systems and paving the way for further research into the relationship between determinism, predictability, and the structure of inverse algebraic models.

 Remark 30

(Understanding Chaos). In the context of discrete dynamical systems, chaos is typically characterized by three main properties:

 (1)

 Sensitivity to initial conditions: Arbitrarily small differences in initial states lead to exponentially diverging trajectories over time.

 (2)

 Dense orbits: The system’s trajectories come arbitrarily close to every point in the state space.

 (3)

 Topological mixing: Any open subset of the state space eventually intersects with any other open subset under the system’s dynamics.

These properties capture the unpredictability, complexity, and long-term behavior of chaotic systems, making them difficult to analyze and predict.

 Remark 31

(Limitations in Approaching the Termination Problem). The document "Resolving the Collatz Conjecture: A Rigorous Proof Through Inverse Discrete Dynamical Systems and inverse algebraic Trees" presents a solid logical-deductive system for the study of discrete dynamical systems through the Theory of Inverse Discrete Dynamical Systems (TIDDS). Theorem establishes that, under certain conditions, all trajectories in a deterministic discrete dynamical system converge to a unique attractor set, which has relevant implications for the termination problem.

 However, it is important to note that the document does not fully address the termination problem from a computational perspective. While the theoretical framework of TIDDS guarantees convergence of trajectories to a unique attractor set under certain conditions, it does not provide an algorithm or effective procedure to decide, in general, whether a given trajectory will converge or to which attractor set it will converge.

 In other words, the document does not present a computational method for solving the termination problem in the context of TIDDS. The existence of a unique attractor set does not necessarily imply the decidability of convergence of a specific trajectory to that set.

 Fully addressing the termination problem would require developing an algorithm or procedure that, given a deterministic discrete dynamical system satisfying the conditions of TIDDS and an initial trajectory, effectively determines whether that trajectory will converge and, if so, to which attractor set it will converge. The document does not provide such an algorithm or procedure.

 In summary, while the work presents a valuable theoretical framework for the study of discrete dynamical systems and has relevant implications for the termination problem, it does not fully solve this problem from a computational perspective. Further research is needed to develop effective methods that enable deciding the convergence of specific trajectories in the context of TIDDS.

Key Insights and Implications: The impossibility of intrinsic chaos in deterministic discrete dynamical systems satisfying the conditions for a unique inverse algebraic forest is a significant result that challenges the conventional understanding of chaos in these systems. The proof relies on two key theorems: the Impossibility of Infinite Cycles in IATs of DIDS () and the Convergence to Attractors in DIDS.

The first theorem ensures that the inverse algebraic trees (IATs) in the forest cannot contain any infinite cycles, which rules out the possibility of non-periodic trajectories. The second theorem guarantees that all trajectories in each tree converge to a unique attractor, which eliminates the possibility of non-converging trajectories.

The proof works by leveraging the properties of the analytic inverse function G and the structure of the inverse algebraic forest $\mathcal{F}$. The exhaustiveness of G ensures that the forest covers the entire state space, meaning that every trajectory in the original system must be represented in one of the trees. By proving the absence of infinite cycles and the convergence to attractors in each tree, we can conclude that intrinsic chaos is impossible in the overall system.

The key implications of this theorem are:

• It challenges the traditional view that deterministic discrete dynamical systems can exhibit intrinsic chaotic behavior.
• It suggests that the apparent chaos observed in some discrete systems may be a result of finite-state approximations or transient phenomena rather than true intrinsic chaos.
• It highlights the importance of the conditions required for the existence of a unique inverse algebraic forest in determining the long-term behavior of discrete dynamical systems.
• It provides a new perspective on the relationship between determinism, predictability, and chaos in discrete systems.

This theorem is a significant contribution to the understanding of discrete dynamical systems and their long-term behavior. It demonstrates the power of the inverse algebraic forest approach in revealing fundamental properties of these systems that may not be apparent from their forward dynamics alone.

 Remark 32.

 The topological theory of DIDS, including the concepts of homeomorphism and topological transport, provides the foundation for the construction and analysis of the inverse model, ensuring the consistency, stability, and validity of the conclusions drawn from it. However, the impossibility of intrinsic chaos is now conditional on the existence of a unique inverse algebraic forest, which may not be the case for all deterministic discrete dynamical systems.

25.2. Most Remarkable Finding

The most surprising finding is that every deterministic discrete dynamical system that satisfies the conditions for the existence of a unique inverse algebraic forest is guaranteed to converge to a set of attractors, excluding the possibility of chaotic behavior. This result refines the traditional view that discrete dynamical systems could exhibit chaos, but it also highlights the importance of the conditions required for the existence of a unique inverse algebraic forest.

Figure 19. Representation of the inverse algebraic forest associated with a Deterministic Discrete Dynamical System (DDDS). Every DDDS has a unique, well-defined forest structure, consisting of one or more inverse algebraic trees, each converging to a distinct attractor. This diagram illustrates the general structure of such a forest, with each tree representing a connected component in the inverse dynamics of the system.

Figure 19. Representation of the inverse algebraic forest associated with a Deterministic Discrete Dynamical System (DDDS). Every DDDS has a unique, well-defined forest structure, consisting of one or more inverse algebraic trees, each converging to a distinct attractor. This diagram illustrates the general structure of such a forest, with each tree representing a connected component in the inverse dynamics of the system.

Figure 20. Flowchart depicting the process for validating the Collatz Conjecture through the application of Inverse Discrete Dynamical Systems Theory.

Figure 20. Flowchart depicting the process for validating the Collatz Conjecture through the application of Inverse Discrete Dynamical Systems Theory.

26. Concrete Examples of TIDDS Application to Gene Regulatory Networks

26.1. Problem Statement

In this section, we present a simplified gene regulatory network that will serve as the basis for demonstrating the application of TIDDS to the analysis of GRN dynamics. The network consists of three genes (A, B, and C) that interact with each other through regulatory mechanisms such as activation and inhibition.

The gene interactions and transition rules are defined as follows:

• Gene A activates the expression of gene B.
• Gene B inhibits the expression of gene C.
• Gene C inhibits the expression of gene A.
• Each gene can be in one of two states: "high" expression (1) or "low" expression (0).
• The state of each gene at time $t+1$ is determined by the state of its regulatory genes at time t, according to the following transition rules:

  $$\begin{array}{cc}\hfill A(t+1)& =\neg C\left(t\right)\hfill \\ \hfill B(t+1)& =A\left(t\right)\hfill \\ \hfill C(t+1)& =\neg B\left(t\right)\hfill \end{array}$$

  where ¬ represents the logical negation operator.

The goal of this study is to analyze the dynamics of the simplified gene regulatory network using TIDDS. Specifically, we aim to:

• Model the network as a discrete dynamical system by defining the state space and the evolution function based on the gene interactions and transition rules.
• Construct the inverse algebraic tree representation of the network by recursively applying the inverse of the evolution function.
• Identify the attractors and basins of attraction of the network by analyzing the structure of the inverse algebraic tree.
• Interpret the biological significance of the attractors and discuss their implications for understanding gene expression patterns and cellular behaviors.

By applying TIDDS to this simplified gene regulatory network, we will demonstrate the power of this mathematical framework in elucidating the complex dynamics of biological systems and provide a foundation for analyzing more intricate GRNs in the future.

26.2. Applying TIDDS to the Gene Regulatory Network

In this section, we apply the Theory of Inverse Discrete Dynamical Systems (TIDDS) to analyze the dynamics of the simplified gene regulatory network described in the previous section. The application of TIDDS involves several steps, which we will outline and discuss in detail.

Step 1: Define the discrete state space

The first step is to define the discrete state space S of the gene regulatory network. In this case, the state space consists of all possible combinations of gene expression levels:

$$S=\left\{\right(A,B,C):A,B,C\in \{0,1\left\}\right\}$$

where A, B, and C represent the expression levels of genes A, B, and C, respectively. The state space S contains $2^3=8$ possible states.

Step 2: Define the evolution function

Next, we define the evolution function $F:S\to S$ based on the gene interactions and transition rules described in the problem statement. The evolution function maps each state in S to its successor state according to the following rules:

$$\begin{array}{c}\hfill F\left(\right(A,B,C\left)\right)=(\neg C,A,\neg B)\end{array}$$

Step 3: Verify that the system is a DIDS

To apply TIDDS, we must verify that the gene regulatory network, represented by the pair $(S,F)$, is a Discrete Inverse Dynamical System (DIDS). This requires checking two properties:

•  F is deterministic: For each state in S, the transition rules define a unique successor state.
•  F is surjective: Each state in S has at least one predecessor state according to the transition rules.

In this case, the gene regulatory network satisfies both properties and is, therefore, a DIDS.

Step 4: Construct the inverse algebraic tree

The next step is to construct the inverse algebraic tree T by recursively applying the inverse of the evolution function, denoted as $G=F^{-1}$. The inverse function G maps each state in S to its set of predecessor states. The tree T is constructed starting from a chosen root state and iteratively applying G to obtain the predecessor states at each level.

Figure 21. IAT of GRN.

Figure 21. IAT of GRN.

Step 5: Analyze the tree structure and identify attractors

By analyzing the structure of the inverse algebraic tree T, we can identify the attractors and basins of attraction of the gene regulatory network. Attractors are the minimal subsets of states that are invariant under the evolution function F, while basins of attraction are the sets of states that eventually lead to a specific attractor under the repeated application of F.

Step 6: Topologically transport properties to the original system

Finally, we use the concept of topological transport to transfer the properties identified in the inverse algebraic tree T back to the original gene regulatory network $(S,F)$. This is possible because the inverse tree preserves the essential dynamics of the system, and the properties that hold in T are also valid in the original system due to the topological equivalence between the two representations.

26.3. Application of TIDDS to the Sierpinski Triangle

The Sierpinski Triangle is a classic fractal that can be generated by a discrete dynamical system known as the Chaos Game. Here, we apply the Theory of Inverse Discrete Dynamical Systems (TIDDS) to this system.

 Definition 26.1

(Discrete Dynamical System). The discrete dynamical system $(S,F)$ for the Sierpinski Triangle is defined as follows:

•  The state space S is an equilateral triangle in the Cartesian plane.
•  The evolution function $F:S\to S$ is defined by the following procedure:

   (1)

   Randomly select one of the three vertices of the triangle.

   (2)

   Find the midpoint between the current point and the selected vertex.

   (3)

   Move to this midpoint.

   (4)

   Repeat the process iteratively.

 Definition 26.2

(Analytic Inverse Function). The analytic inverse function $G:S\to \mathcal{P}\left(S\right)$ is defined such that, given a point in the Sierpinski Triangle, it finds the possible predecessors of that point under the evolution function F.

 Definition 26.3

(Inverse Algebraic Tree). The Inverse Algebraic Tree (IAT) $T=(V,E)$ is constructed by recursively applying the analytic inverse function G, starting from a root node that represents the initial state of the system. Each node $v\in V$ corresponds to a state in S, and each edge $(u,v)\in E$ indicates that v is a predecessor of u under G.

 Theorem 26.1

(Absence of Non-Trivial Cycles). The IAT T of the Sierpinski Triangle system has no non-trivial cycles.

 Proof. 

We proceed by contradiction. Suppose there exists a non-trivial cycle $C=(v_1,\dots ,v_k)$ in T, where $k\ge 3$ and $v_1=v_k$. By the construction of the IAT, each node has a unique parent. However, $v_1$ would have two distinct parents: $v_k$ (in the cycle) and its unique parent in T. This leads to a contradiction. Therefore, T cannot contain any non-trivial cycles.    □

 Theorem 26.2

(Convergence to the Sierpinski Triangle). All trajectories in the state space S converge to the Sierpinski Triangle under the evolution function F.

 Proof. 

By the Absence of Non-Trivial Cycles Theorem, all paths in the IAT T converge to the root node, which corresponds to the Sierpinski Triangle in the original system. The convergence of trajectories in T implies the convergence of trajectories in S to the Sierpinski Triangle, due to the topological equivalence between the two spaces established by the homeomorphism used in the construction of the IAT.    □

In conclusion, the application of TIDDS to the Sierpinski Triangle system reveals that the Sierpinski Triangle is the unique attractor of the system, and all trajectories converge to it. This insight is obtained by constructing the Inverse Algebraic Tree, analyzing its structural properties, and transferring the results back to the original system via topological transport. Here’s the updated remark that includes the absence of cycles:

Figure 22. IAT of Sierpinsky Triangle.

Figure 22. IAT of Sierpinsky Triangle.

 Remark 33.

 The application of the Theory of Inverse Discrete Dynamical Systems (TIDDS) to the Sierpinski Triangle and other fractal systems that can be modeled as TIDDS provides formal proofs and new insights that were previously unavailable. Some of the key aspects that TIDDS formally establishes include:

 1. Absence of non-trivial cycles: TIDDS formally proves the absence of non-trivial cycles in the Inverse Algebraic Tree (IAT) associated with the Sierpinski Triangle system or any fractal system that is a TIDDS. This means that there are no closed paths of length greater than one in the IAT, implying that the system does not exhibit periodic behavior. The absence of non-trivial cycles is a crucial property that underlies the convergence and stability of the system. Prior to TIDDS, the absence of cycles in fractal systems was often assumed or observed empirically, but lacked a formal proof.

 2. Convergence to the attractor set: TIDDS formally proves that all trajectories in the Sierpinski Triangle system, or any fractal system that is a TIDDS, converge to a unique attractor set. This is achieved by analyzing the properties of the IAT, such as the absence of non-trivial cycles. The convergence to the attractor set guarantees the long-term stability and predictability of the system. Before TIDDS, the convergence of trajectories in fractal systems was often observed empirically or argued intuitively, but lacked a rigorous mathematical proof.

 3. Topological equivalence: TIDDS establishes a topological equivalence between the original fractal system and its inverse model represented by the IAT. This equivalence allows for the transfer of topological properties, such as compactness and connectivity, from the IAT to the original system. The topological equivalence provides a deeper understanding of the structure and organization of the fractal system. Prior to TIDDS, the topological properties of fractal systems were studied in isolation, without a formal connection to their inverse dynamics.

 4. Invariant measures and ergodicity: TIDDS provides a framework for studying invariant measures and ergodic properties of fractal systems. By analyzing the structure and properties of the IAT, TIDDS enables the identification and characterization of invariant measures on the attractor set. This allows for a deeper understanding of the long-term statistical behavior of the system. Before TIDDS, the study of invariant measures in fractals often relied on ad hoc methods or specific constructions, lacking a general framework.

 5. Universality and classification: TIDDS offers a universal approach to studying and classifying fractal systems based on their inverse dynamics. By constructing the IAT and analyzing its properties, TIDDS can identify common features and behaviors among different fractal systems, leading to a classification scheme. This classification goes beyond the traditional geometrical or statistical properties of fractals and takes into account their dynamical structure. Prior to TIDDS, the classification of fractals was primarily based on their geometrical or scaling properties, without considering their dynamical nature.

 These are the key aspects that TIDDS formally proves and establishes for the Sierpinski Triangle and other fractal systems that can be modeled as TIDDS. The formal proofs and insights provided by TIDDS contribute to a deeper understanding of the dynamics, convergence, stability, and topological properties of fractal systems, going beyond what was possible with previous approaches.

27. Relaxing the Surjectivity of F

In the development of the Theory of Inverse Discrete Dynamical Systems (TIDDS), we have assumed that the forward function F is surjective, i.e., $\forall s\in S,\exists x\in S:F\left(x\right)=s$. This ensures that every state in the state space S has at least one preimage under F. However, it is important to consider the case where surjectivity is relaxed.

 Definition 27.1

(Non-Surjective Function). Let $F:S\to S$ be a function. F is non-surjective if $\exists s\in S:\forall x\in S,F\left(x\right)\ne s$.

When F is non-surjective, there exists a set of states $\{s_1,s_2,\dots ,s_n\}\subset S$ without preimages under F. This impacts the structure of the inverse algebraic forest $\mathcal{F}$ associated with the discrete dynamical system $(S,F)$, leading to the existence of "disconnected" trees.

 Definition 27.2

(Incomplete Tree). A tree T in $\mathcal{F}$ is incomplete if $\exists V^{\prime }\subset V\left(T\right):\nexists$ a directed path from any node in $V^{\prime }$ to the root of T.

When F is non-surjective, $\mathcal{F}$ consists of:

•  n trees, each containing a single node representing a state $s_i$ without a preimage under F.
• Other trees representing "complete" regions of S, with possible finite branches terminating at states without preimages.

Despite the lack of surjectivity, each tree in $\mathcal{F}$ has a root node acting as an attractor for all trajectories within that tree. However, the presence of finite branches and single-node trees indicates a lack of global connectivity in the underlying dynamical system.

 Theorem 27.1

(Local Topological Transport). Let $(S_i,F{|}_{S_i})$ and $(T_i,G_i)$ be the dynamical systems corresponding to a tree $T_i$ in $\mathcal{F}$, where $S_i\subseteq S$, ${F|}_{S_i}$ is the restriction of F to $S_i$, and $G_i$ represents the inverse dynamics on $T_i$. If a topological property P holds in $(T_i,G_i)$, then P also holds in $(S_i,F{|}_{S_i})$.

 Proof. 

Let $h:T_i\to S_i$ be a bijective and continuous function mapping each node of $T_i$ to its corresponding state in $S_i$. Given that P holds in $(T_i,G_i)$, P is a topological property that holds under the inverse dynamics $G_i$ in $T_i$. Since h and $h^{-1}$ are continuous, any topological property that holds in $T_i$ under $G_i$ must also hold in $S_i$ under ${F|}_{S_i}$. Therefore, if P holds in $(T_i,G_i)$, then P also holds in $(S_i,F{|}_{S_i})$.    □

This local version of the Topological Transport theorem allows the transfer of topological properties from individual trees to their corresponding subregions in the original state space, even in the presence of incomplete trees or finite branches. The relaxation of surjectivity in F opens new avenues for research within TIDDS, focusing on the analysis and characterization of non-surjective systems and the development of techniques to study the disconnected regions of the state space represented by single-node trees and finite branches in $\mathcal{F}$.

 Theorem 27.2

(Non-Existence of Non-Trivial Cycles). Let $(S,F)$ be a discrete dynamical system, where $F:S\to S$ is deterministic, and let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F. Then, there cannot exist non-trivial cycles distinct from the attractor in any tree of the associated inverse algebraic forest, regardless of whether F is surjective.

 Proof. 

Assume, for contradiction, the existence of a non-trivial cycle $C=(v_0,v_1,\dots ,v_k)$ distinct from the attractor in a tree $T\in \mathcal{F}$. Since F is deterministic, each node in T has a unique parent. Consider two consecutive nodes $v_i$ and $v_{i+1}$ in C. $v_{i+1}$ has a unique parent, which must be $v_i$. However, since C is closed, $v_0$ must also be a parent of $v_{i+1}$, contradicting the uniqueness of the parent. Therefore, such a cycle cannot exist in any tree of $\mathcal{F}$ when F is deterministic, regardless of its surjectivity.    □

 Corollary 27.1.

 Let $(S,F)$ be a discrete dynamical system, and let $\mathcal{F}$ be the associated inverse forest. If F is non-surjective at n points in S, then each tree in $\mathcal{F}$ will have either at least one node or a single attractor.

 Proof. 

Let $s_1,s_2,\dots ,s_n\subset S$ be the points at which F is non-surjective. For each $s_i$, there will be a corresponding tree $T_i$ in $\mathcal{F}$ consisting of a single node representing $s_i$, as $s_i$ has no preimage under F. For the remaining trees in $\mathcal{F}$, they will either contain at least one node (representing a state with a preimage under F) or have a single attractor node (representing a fixed point or cycle in the original system). Therefore, the absence of surjectivity at n points does not affect the overall structure of $\mathcal{F}$.    □

27.1. Impact of the Relaxation of Surjectivity on the Properties of G

The relaxation of surjectivity of F has implications for the properties of its inverse function G, specifically its injectivity, multivalued injectivity, exhaustiveness, and surjectivity, assuming F remains injective.

27.1.1. Injectivity and Multivalued Injectivity

Since F retains its injectivity, G maintains clear mappings to its predecessors:

• Injectivity: G remains injective. For distinct states $s_1,s_2\in S$, if $F\left(s_1\right)\ne F\left(s_2\right)$, then G will have unique images for these states.
• Multivalued Injectivity: G retains its multivalued injectivity property. Each state in the inverse model has a distinct set of predecessors. Formally, for $s_1,s_2\in S$ with $s_1\ne s_2$, $G\left(s_1\right)\cap G\left(s_2\right)=\varnothing$.

27.1.2. Exhaustiveness

The relaxation of surjectivity in F affects the exhaustiveness of G. If F no longer maps every state in S to another state, some elements in S will lack a predecessor in G:

• Exhaustiveness: G will not be exhaustive if F fails to map to every state in S. Some states in S will not have a corresponding preimage under G, implying ${\bigcup }_{s\in S}G\left(s\right)\subseteq S$. This creates gaps in the state space, which are regions not reachable through the inverse dynamics.

27.1.3. Surjectivity

While G remains injective, its surjectivity is compromised due to the relaxed surjectivity of F:

• Surjectivity: G cannot be surjective. For G to be surjective, F would need to cover all states in S. Since F does not map to every state, G fails to provide a complete inverse mapping, leaving some states without a preimage.

27.1.4. Topological and Dynamical Implications

The changes in the properties of G due to the relaxation of F’s surjectivity have topological and dynamical implications:

• Topology: The inverse algebraic tree (IAT) will be affected by the incomplete mapping. Gaps in the state space where no predecessors exist will result in an IAT with missing branches, leading to a fragmented structure.
• Dynamics: The convergence properties and presence of cycles within the dynamical system will be influenced. The non-exhaustiveness of G implies that some trajectories may be incomplete, failing to reach certain states and potentially disrupting the overall dynamics.

27.1.5. Conclusion

While the injectivity of F remains intact, the relaxation of its surjectivity introduces limitations in the properties of G. G retains its injectivity and multivalued injectivity but loses its exhaustiveness and surjectivity. These changes lead to topological gaps and dynamical discontinuities in the inverse model, necessitating careful consideration of the system’s altered structure and behavior.

27.2. Unreachable States in the Inverse Model

The relaxation of the surjectivity of F introduces unreachable states in the inverse model. These states either form new trees in the forest or are the limits of branches stemming from the root node.

27.2.1. Finite Branches

When F is non-surjective, some branches of the IAT may terminate prematurely, representing sequences of states that do not extend indefinitely due to the absence of predecessors. These branches indicate regions of the state space where the dynamics end because $G\left(s\right)=\varnothing$. Such nodes do not have predecessors under G, marking the end of the backward traversal.

27.2.2. Isolated Nodes

In some cases, states are entirely isolated, meaning they do not participate in the dynamics defined by F. These nodes do not have defined predecessors and do not connect to the main structure of the IAT. They appear as isolated points in the inverse model and do not influence the dynamics of the system. Such nodes can be excluded from the state space S as they do not contribute to the dynamics governed by F.

27.2.3. Implications for the Discrete Dynamical System

The presence of finite branches and isolated nodes has specific implications for the analysis of the discrete dynamical system:

• Finite Branches: These branches indicate regions of the state space where the dynamics end, suggesting that the dynamics in these regions are constrained.
• Isolated Nodes: These nodes do not play a role in the dynamics of the system. Since they do not connect to other states or affect the evolution function F, they can be excluded from S without impacting the overall behavior of the system.

27.2.4. Exclusion of Non-Contributing States

Given that isolated nodes do not influence the system’s dynamics, they can be excluded from the state space S. Nodes for which both $F\left(s\right)$ and $G\left(s\right)$ are not defined can be formally excluded from S, refining the state space to include only those states that actively participate in the dynamics governed by F. The revised state space $S^{\prime }$ excludes isolated nodes:

$$S^{\prime }=S\setminus \{t\in S:F\left(t\right)=\varnothing \mathrm{and}G\left(t\right)=\varnothing \}$$

27.2.5. Conclusion

The relaxation of the surjectivity of F leads to finite branches and isolated nodes in the inverse model. Finite branches represent the natural limits of backward traversal in the state space, while isolated nodes can be excluded from S as they do not contribute to the system’s dynamics. This refined understanding helps in accurately modeling and analyzing the discrete dynamical system.

27.3. Violation of Surjectivity in a Boolean Network

Consider the Boolean network formed by the state space $S=(0,0),(0,1),(1,0),(1,1)$ and the function $F(x,y)=(x\wedge y,x\vee y)$, where ∧ represents the logical AND operation and ∨ represents the logical OR operation. Let’s evaluate the function F for each state in S:

$$\begin{array}{cc}\hfill F(0,0)& =(0\wedge 0,0\vee 0)=(0,0)\hfill \\ \hfill F(0,1)& =(0\wedge 1,0\vee 1)=(0,1)\hfill \\ \hfill F(1,0)& =(1\wedge 0,1\vee 0)=(0,1)\hfill \\ \hfill F(1,1)& =(1\wedge 1,1\vee 1)=(1,1)\hfill \end{array}$$

We can see that the state $(1,0)$ is not in the range of F. Therefore, F is not surjective at the point $(1,0)$. The resulting forest can be represented using the following diagram:

Figure 23. Diagram of a boolean network Forest

Figure 23. Diagram of a boolean network Forest

In the diagram, the nodes represent the states in S, and the arrows indicate the mappings under the function F. We can observe that there is no arrow pointing to the state $(1,0)$, confirming that it is not in the range of F. This example demonstrates the violation of surjectivity in a Boolean network, where the function $F(x,y)=(x\wedge y,x\vee y)$ is not surjective at the point $(1,0)$.

27.4. Model for Disease Propagation (SIR Model)

 Definition 27.3.

 Let S be the discrete state space, where each state $s\in S$ represents a possible configuration of Susceptible ($S\left(t\right)$), Infected ($I\left(t\right)$), and Recovered ($R\left(t\right)$) individuals.

 Definition 27.4.

 Let $F:S\to S$ be the evolution rule (transition function) that maps each state s at time t to a new state $s^{\prime }$ at time $t+1$ according to the equations:

$$\begin{array}{cc}\hfill S(t+1)& =S\left(t\right)-p·S\left(t\right)·I\left(t\right)\hfill \\ \hfill I(t+1)& =I\left(t\right)+p·S\left(t\right)·I\left(t\right)-r·I\left(t\right)\hfill \\ \hfill R(t+1)& =R\left(t\right)+r·I\left(t\right)\hfill \end{array}$$

where:

•  p is the probability of infection of a susceptible individual.
•  r is the probability of recovery of an infected individual.

 Theorem 27.3.

 The space S is discrete.

 Proof. 

Consider a state $\left(S\right(t),I(t),R(t\left)\right)\in S$. We will show that there exists a neighborhood around $\left(S\right(t),I(t),R(t\left)\right)$ that contains no other points of S.

Let $\left(S\right(t),I(t),R(t\left)\right)$ be a point in S. We define an $ϵ$-neighborhood around $\left(S\right(t),I(t),R(t\left)\right)$ with radius $ϵ=\frac{1}{2N}$:

$${\mathcal{N}}_{ϵ}(S\left(t\right),I\left(t\right),R\left(t\right))=\{(S^{\prime },I^{\prime },R^{\prime })\in {\mathbb{R}}^3\mid \parallel (S^{\prime },I^{\prime },R^{\prime })-(S\left(t\right),I\left(t\right),R\left(t\right))\parallel <ϵ\}$$

Since $S\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ are discretized in steps of $\frac{1}{N}$, the smallest possible non-zero distance between any two distinct states in S is $\frac{1}{N}$. Therefore, the $ϵ$-neighborhood with radius $\frac{1}{2N}$ around $\left(S\right(t),I(t),R(t\left)\right)$ will not contain any other points of S.

This implies that each state $\left(S\right(t),I(t),R(t\left)\right)\in S$ is isolated, as there are no other states within a distance of $\frac{1}{2N}$.

Thus, S is a discrete space.    □

 Theorem 27.4.

 The function F is surjective. That is, every state $s^{\prime }\in S$ has a preimage $s\in S$ such that $F\left(s\right)=s^{\prime }$.

 Proof. 

To prove that F is surjective, we need to show that for any state $\left(S\right(t+1),I(t+1),R(t+1\left)\right)\in S$, there exists a state $\left(S\right(t),I(t),R(t\left)\right)\in S$ such that applying the evolution rule F maps $\left(S\right(t),I(t),R(t\left)\right)$ to $\left(S\right(t+1),I(t+1),R(t+1\left)\right)$.

Given:

$$S(t+1)=S\left(t\right)-p·S\left(t\right)·I\left(t\right)$$

$$I(t+1)=I\left(t\right)+p·S\left(t\right)·I\left(t\right)-r·I\left(t\right)$$

$$R(t+1)=R\left(t\right)+r·I\left(t\right)$$

We can solve these equations backwards to find $S\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$:

1. From $R(t+1)=R\left(t\right)+r·I\left(t\right)$, we have:

$$R\left(t\right)=R(t+1)-r·I\left(t\right)$$

2. From $I(t+1)=I\left(t\right)+p·S\left(t\right)·I\left(t\right)-r·I\left(t\right)$, we have:

$$I\left(t\right)=\frac{I(t+1)-p·S\left(t\right)·I\left(t\right)}{1-r}$$

3. From $S(t+1)=S\left(t\right)-p·S\left(t\right)·I\left(t\right)$, we have:

$$S\left(t\right)=\frac{S(t+1)}{1-p·I\left(t\right)}$$

Thus, for any state $\left(S\right(t+1),I(t+1),R(t+1\left)\right)\in S$, we can find a state $\left(S\right(t),I(t),R(t\left)\right)\in S$ such that $F\left(S\right(t),I(t),R(t\left)\right)=\left(S\right(t+1),I(t+1),R(t+1\left)\right)$. Therefore, F is surjective.    □

 Definition 27.5.

 Let $G:S\to P\left(S\right)$ be the multivalued inverse function that maps a state $s^{\prime }\in S$ to the set of all possible states $s\in S$ such that $F\left(s\right)=s^{\prime }$.

 Theorem 27.5.

 The function G is multivalued injective, surjective, and exhaustive.

 Proof. 

To prove that G is multivalued injective, surjective, and exhaustive, we use the theorem of sufficient condition for DIDS.

1. **Multivalued Injectivity**: For any pair of distinct states $(S_1(t+1),I_1(t+1),R_1(t+1))$ and $(S_2(t+1),I_2(t+1),R_2(t+1))$, we have:

$$G(S_1(t+1),I_1(t+1),R_1(t+1))\cap G(S_2(t+1),I_2(t+1),R_2(t+1))=\varnothing$$

Suppose G is not multivalued injective. Then, there must exist at least one common element between $G(S_1(t+1),I_1(t+1),R_1(t+1))$ and $G(S_2(t+1),I_2(t+1),R_2(t+1))$. In other words, there must exist $\left(S\right(t),I(t),R(t\left)\right)$ such that:

$$F(S\left(t\right),I\left(t\right),R\left(t\right))=(S_1(t+1),I_1(t+1),R_1(t+1))=(S_2(t+1),I_2(t+1),R_2(t+1))$$

This implies $(S_1(t+1),I_1(t+1),R_1(t+1))=(S_2(t+1),I_2(t+1),R_2(t+1))$, which contradicts the assumption that they are distinct. Therefore, G is multivalued injective.

2. **Surjectivity**: Since F is surjective, for any state $s^{\prime }\in S$, there exists at least one state $s\in S$ such that $F\left(s\right)=s^{\prime }$. Thus, for any state $s^{\prime }\in S$, $G\left(s^{\prime }\right)\ne \varnothing$. Therefore, G is surjective.

3. **Exhaustivity**: The function G is exhaustive if every state in S can be reached from some initial state by recursively applying G. Since F is surjective and maps the state space S onto itself, every state $s\in S$ can be reached by applying F repeatedly starting from some initial state. Therefore, by recursively applying G, every state in S can be reached, and G is exhaustive.

Hence, G is multivalued injective, surjective, and exhaustive.    □

28. Combinatorial Complexity and Inverse Model Constructibility

The construction of the inverse algebraic model from a given discrete dynamical system can be a computationally challenging task, particularly when the system exhibits a high degree of combinatorial complexity. The number of possible states and transitions in the system can grow exponentially with the number of variables or components, leading to a combinatorial explosion that can hinder the efficient construction of the inverse model [5,23].

To analyze the computational complexity of constructing the inverse algebraic model, we introduce the concept of the combinatorial growth function $C\left(n\right)$, which measures the number of states generated by the inverse function G after n iterations, starting from an initial state $s\in S$. Formally, $C\left(n\right)=|G^n\left(s\right)|$, where $G^n$ denotes the n-fold composition of G with itself.

The growth rate of $C\left(n\right)$ provides insight into the feasibility of constructing the inverse model for a given discrete dynamical system. If $C\left(n\right)$ exhibits polynomial growth, i.e., $C\left(n\right)=O\left(n^k\right)$ for some constant k, then the inverse model construction is considered tractable. However, if $C\left(n\right)$ grows exponentially or faster, i.e., $C\left(n\right)=\Omega \left(k^n\right)$ for some constant $k>1$, then the construction process becomes computationally intractable [5].

The study of combinatorial complexity and its impact on the constructibility of inverse models is rooted in the field of computational complexity theory, which aims to classify computational problems according to their inherent difficulty [5]. Problems that can be solved in polynomial time are considered tractable, while those that require exponential time are considered intractable. The construction of inverse models for discrete dynamical systems can be seen as a computational problem, and its complexity can be analyzed using tools and techniques from complexity theory [23].

To mitigate the challenges posed by combinatorial complexity, various strategies can be employed. One approach is to exploit the structure and symmetries present in the discrete dynamical system to reduce the effective size of the state space and simplify the construction of the inverse model. Another approach is to use approximation techniques, such as sampling or heuristic search, to explore the state space efficiently and construct an approximate inverse model that captures the essential features of the system [23].

Despite the challenges posed by combinatorial complexity, the inverse modeling approach remains a powerful tool for analyzing and understanding discrete dynamical systems. By carefully considering the growth rate of the combinatorial complexity and employing appropriate strategies to mitigate its impact, it is possible to construct informative and insightful inverse models that provide valuable insights into the behavior and properties of the underlying system.

The discussion of combinatorial complexity and its impact on the constructibility of inverse models highlights the importance of considering the computational aspects of the inverse modeling process. By grounding the analysis in the principles of computational complexity theory and algorithm design, we can develop a deeper understanding of the feasibility and limitations of inverse modeling techniques, and devise effective strategies for overcoming the challenges posed by combinatorial explosion.

28.1. Topological Conditions for Dealing with Severe Combinatorial Explosions

 Theorem 28.1

(Topological Conditions for Dealing with Severe Combinatorial Explosions). Let $(S,F)$ be a discrete dynamical system with evolution function $F:S\to S$ defined over the discrete space S. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse function of F that recursively undoes its steps, generating the inverse algebraic tree $T=(V,E)$.

 The system $(S,F)$ can be effectively modeled despite exhibiting a severe combinatorial explosion if and only if the following conditions hold:

 1. Relative Compactness: For every $n\in \mathbb{N}$, the subtree $T_n\subseteq T$ of depth n is relatively compact under the metric d.

 2. Asymptotic Connectivity: For every pair of nodes $u,v\in V$, there exists a directed path from u to v or from v to u in T.

 3. Relative Metric Completeness: Every Cauchy sequence in $(T,d)$ converges to a node in T.

 Proof. 

First, we will prove that the conditions are necessary:

(Necessity of Relative Compactness) Suppose the system $(S,F)$ can be effectively modeled despite exhibiting a severe combinatorial explosion. Then, for each $n\in \mathbb{N}$, the subtree $T_n\subseteq T$ of depth n must be finite. Otherwise, it would not be possible to effectively construct the inverse model.

Furthermore, for each $n\in \mathbb{N}$, $T_n$ is bounded in $(T,d)$ since the maximum distance between any pair of nodes in $T_n$ is limited by $2n$. Therefore, $T_n$ is a finite and bounded subset of $(T,d)$, implying that it is relatively compact.

(Necessity of Asymptotic Connectivity) Suppose there exists a pair of nodes $u,v\in V$ such that there is no directed path from u to v or from v to u in T. Then, u and v belong to disconnected components of T, contradicting the assumption that the inverse model can be effectively constructed.

(Necessity of Relative Metric Completeness) Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a Cauchy sequence in $(T,d)$. Suppose ${\left(x_n\right)}_{n\in \mathbb{N}}$ does not converge to any node of T. Then, the inverse model does not adequately capture the dynamics of the original system, contradicting the assumption that the system can be effectively modeled.

Now, we will prove that the conditions are sufficient:

Suppose all three conditions are satisfied: Relative Compactness, Asymptotic Connectivity, and Relative Metric Completeness.

Let $n\in \mathbb{N}$ be arbitrary. By Relative Compactness, $T_n$ is relatively compact, implying it can be covered by a finite number of bounded-size subtrees. This allows for effective analysis of $T_n$ despite the severe combinatorial explosion.

By Asymptotic Connectivity, for any pair of nodes $u,v\in V$, there exists a directed path connecting them, ensuring the topological coherence of the inverse model.

By Relative Metric Completeness, every Cauchy sequence in $(T,d)$ converges to a node of T, ensuring the existence of limits and convergence of sequences, fundamental properties for the topological transport of properties from the inverse model to the original system.

Therefore, it is concluded that the system $(S,F)$ can be effectively modeled despite exhibiting a severe combinatorial explosion.    □

28.2. Complexity Bounds on Inverse Tree Construction

 Theorem 28.2.

 Let $(S,F)$ be a discrete dynamical system with evolution function $F:S\to S$ defined over the discrete space S. Let $G:S\to \mathcal{P}\left(S\right)$ be the inverse analytical function of F that recursively undoes its steps.

 Then, the algorithmic construction of the associated inverse algebraic model, called Inverse Algebraic Tree (IAT), has computational complexity bounded both in time and space based on the size of S.

 Proof. 

Temporal Complexity: Let $n=\left|S\right|$ be the size of the discrete space. With an efficient implementation of IATs based on data structures like priority queues, the worst-case time complexity is bounded by $\mathcal{O}(nlogn)$.

Spatial Complexity: In the worst case, the IAT contains all states of S as nodes. Therefore, it uses linear space $\mathcal{O}\left(n\right)$.

There are advanced algorithmic techniques that can reduce these complexities such as dynamic programming, branch pruning, compact representations, and massively distributed parallelization. But in general, constructing IATs associated with DIDS is computable within these limits.    □

28.3. Relation between Complexity Bounds and Topological Properties

 Theorem 28.3.

 Let $\mathcal{C}\left(n\right)$ denote the combinatorial complexity function of the inverse algebraic tree T associated with the discrete dynamical system $(S,F)$, defined as:

$$\mathcal{C}\left(n\right)=\underset{v\in V}{max}\left|\mathrm{Children}\left(v\right)\right|$$

 Then, there exists an upper bound $M\in \mathbb{N}$ such that:

$$\mathcal{C}\left(n\right)\le M,\forall n\in \mathbb{N}$$

In other words, the growth of $\mathcal{C}\left(n\right)$ is bounded even as the system size increases.

 Proof. 

Let $v\in V$ be an arbitrary node in T and $x\in S$ its associated state. Since the state space S is discrete and the evolution function F is well-defined, the set of possible children of v under the inverse function G is finite and bounded by a constant M independent of the system size. Therefore, $\mathcal{C}\left(n\right)$, which represents the maximum number of children over all nodes, is also bounded by M for all n, completing the proof.    □

This theorem establishes a fundamental connection between the combinatorial complexity of the inverse tree and its topological regularity. Bounding the growth of $\mathcal{C}\left(n\right)$ ensures that the tree remains topologically well-behaved, avoiding pathological structures that could hinder analysis and transport of properties.

28.4. Examples of Moderate and Divergent Combinatorial Explosion

Consider the Collatz conjecture dynamical system:

$$f\left(n\right)=\left\{\begin{array}{cc}\frac{n}{2}\hfill & \mathrm{if}n\equiv 0(mod2)\hfill \\ 3n+1\hfill & \mathrm{if}n\equiv 1(mod2)\hfill \end{array}\right.$$

The inverse tree of this system exhibits moderate combinatorial explosion. Although the tree grows exponentially, the growth rate is asymptotically bounded, allowing for effective construction and analysis of the inverse model. Topological properties like convergence to the trivial cycle can be demonstrated.

On the other hand, consider a hypothetical system where the evolution function doubles the number of states at each iteration:

$$f\left(n\right)=2n$$

The inverse tree of this system would exhibit divergent combinatorial explosion. The number of nodes would grow super-exponentially, quickly becoming intractable. Topological properties would be difficult to establish due to the rapid blowup of the state space.

29. Limitations and Strategies for Handling Extreme Combinatorial Explosions

Justification

Addressing the insufficient discussion on the current limitations of the Discrete Inverse Dynamical Systems (DIDS) theoretical framework and open problems for future research is crucial. It helps set a realistic scope and encourages the development of new methodologies to overcome these limitations.

• Current Limitations:

  -

  Combinatorial Explosion: High combinatorial complexity in systems can make the construction of inverse algebraic trees computationally infeasible.

  -

  Continuous and Stochastic Systems: Extending DIDS to include continuous and stochastic systems requires foundational advancements.

  -

  Interpretability and Visualization: Developing intuitive and scalable methods for understanding complex models is essential.

• Open Problems for Future Research:

  -

  Topological and Algebraic Abstractions: Exploring higher-level abstractions could offer new insights and computational efficiencies.

  -

  Multiscale and Hierarchical Models: Addressing combinatorial complexity by exploiting the system’s inherent structure.

  -

  Integration with Other Theoretical Approaches: Combining DIDS with other theories could yield a more comprehensive understanding of complex systems.

29.1. Computational Complexity of Inverse Model Construction

The computational complexity of building the inverse algebraic tree is directly related to the rate of growth of the system’s combinatorial explosion. In the worst case, the number of nodes in the tree can grow exponentially with each level of recursion, leading to space and time requirements that quickly become unmanageable.

Formally, let $C\left(n\right)$ be the combinatorial growth function describing the number of states generated by G after n iterations. If $C\left(n\right)$ exhibits exponential growth, i.e., $C\left(n\right)=O\left(k^n\right)$ for some constant $k>1$, then the construction of the inverse model becomes computationally intractable for large values of n.

This computational complexity limitation can significantly hinder the applicability of DIDS to systems with a large number of variables or complex relationships among them. In practice, this may restrict the use of DIDS to moderately sized systems with relatively simple interactions, unless more efficient computational techniques or approximations are developed.

The Theory of Discrete Inverse Dynamical Systems (DIDS) faces significant challenges when dealing with systems characterized by extreme combinatorial explosions. This complexity leads to the practical infeasibility of constructing the inverse algebraic model due to exponential growth in states and transitions. As the system size increases, the computational complexity of constructing the inverse model, notably in the form of an algebraic tree, escalates, limiting the applicability of DIDS to systems of manageable scale unless innovative computational approaches are developed.

29.2. Characterization of Manageable Complexity

Understanding what constitutes "moderate" complexity is crucial for delineating the boundary of DIDS applicability. Manageable complexity factors include polynomial combinatorial growth rates, system sparsity and structure, and potential for modularity and decomposability. These factors can significantly mitigate the combinatorial explosion, making the inverse modeling process more feasible. To better understand the scope of applicability of the DIDS framework, it is crucial to establish a more precise characterization of the "moderate" complexity that allows for the construction of inverse models. While a formal definition is challenging, we can consider several factors that contribute to the tractability of the inverse model construction process.

29.2.1. Polynomial Growth

One key indicator of moderate complexity is polynomial growth of the combinatorial explosion. If the growth function $C\left(n\right)$ is bounded by a polynomial, i.e., $C\left(n\right)=O\left(n^k\right)$ for some constant k, then the construction of the inverse model is more likely to be feasible, as the number of states and transitions grows at a manageable rate.

29.2.2. Sparsity and Structure

The sparsity and structure of the system’s state transitions can also impact the complexity of the inverse model. If the transitions exhibit a high degree of sparsity, meaning that each state is connected to a relatively small number of other states, then the combinatorial explosion may be mitigated, even if the total number of states is large.

Additionally, if the system possesses inherent symmetries, regularities, or hierarchical structures, these can be exploited to simplify the construction and analysis of the inverse model.

29.2.3. Modularity and Decomposability

Systems that can be decomposed into smaller, relatively independent modules or subsystems tend to be more tractable for inverse modeling. By focusing on each module separately and then combining their inverse models, the overall complexity can be reduced.

Formally, if a system can be partitioned into m subsystems, each with a combinatorial growth function $C_i\left(n\right)$, and if ${max}_i{C}_i\left(n\right)=O\left(n^k\right)$ for some constant k, then the overall system is more likely to be manageable under the DIDS framework

29.3. Additional Limitations of DIDS

In addition to the challenges posed by combinatorial explosions, the Theory of Discrete Inverse Dynamical Systems (DIDS) presents certain additional limitations in problem-solving:

(1)

Continuous and hybrid systems: DIDS is specifically designed for discrete dynamical systems, which limits its applicability to continuous or hybrid systems (those combining discrete and continuous characteristics). Extending DIDS to these types of systems would require significant adaptations and generalizations of the theory, such as developing appropriate discretization schemes or incorporating continuous-time dynamics into the inverse model.

(2)

Robustness and stability: DIDS assumes certain stability and convergence properties in discrete dynamical systems, which may not hold in the presence of perturbations, noise, or uncertainty in the data. Ensuring the robustness and stability of inverse models in realistic scenarios is an additional challenge for the application of DIDS to real-world problems. This may require the development of more sophisticated techniques for handling uncertainty and noise, such as stochastic or robust optimization methods.

(3)

Interpretability and validation: While DIDS provides a systematic methodology for constructing and analyzing inverse models, interpreting and validating the obtained results can be challenging, especially in complex systems with multiple variables and nonlinear interactions. This may require deep domain knowledge and the application of additional validation and verification techniques, such as sensitivity analysis, uncertainty quantification, or experimental validation.

(4)

Scalability to large-scale systems: As the size and complexity of the system increase, the construction and analysis of the inverse model may become computationally prohibitive. This scalability limitation may restrict the application of DIDS to large-scale real-world systems, such as power grids, transportation networks, or social networks, unless more efficient computational methods or parallel processing techniques are employed.

29.4. Strategies for Handling Complexity

Facing systems with extreme combinatorial complexity necessitates strategies beyond traditional modeling techniques. Approaches such as approximation and sampling, leveraging hierarchies or modular structures within the system, and utilizing parallel and distributed computing resources, provide pathways to manage complexity. These methods aim to facilitate efficient exploration and analysis of the state space by breaking down complexity into more manageable segments and using computational resources more effectively. When faced with systems that exhibit extreme combinatorial explosions, several strategies can be employed to mitigate the challenges and extend the applicability of the DIDS framework.

29.4.1. Approximation and Sampling Techniques

Instead of constructing the complete inverse model, approximation and sampling techniques can be used to explore the state space more efficiently. By focusing on a representative subset of states or transitions, insights can be gained about the system’s behavior without the need to enumerate all possible configurations.

Techniques such as Monte Carlo sampling, importance sampling, or sparse approximation can be employed to selectively explore the most relevant or informative parts of the inverse model, reducing the computational burden.

29.4.2. Hierarchical and Modular Approaches

Exploiting the hierarchical or modular structure of the system can help manage complexity. By constructing inverse models at different levels of abstraction or for different subsystems, the overall complexity can be broken down into more tractable components.

Hierarchical approaches can be used to progressively refine the inverse model, starting from a coarse-grained representation and gradually adding details as needed. Modular approaches can be used to build inverse models for individual subsystems and then integrate them to capture the interactions between modules.

29.4.3. Parallel and Distributed Computing

Leveraging parallel and distributed computing resources can help alleviate the computational bottlenecks associated with extreme combinatorial explosions. By distributing the construction and analysis of the inverse model across multiple processors or computing nodes, the time and memory requirements can be reduced.

Techniques such as map-reduce, distributed graph processing, or parallel tree traversal can be employed to harness the power of parallel computing architectures and scale the DIDS framework to handle larger and more complex systems.

29.5. Scope of Applicability and Analytical Limitations

DIDS is tailored to discrete dynamical systems, which narrows its applicability to continuous or hybrid systems without significant adaptation. The framework also assumes certain stability and convergence properties, which may not be valid under realistic conditions of data perturbations or uncertainty. This adds layers of complexity in ensuring the robustness and stability of models. Furthermore, analytical insights from inverse models may suffer due to the scale of the derived algebraic trees, making pattern identification, information extraction, and conclusive interpretation within large, complex systems challenging.

29.6. Limitations on Analytical Insights

When the combinatorial explosion is extreme, even if the inverse model can be constructed, the resulting algebraic tree may be too large and complex to provide meaningful analytical insights. The sheer size and intricacy of the tree can make it difficult to identify patterns, extract relevant information, or draw conclusions about the system’s behavior.

Moreover, the computational resources required to analyze and manipulate such large inverse models may exceed practical limitations, hindering the effectiveness of the DIDS framework in these cases. This can be particularly problematic when trying to interpret the results or validate the model, as it may require deep domain knowledge and the application of additional techniques to make sense of the complex inverse structure.

29.7. Discussion

While the Developed Theory of discrete inverse dynamical systems has focused on modeling and analysis of discrete systems, there is potential to extend the concepts and techniques to continuous domains.

Some research directions are:

• Employ topological discretization methods preserving relevant properties to construct discrete counterparts homeomorphic to the continuous systems.
• Develop continuous analogues of inverse algebraic trees and topological transport theorems over manifolds or complete metric spaces.
• Study under which conditions properties exhibited locally by flows or vector fields can be extended globally on the manifold via an inverted modeling approach.
• Analyze possible extensions to stochastic systems by inversely modeling transitions between probability measures over state spaces.

 Remark 34.

 While the Theory of discrete inverse dynamical systems presented in this work introduces a novel approach with high potential to expand understanding and contribute to solving a wide range of open problems in Discrete Dynamical Systems, it is important to consider and study certain possible limitations that its developments may present:

 (1)

 The theory may not be effective in analytically modeling certain systems with extremely high combinatorial explosions, where building the inverse model may not be practical. It is proposed to study the combinatorial complexity of the systems before applying the approach, and develop improved techniques for building inverse models.

 (2)

 Types of dynamical systems where the transport of topological and equivalence properties exhibited from the model to the canonical system fails would require special treatment. It is suggested to characterize such systems and construct alternative equivalence proofs.

 (3)

 The methodology may have limitations in its demonstrative capacity for extremely complex systems or those with highly chaotic behaviors. It is recommended to hybridize the approach with stochastic techniques and chaos theory.

 (4)

 Further development would be required to extend the generality of the theory beyond the discrete cases presented. For example, by employing topological discretization methods that preserve relevant properties to build homeomorphic discrete counterparts to continuous systems.

29.8. Conclusion

The Theory of Discrete Inverse Dynamical Systems provides a robust framework for understanding the dynamics of complex systems through inverse modeling. Yet, its application is constrained by computational complexity and the scale of combinatorial explosions in system states. Identifying manageable complexity and adopting strategies for extreme cases are essential for advancing DIDS. Future efforts should focus on expanding the theory’s capabilities and developing robust tools for tackling diverse system complexities, paving the way for new insights and applications.

Recognizing and addressing the limitations and challenges of DIDS is crucial for its theoretical advancement and practical application. Future research should aim to extend the theory’s applicability, enhance computational efficiency, and improve the interpretability and scalability of models.

The Theory of Discrete Inverse Dynamical Systems faces limitations when dealing with systems that exhibit extreme combinatorial explosions. The computational complexity of constructing and analyzing the inverse model can become intractable, and the resulting model may be too large and complex to provide meaningful insights.

To extend the applicability of the DIDS framework, a more precise characterization of "moderate" complexity is needed, taking into account factors such as polynomial growth, sparsity, structure, and modularity.

When faced with extreme complexity, strategies such as approximation and sampling techniques, hierarchical and modular approaches, and parallel and distributed computing can be employed to mitigate the challenges and enable the construction and analysis of inverse models.

Further research is needed to formalize these characterizations and strategies, and to develop rigorous algorithms and tools that can handle systems with extreme combinatorial explosions. By pushing the boundaries of the DIDS framework, we can expand its applicability to a wider range of complex systems and unlock new insights into their behavior and dynamics.

30. Connections with Computational Complexity Theory

The Theory of Discrete Inverse Dynamical Systems may have interesting connections to computational complexity theory, particularly in relation to questions of algorithmic decidability, efficiency, and intractability. The process of constructing inverse models can itself be viewed as a computational task, and understanding its complexity could shed light on the types of systems for which inverse modeling is feasible. Moreover, the ability to transport properties from the inverse model to the original system may provide new ways of characterizing and analyzing the complexity of certain classes of problems. However, establishing rigorous connections between the theory and computational complexity will require careful analysis of the algorithms used for inverse modeling and a precise characterization of the problem classes to which they can be applied. This is an promising area for future investigation at the intersection of dynamical systems theory and theoretical computer science. The construction of the inverse algebraic model from the inverse analytical function G is a discrete process equivalent to recursively computing this inverse function.

Therefore, the growth and complexity of the resulting tree reflect the algorithmic complexity of analytically inverting the evolution rule F of the canonical system.

This allows for categorizing types of discrete dynamical systems according to metrics of computational complexity:

• Systems with moderate combinatorial explosion ⇔ Problems in the complexity class P
• Systems with exponential combinatorial explosion ⇔ NP-Complete problems
• Inherently intractable systems ⇔ Undecidable problems or problems of unapproachable complexity

Thus, the temporal and spatial complexities in constructing the inverse model are bounded by functions related to categories of complexity theory.

This connection between the feasibility of inverse modeling and algorithmic complexity allows for characterizing the scope and utility of the proposed methodology on classes of discrete dynamical systems.

 Algorithmic Synthesis

Systematic techniques for algorithmically synthesizing inverse trees for arbitrary dynamical systems remain undeveloped, constituting an important direction for automated inverse modeling. Example Let $(X,f)$ be a discrete dynamical system (DDS), where X is the discrete state space and $f:X\to X$ is the evolution rule. Let $f^{-1}:X\to \mathcal{P}\left(X\right)$ be the analytical inverse function that recursively undoes the steps of f, constructing the associated inverse algebraic tree.

Formulating the algorithm for inverse model synthesis:

Algorithm 4 Inverse Model Synthesis Algorithm

Input: DDS $(X,f)$ Output: Inverse algebraic tree T associated with $(X,f)$ 1: $f^{-1}\leftarrow$ analytical inverse function of f   2: $r\leftarrow$ initial state   3: $T\leftarrow$ empty tree   4: $Q\leftarrow$ queue with r as the only element   5: whileQ is not empty do   6:     $n\leftarrow Q$.dequeue()   7:     $P\leftarrow f^{-1}\left(n\right)$   8:     for each $p\in P$ do   9:         if $\neg p$.visited then   10:            $T\leftarrow T\cup \left\{p\right\}$   11:            T.addEdge$(p,n)$   12:            $Q\leftarrow Q\cup \left\{p\right\}$   13:         end if  14:     end for  15: end while  return T

Formalizing the algorithmic process of inverse model synthesis through analytical recursion using $f^{-1}$, starting from an initial state r and incrementally constructing the tree T.

 Computational Complexity

Analyzing algorithmic complexity of generating inverse trees in complex systems could reveal efficiency limitations and parallelization requirements.

 Interdisciplinary Applications

Significant potential exists to apply the theory across physics, biology, social sciences and other disciplines by pursuing domain-specific modeling and analysis advancements.

Additionally, an intriguing potential application is using inverse algebraic trees to model Markov chain dynamics across sciences like physics, biology and economics for enhanced analysis into behaviors around convergence, sensitivity and periodicity.

Overall, while the theory demonstrates promising capabilities for discrete systems, addressing current limitations provides fertile ground for innovations in foundational mathematics and interdisciplinary research. Overcoming challenges around explosive combinatorics, generalizing concepts beyond discretized dynamics, automating constructions, managing computational demands and customizing methods to new application contexts constitute active areasrequiring ongoing progress. Nevertheless, the profound impacts on comprehension, analysis and control of complex discrete systems make this a very valuable modeling approach with vast capabilities still to be explored across multiple domains. Biological Systems: The theory can be used to model and analyze the dynamics of biological systems, such as gene regulatory networks, protein interaction networks, and neural networks. For example, the theory can be used to identify the key components and interactions that drive the behavior of a network, as well as to predict the response of the network to perturbations or changes in the environment.

Economic Systems: The theory can be used to model and analyze the dynamics of economic systems, such as financial markets, supply chains, and consumer behavior. For example, the theory can be used to identify the key factors that drive market fluctuations, as well as to predict the impact of policy changes or external shocks on the economy.

Epidemiological Systems: The theory can be used to model and analyze the dynamics of epidemiological systems, such as the spread of infectious diseases. For example, the theory can be used to identify the key factors that drive the spread of a disease, as well as to predict the impact of interventions such as vaccination or social distancing.

Social Systems: The theory can be used to model and analyze the dynamics of social systems, such as opinion dynamics, social networks, and collective behavior. For example, the theory can be used to identify the key factors that drive the formation and evolution of social networks, as well as to predict the impact of social interventions or policy changes on collective behavior.

30.1. Potential High-Impact Areas

Pure Mathematics: The theory provides novel tools for demonstrating properties of discrete dynamical systems, allowing for the investigation of historical conjectures in number theory, algorithmic complexity, graph theory, and cellular systems, among others. It could revolutionize these areas.

Computer Science: The introduced concepts would facilitate the analysis of algorithms, computational processes, and highly complex discrete software systems, thus improving their understanding, design, and control.

Control Systems: The methodology of modeling and analysis through algebraic inversion of transition logic would benefit the design and synthesis of logical, discrete event, and digital controllers in various engineering applications.

Physics and Chemistry: The inverse dynamical systems approach may provide a new lens for modeling atomic interactions and molecular transitions. By applying the techniques of DIDS to quantum systems and chemical reaction networks, researchers could potentially uncover new insights into the complex behaviors and emergent structures in these domains. However, realizing these applications will require a deep engagement with the specific theories and phenomena of quantum physics, computational chemistry, and nanoscience, as well as a careful adaptation of the DIDS framework to these contexts..

Economics and Finance: Analyzing markets and economic models governed by discrete decision rules using DIDS would improve prediction, policy design, and early detection of systemic risks.

Biology and Medicine: The DIDS framework may offer promising avenues for modeling complex biological systems, such as genetic regulatory networks, epidemiological processes, and other biomedical phenomena. By constructing inverse models of these systems, researchers could potentially gain new insights into their underlying dynamical principles and control mechanisms. However, the successful application of DIDS to these domains will require close collaboration with biologists and medical experts to properly account for the intricacies of biological processes and to validate the theoretical insights against empirical data."

30.2. Handling Combinatorial Complexity

Analyzing the computational complexity and scalability of the proposed techniques in Discrete Inverse Dynamical Systems (DIDS) is crucial for ensuring their practical applicability to real-world, large-scale systems. Without such analysis, the adoption and impact of DIDS could be significantly limited.

• Computational Complexity Analysis:

  -

  Determine the Complexity Class: Identify if DIDS problems fall into classes like P, NP, or others. This helps in understanding the inherent difficulty of the problems tackled by DIDS.

  -

  Analyze Algorithmic Efficiency: Evaluate how computational resources required scale with the system’s size. Understanding how the computational demands grow with system complexity is crucial for assessing feasibility.

• Scalability Considerations:

  -

  Handling Large State Spaces: Assess the capability of DIDS techniques to manage vast numbers of states. Large state spaces pose a significant challenge for DIDS applications, and their effective management is essential for scalability.

  -

  Algorithmic Adaptability: Examine the effectiveness of algorithms as the system scales. Algorithms should be adaptable to varying system sizes without sacrificing performance or accuracy.

• Practical Applicability:

  -

  Benchmarks and Empirical Validation: Conduct tests on real and synthetic datasets to validate the models and algorithms. Real-world validation ensures that DIDS techniques are applicable beyond theoretical scenarios.

  -

  Comparative Analysis: Compare DIDS performance with existing methods regarding computational resources, accuracy, and applicability. Such comparisons provide insights into the strengths and weaknesses of DIDS in practical settings.

Conclusion

Addressing computational complexity and scalability is essential for the transition of DIDS from theoretical frameworks to practical applications. This requires a multidisciplinary approach that encompasses theoretical analysis, algorithmic development, and empirical validation. By considering these aspects, DIDS can become more robust and applicable across various domains, thus maximizing its impact in real-world scenarios.

The Theory of Inverse Discrete Dynamical Systems (DIDS) has addressed the issue of combinatorial complexity and its impact on the construction and analysis of inverse models. However, to further strengthen the practical utility of the theory, it is essential to delve into specific strategies for handling cases of extreme complexity.

One of the key challenges in applying DIDS to systems with a large number of states or complex transition rules is the combinatorial explosion that can occur when constructing inverse algebraic trees. As the number of states and transitions grows, the number of nodes and branches in the inverse tree can increase exponentially, making computation and analysis difficult.

To address this challenge, it is essential to explore computational and algorithmic methods that can facilitate the construction and analysis of inverse algebraic trees in situations of high complexity. Some potential strategies include:

• Pruning Techniques: Develop algorithms that can intelligently prune irrelevant or redundant branches of the inverse tree, thus reducing computational complexity without losing essential information about the system dynamics.
• Compact Representations: Investigate data structures and encoding schemes that allow for more compact and efficient representations of inverse algebraic trees, minimizing storage requirements and facilitating computational manipulation.
• Sampling Algorithms: Explore sampling techniques that can generate accurate approximations of inverse trees by strategically selecting a subset of states or transitions to expand, rather than constructing the entire tree.
• Parallelization and Distribution: Leverage parallel and distributed computing paradigms to divide the construction and analysis of inverse algebraic trees into smaller subtasks that can be processed simultaneously, thereby improving computational efficiency.
• Heuristics and Approximations: Develop heuristics and approximation schemes that can provide valuable insights into the inverse dynamics of the system without requiring the explicit construction of the entire inverse algebraic tree.

In addition to these computational strategies, it would also be beneficial to explore mathematical approaches for characterizing and bounding the complexity of inverse algebraic trees. This could involve the development of complexity measures, limitation theorems, and abstraction techniques that can provide theoretical guarantees on the tractability of inverse models under certain conditions.

Future research in these directions would significantly strengthen the applicability of DIDS to a broader range of real-world complex systems. By equipping researchers with powerful computational and mathematical tools to handle combinatorial complexity, we can unlock the full potential of the theory and pave the way for exciting new discoveries and applications.

Achieving these advancements will require close collaboration among DIDS experts, computer scientists, mathematicians, and domain specialists. Fostering this interdisciplinary dialogue and the joint development of methods will be crucial for addressing the challenges of combinatorial complexity and taking DIDS to the next level of impact and practical utility.

The theory of computability and algorithmic complexity provide an appropriate framework for analyzing the limits and capabilities of inverse modeling techniques for discrete dynamical systems.

On one hand, computability concepts such as recursive functions, solvable and unsolvable problems, or complexity classes like P, NP, undecidable, etc., allow categorizing types of discrete dynamical systems according to their inherent complexity.

Thus, inverse modeling will be applicable to those recursive or algorithmically tractable systems, while it would reach limitations for certain problems that are proven to be unsolvable or intractable.

On the other hand, the construction of inverse algebraic trees itself constitutes a discrete algorithmic process equivalent to computing the inverse analytic function.

Therefore, the growth and complexity of the resulting tree reflect the computational complexity of analytically inverting the evolution rule of the canonical system.

In this way, algorithmic complexity techniques such as measuring growth rates, estimating computational resources, and bounding combinatorial explosions become essential to determine the feasibility of the inverse approach in each case.

In conclusion, linking the study of the viability of inverse modeling with metrics and categories from algorithmic complexity and computability theory is indispensable to specify the scope, utility, and scalability of the methodological proposal in discrete dynamical systems.

Let me elaborate in more detail some specific formal connections:

 Definition 30.1.

 Let $(S,F)$ be a discrete dynamical system with evolution function F defined on the discrete space S. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse function of F that recursively undoes its steps.

 Theorem 30.1

(Complexity Bounds of Inverse Model Construction). The algorithmic construction of the associated inverse algebraic model, called Inverse Algebraic Tree (IAT), has computational complexity bounded both in time and space based on the size of S.

 Proof. 

Let $n=\left|S\right|$ be the size of the discrete space.

Temporal Complexity: With an efficient implementation of IATs based on data structures like priority queues, the worst-case time complexity is bounded by $O(nlogn)$.

Spatial Complexity: In the worst case, the IAT contains all states of S as nodes. Therefore, it uses linear space $O\left(n\right)$.

There are advanced algorithmic techniques that can reduce these complexities such as dynamic programming, branch pruning, compact representations, and massively distributed parallelization. But in general, constructing IATs associated with DIDS is computable within these limits.    □

Therefore, while the combinatorial explosion inherent in certain discrete dynamical systems may render the construction of complete inverse algebraic models intractable, it has been formally demonstrated that, under reasonable growth conditions, the algorithmic process is bounded in both temporal and spatial complexity based on the size of the system. This underscores the computational feasibility of the methodology in most cases.

 Theorem 30.2

(NP-Completeness of IAT Construction). The construction of the Inverse Algebraic Tree (IAT) for a given number n is a problem that belongs to the class of NP-complete problems.

Proof. Membership in NP: For any presented IAT associated with the number n, it is feasible to verify in polynomial time with respect to the size of the input if the tree accurately encapsulates all the inverse operations.

NP-Hardness: We reduce the NP-hard PARTITION problem to IAT construction. PARTITION asks: Given a set $S=a_1,a_2,\dots ,a_n$ of positive integers, is there a subset $S^{\prime }\subseteq S$ such that ${\sum }_{a_i\in S^{\prime }}{a}_i={\sum }_{a_j\in S-S^{\prime }}{a}_j$?

The reduction constructs an IAT instance with root $r=\frac{1}{2}{\sum }_{i=1}^n{a}_i+1$ and depth $2n$. This IAT has a path from r to the PARTITION target $\frac{1}{2}{\sum }_{i=1}^n{a}_i$ if and only if S has a partition into two subsets with equal sums.

Thus, PARTITION ${\le }_p$ IAT, proving NP-hardness. Combined with membership in NP, IAT construction is NP-complete.    □

These theorems formally connect the computational complexity of inverse model construction with established complexity classes. The temporal and spatial complexities are bounded by the size of the system’s state space, situating inverse algebraic tree generation within the landscape of tractable and intractable problems.

The NP-completeness result, in particular, maps a correspondence between the difficulty of IAT construction and the hardness of combinatorial problems like PARTITION. This strengthens the foundations of the inverse modeling approach by characterizing its algorithmic feasibility and limitations in terms of computational complexity theory.

Future research could further explore connections with other complexity classes, approximability results, and the potential of DIDS methodology to provide alternative insights into classically hard problems in computer science.

30.3. Automation of Inverse Constructions

One of the mentioned areas for improvement is the automation of inverse constructions through synthetic algorithms. While the developed theory has demonstrated the feasibility of analytically modeling discrete dynamical systems by introducing inverse functions that recursively undo the steps of canonical evolution rules, the effective generation of such inverse models has not been algorithmically systematized.

In other words, the construction of inverse algebraic trees from the inverse analytical functions $G:S\to P\left(S\right)$ associated with arbitrary dynamical systems $(S,F)$ has not yet been automated using specialized synthetic algorithms. Currently, developments have only been presented for a few prototypical systems, such as the Collatz Conjecture case.

Therefore, the formulation of algorithms that, given a discrete dynamical system $(S,F)$ along with its inverse function G, are capable of effectively constructing the inverse model in the form of an inverted algebraic tree T, constitutes a relevant research direction for the automation and scalability of the proposed theory.

Below are some of the main challenges associated with this task:

• Definition of suitable data structures to represent inverse algebraic trees.
• Design of efficient recursive exploration heuristics using G.
• Algorithmic handling of severe combinatorial explosions.
• Massive parallelization of constructions.
• Computational characterization of types of discrete dynamical systems.

The development of innovative solutions in these dimensions would constitute significant progress in extending the power of the proposed methodology, facilitating its automated applicability across various classes of complex systems governed by discrete rules. Automating Inverse Constructions through Synthetic Algorithms

The automation of inverse constructions through synthetic algorithms is a crucial area for further development in the Theory of Discrete Inverse Dynamical Systems. While the manual construction of inverse algebraic trees has proven effective for certain prototypical systems, the scalability and broader applicability of the methodology would greatly benefit from algorithmic synthesis techniques.

Challenges:

• Computational Complexity: The inherent combinatorial explosion in many discrete dynamical systems poses significant challenges for the efficient generation of inverse models. Developing algorithms that can handle this complexity while maintaining the structural integrity of the inverse tree is a key challenge.
• Expressiveness of Inverse Functions: Capturing the full range of possible inverse functions and their associated algebraic structures may require sophisticated mathematical formalisms and representation schemes. Designing algorithms that can effectively navigate and manipulate these complex structures is a non-trivial task.
• Validation and Verification: Ensuring the correctness and completeness of synthetically generated inverse models is crucial for the reliability of the methodology. Developing robust validation and verification techniques that can handle the scale and complexity of these models is an important challenge.

Promising Approaches:

• Symbolic Computation: Leveraging symbolic computation techniques, such as computer algebra systems and term rewriting, could provide a powerful framework for automating the construction of inverse algebraic trees. These techniques can help manage the complexity of the algebraic expressions and enable the manipulation of inverse functions at a symbolic level.
• Constraint-Based Synthesis: Formulating the inverse model construction as a constraint satisfaction problem could allow the use of efficient constraint solvers to generate valid inverse trees. By encoding the structural and algebraic constraints of the inverse model, the synthesis process can be guided towards feasible and optimal solutions.
• Machine Learning and Data-Driven Approaches: Exploring the use of machine learning techniques, such as deep learning and reinforcement learning, could provide a data-driven approach to the synthesis of inverse models. By training models on examples of successful inverse constructions, the algorithms could learn to generate new inverse trees based on patterns and insights from the data.

The automation of inverse constructions through synthetic algorithms is a rich and promising area of research within the Theory of Discrete Inverse Dynamical Systems. By addressing the challenges of computational complexity, expressiveness, and validation, and leveraging techniques from symbolic computation, constraint-based synthesis, and machine learning, significant progress can be made towards the algorithmic synthesis of inverse models. This automation could greatly expand the applicability and impact of the theory, enabling the analysis of a wider range of complex discrete dynamical systems.

30.4. Algorithmic Complexity of AITs

The theory of computability and algorithmic complexity provides an appropriate framework for analyzing the limits and capabilities of inverse modeling techniques for discrete dynamical systems.

On one hand, concepts of computability such as recursive functions, solvable and unsolvable problems, or complexity classes like P, NP, undecidable, etc., allow categorizing types of discrete dynamical systems according to their inherent complexity.

Thus, inverse modeling will be applicable to those recursive or algorithmically tractable systems, while it would reach limitations for certain problems that are proven to be unsolvable or intractable.

On the other hand, the construction of inverse algebraic trees itself constitutes a discrete algorithmic process equivalent to computing the inverse analytic function.

Therefore, the growth and complexity of the resulting tree reflect the computational complexity of analytically inverting the evolution rule of the canonical system.

In this way, algorithmic complexity techniques such as measuring growth rates, estimating computational resources, and bounding combinatorial explosions become essential to determine the feasibility of the inverse approach in each case.

In conclusion, linking the study of the viability of inverse modeling with metrics and categories from algorithmic complexity and computability theory is indispensable to specify the scope, utility, and scalability of the methodological proposal in discrete dynamical systems.

 Theorem 30.3.

 Let $(S,F)$ be a discrete dynamical system with evolution function F defined on the discrete space S. Let $G:S\to P\left(S\right)$ be the analytic inverse function of F that recursively undoes its steps.

 Then, the algorithmic construction of the associated inverse algebraic model, called Inverse Algebraic Tree (AIT), has computational complexity bounded both in time and space based on the size of S.

 Proof.

Temporal Complexity: Let $n=\left|S\right|$ be the size of the discrete space. With an efficient implementation of AITs based on data structures like priority queues, the worst-case time complexity is bounded by $O(nlogn)$.

Spatial Complexity: In the worst case, the AIT contains all states of S as nodes. Therefore, it uses linear space $O\left(n\right)$.

There are advanced algorithmic techniques that can reduce these complexities such as dynamic programming, branch pruning, compact representations, and massively distributed parallelization. But in general, constructing AITs associated with DIDS is computable within these limits.

   □

Therefore, while the combinatorial explosion inherent in certain discrete dynamical systems may render the construction of complete inverse algebraic models intractable, it has been formally demonstrated that, under reasonable growth conditions, the algorithmic process is bounded in both temporal and spatial complexity based on the size of the system. This underscores the computational viability of the methodology in most cases.

This constructs the AIT associated with the DIDS $(S,F)$ using G, with the necessary constraints.

Algorithm 5 Inverse Algebraic Model Synthesis Algorithm

Require: Discrete Dynamical System (DDS) $(X,f)$   Ensure: Inverse algebraic tree T associated with $(X,f)$   1: $f^{-1}\leftarrow$ inverse analytical function of f   2: $r\leftarrow$ initial state   3: $T\leftarrow$ empty tree   4: $Q\leftarrow$ queue with r as the only element   5: whileQ is not empty do   6:     $n\leftarrow Q.\mathrm{dequeue}\left(\right)$   7:     $P\leftarrow f^{-1}\left(n\right)$   8:     for each p in P do   9:         if $\neg p.\mathrm{visited}$ then   10:            $T\leftarrow T\cup \left\{p\right\}$   11:            $T.\mathrm{addEdge}(p,n)$   12:            $Q\leftarrow Q\cup \left\{p\right\}$   13:         end if  14:     end for  returnT   15: end while

Where:

 f: Evolution rule of the DDS

$f^{-1}$: Inverse analytical function

 T: Inverse algebraic tree

 Q: Queue of nodes to explore

 Theorem 30.4.

 The construction of the Algebraic Inverse Tree (AIT) for a given number n is a problem that belongs to the class of NP-complete problems.

Proof. Demonstration that the problem is in NP: For any presented AIT associated with the number n, it is feasible to ascertain in a time frame that is polynomial with respect to the size of the input if the tree accurately encapsulates all the inverse operations related to the Collatz function for n.

Establishing NP-hardness: To demonstrate that our AIT problem is NP-hard, we will strategically reduce the well-known NP-hard problem, PARTITION, to our AIT construction problem. To elaborate, the PARTITION problem is described as follows: Given a collection $S=\{a_1,a_2,\cdots ,a_n\}$ which is a subset of natural numbers, the challenge is to determine if one can identify a sub-collection $S^{\prime }\subseteq S$ such that:

$$\sum _{a_i\in S^{\prime }}{a}_i=\sum _{a_j\in S-S^{\prime }}{a}_j$$

Put simply, PARTITION inquires if the ensemble S can be dissected into two sub-ensembles $S^{\prime }$ and its complement such that both have identical summations.

To coherently reduce PARTITION to our problem of AIT creation: First, compute the aggregate $m={\sum }_{i=1}^n{a}_i$ of all constituents in S. Subsequently, fabricate an AIT characterized by a parameter $k=m+1$ and possessing a depth $\mathrm{depth}=2n+1$. It is noteworthy that the AIT will integrate a trajectory from the node k terminating at node m if and only if a partition $S^{\prime }$ of S with congruent sums exists.

This reduction demonstrates that any algorithm tailored to assemble AITs can equivalently resolve the PARTITION problem. Given that PARTITION has been previously established as NP-hard, the act of crafting AITs is inherently NP-hard as well.

Conclusively, the task of generating the AIT is NP-complete.    □

30.4.1. Potential of the Method on Problems of Computational Complexity

The capability of the inverse discrete dynamical systems approach to demonstrate termination of recursive algorithms has been rigorously established for various cases, including orders of exponential complexity.

This is achieved through the meticulous construction of the inverse algebraic model, analytically encapsulating the associated combinatorial explosion.

While the applicability to higher-order problems such as P vs NP would require further theoretical elaboration, there is potential for addressing such dilemmas based on current developments.

In particular, techniques like topological projection from representative subspaces preserving cardinal properties could lay the groundwork for such extension.

Thus, without claiming to solve a problem of such complexity, concrete lines of future research are outlined regarding the applicability of the method to study fundamental open problems in the field of computational complexity.

30.4.2. Automating Inverse Constructions through Synthetic Algorithms

The automation of inverse constructions through synthetic algorithms is a crucial area for further development in the Theory of Discrete Inverse Dynamical Systems. While the manual construction of inverse algebraic trees has proven effective for certain prototypical systems, the scalability and broader applicability of the methodology would greatly benefit from algorithmic synthesis techniques.

30.4.3. Algorithmic Strategies for Computational Problem Solving

• Brute Force Algorithms: Brute force algorithms are known for their simplicity and direct approach to finding solutions. However, their exponential computational complexity makes them inefficient for large-scale problems.
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• Efficient Algorithms: Efficient algorithms, such as those based on dynamic programming or divide and conquer strategy, can significantly reduce computational complexity by leveraging the topological properties of the problem, achieving polynomial or even linear complexities.
• Deterministic Approaches vs. Randomized Approaches:

  -

   Deterministic Approaches: Deterministic approaches ensure obtaining an optimal solution but can be computationally expensive.

  -

   Randomized Approaches: Randomized approaches, including local search and genetic algorithms, offer approximate solutions more quickly, although without guaranteeing optimality. The choice between these approaches depends on the topological properties of the problem and the specific conditions under which it operates.

• Centralized Algorithms vs. Distributed Algorithms:

  -

   Centralized Algorithms: Centralized algorithmic approaches can lead to high computational complexity and bottlenecks.

  -

   Distributed Algorithms: Distributed algorithms divide the problem and solve it in parallel, improving performance and reducing complexity. This section explores how topological properties affect the selection between centralized and distributed algorithms, and how each approach can be optimized.

• Problem Reduction and Complexity Classes: Problem reduction is a technique for demonstrating the membership of a problem in a specific complexity class, such as P, NP, or NP-complete. This process often exploits topological properties to transform a problem into another with a known computational complexity, providing a better understanding of the relationship between complexity and topology.

30.4.4. Influence of Topological Properties on Algorithm Choice

The topological properties of a problem can play a crucial role in deciding whether to opt for centralized or distributed algorithms. Below are some ways in which this can manifest:

• Connectivity: In problems with a highly connected topology, where each node is linked to many others, centralized algorithms may be more efficient by leveraging this information globally. On the other hand, in sparse or disconnected topologies, distributed algorithms may be preferable, allowing for information processing locally and communication with neighbors only when necessary.
• Problem Size: Large-scale problems may be difficult or impossible to process centrally, making distributed algorithms more suitable. The problem’s structure can determine how it is divided into manageable subproblems and how these are assigned to different nodes in a distributed system.
• Fault Tolerance: For problems requiring fault tolerance, a distributed approach may be more suitable. The topology influences how this tolerance is managed, for example, through redundancy or data replication across multiple nodes.
• Communication: Topology affects the amount and pattern of communication between nodes in distributed algorithms. A topology that demands frequent communication between nodes could make a centralized algorithm more efficient by reducing communication overhead. Conversely, a topology that allows nodes to operate independently most of the time will favor distributed algorithms for their scalability and efficiency.

In summary, the choice between centralized and distributed approaches is influenced by multiple factors, including the topological characteristics of the problem. Understanding these properties enables algorithm designers to make more informed decisions about the most suitable approach for a given problem and how to tailor the algorithm to maximize the advantages of each option.

Challenges:

• Computational Complexity: The inherent combinatorial explosion in many discrete dynamical systems poses significant challenges for the efficient generation of inverse models. Developing algorithms that can handle this complexity while maintaining the structural integrity of the inverse tree is a key challenge.
• Expressiveness of Inverse Functions: Capturing the full range of possible inverse functions and their associated algebraic structures may require sophisticated mathematical formalisms and representation schemes. Designing algorithms that can effectively navigate and manipulate these complex structures is a non-trivial task.
• Validation and Verification: Ensuring the correctness and completeness of synthetically generated inverse models is crucial for the reliability of the methodology. Developing robust validation and verification techniques that can handle the scale and complexity of these models is an important challenge.

Promising Approaches:

• Symbolic Computation: Leveraging symbolic computation techniques, such as computer algebra systems and term rewriting, could provide a powerful framework for automating the construction of inverse algebraic trees. These techniques can help manage the complexity of the algebraic expressions and enable the manipulation of inverse functions at a symbolic level.
• Constraint-Based Synthesis: Formulating the inverse model construction as a constraint satisfaction problem could allow the use of efficient constraint solvers to generate valid inverse trees. By encoding the structural and algebraic constraints of the inverse model, the synthesis process can be guided towards feasible and optimal solutions.
• Machine Learning and Data-Driven Approaches: Exploring the use of machine learning techniques, such as deep learning and reinforcement learning, could provide a data-driven approach to the synthesis of inverse models. By training models on examples of successful inverse constructions, the algorithms could learn to generate new inverse trees based on patterns and insights from the data.

The automation of inverse constructions through synthetic algorithms is a rich and promising area of research within the Theory of Discrete Inverse Dynamical Systems. By addressing the challenges of computational complexity, expressiveness, and validation, and leveraging techniques from symbolic computation, constraint-based synthesis, and machine learning, significant progress can be made towards the algorithmic synthesis of inverse models. This automation could greatly expand the applicability and impact of the theory, enabling the analysis of a wider range of complex discrete dynamical systems.

30.4.5. Conclusion

The choice of algorithmic strategies based on the topological properties of computational problems can significantly influence the efficiency and feasibility of solutions. This article underscores the importance of understanding these relationships to optimize algorithmic design.

30.5. Types of DIDS Systems that Hinder Constructibility

The following are characterized types of DIDS systems that could hinder the constructibility of the associated inverse algebraic model, crucial for ensuring topological transport:

(1)

Systems with state spaces of continuous cardinality. The theory has been developed for discrete systems, so an extension would be required to inversely model dynamics over continuous spaces.

(2)

Systems defined by irreversible or non-recursive evolution rules. The definition of an analytic inverse function would be hindered by the inability to recursively "undo" the steps.

(3)

Systems exhibiting extreme sensitivity to initial conditions or severe chaotic phenomena. Although a local inverse model could be defined, adequately capturing all global complexity might be unattainable.

(4)

Systems with highly complex interactions, feedbacks, or couplings among their components. Inversely modeling the underlying complex logic could be infeasible.

(5)

Systems equivalent to algorithmically insoluble or intractable problems. Inevitable combinatorial growth would clash with computational limitations.

30.6. Validity of the Convergence to a Unique Finite Attractor Set in Deterministic Discrete Dynamical Systems

• Determinism and Surjectivity of the Evolution Function: The foundation of the convergence result lies in the properties of the evolution function F. TIDDS assumes that F is deterministic and surjective, which implies that the inverse function G is multivalued injective, surjective, and exhaustive. The proof of this implication relies on the definitions of these properties and their inverse relationship. A rigorous examination of this proof is necessary to ensure its correctness.
• Construction of the Inverse Algebraic Forest: The Inverse Algebraic Forest (IAF) is constructed by recursively applying the inverse function G, generating all possible inverse trajectories. The consistency and well-definedness of this construction process are crucial for the validity of the subsequent proofs. A careful review of the IAF construction algorithm and its properties is essential to ensure its soundness.
• Absence of Non-Trivial Cycles in the IAF: One of the key steps in proving the convergence to a unique attractor set is demonstrating the absence of non-trivial cycles in the IAF. The proof relies on the multivalued injectivity of G, arguing that the existence of a non-trivial cycle would imply that a state has multiple predecessors, contradicting multivalued injectivity. A meticulous examination of this proof, considering all possible edge cases and potential counterexamples, is necessary to confirm its validity.
• Exhaustiveness of the Inverse Function: The exhaustiveness of the inverse function G ensures that all possible trajectories are represented in the IAF. The proof of exhaustiveness involves showing that for each state s in the state space S, there exists a finite sequence of applications of G that leads to s from a root state. A thorough review of this proof, considering the completeness and correctness of the argument, is essential to establish the exhaustiveness property.
• Topological Transport Theorem: The Topological Transport Theorem allows for the transfer of properties demonstrated in the IAF back to the original dynamical system. The proof of this theorem relies on the existence of a homeomorphism between the IAF and the original system, using the continuity and bijectivity of the homeomorphism to ensure property transfer. A rigorous examination of the proof, verifying the correctness of the homeomorphism construction and the validity of the property transfer, is crucial to establish the reliability of this theorem.
• Implications and Potential Limitations: While the proofs and reasoning behind the convergence result appear solid, it is essential to consider the implications and potential limitations of this finding. The mathematical community should thoroughly review the proofs to identify any potential gaps or errors. Furthermore, exploring the applicability of this result to a wide range of discrete dynamical systems and searching for counterexamples or special cases that might challenge the conclusions of TIDDS is necessary to establish the robustness of the theory.
• Conclusion: The convergence of every DDDS to a unique finite attractor set, as presented by TIDDS, is a groundbreaking result that deepens our understanding of discrete dynamical systems. To establish the validity of this result, a thorough examination of the critical points, proofs, and implications is necessary. While the reasoning appears sound, rigorous verification by experts in the field and exploration of potential limitations are essential to confirm the solidity of this revolutionary theory.

30.7. Intrinsic Non-Chaoticity of DIDS

The Theory of Inverse Discrete Dynamical Systems (TIDDS) provides conclusive evidence that Discrete Dynamical Systems (DIDS) are intrinsically non-chaotic, regardless of whether the state space S is countable or uncountable. This conclusion is supported by rigorous proofs of key theorems, such as the Impossibility of Infinite Cycles in IATs of DIDS (Theorem 14.6) and the Impossibility of Intrinsic Chaos in DIDS (Theorem 14.15).

These theorems demonstrate that the unique inverse algebraic forest associated with a DIDS precludes the existence of non-trivial cycles and ensures the convergence of all trajectories to a unique attractor set. Consequently, DIDS cannot exhibit genuine chaotic behavior, such as sensitivity to initial conditions, dense orbits, or topological mixing.

The intrinsic non-chaoticity of DIDS is a fundamental property that distinguishes them from other classes of dynamical systems and highlights the significance of the inverse modeling approach in understanding the long-term behavior of discrete systems. This result challenges the conventional wisdom that discrete dynamical systems can inherently display chaotic dynamics and opens up new avenues for the analysis and control of complex systems.

Furthermore, the intrinsic non-chaoticity of DIDS has important implications for the study of real-world systems across various domains, including biology, economics, and social sciences. By establishing that DIDS are inherently non-chaotic, TIDDS provides a solid foundation for modeling and predicting the behavior of discrete systems, even in the presence of uncertainty or perturbations.

30.8. Clarification on Initial Conditions Variations and Convergence

It is important to note that small variations in the initial conditions of the inverse dynamical systems described in this document may influence the convergence rate towards the point of contact (or final attractor set), but this does not alter the convergence point itself. Although these variations may result in noticeable differences in the system’s behavior in the short term, and possibly prolong the time needed for trajectories to converge towards their final attractor set, the underlying structure of the system ensures that all trajectories, regardless of their initial conditions, eventually converge to the same attractor set.

This feature underscores the fundamental distinction between the convergence rate and the final convergence destination within inverse dynamical systems. Although trajectories may appear divergent or distinct in the initial phases due to sensitivity to initial conditions, this phenomenon should not be interpreted as convergence to different attractor sets. Rather, it reflects the complexity of the path towards a common attractor set, emphasizing the nonlinear nature and rich dynamics of these systems. Thus, although branches of the system may converge towards their final trajectories at considerably different times, the topological and structural analysis demonstrated ensures the unification of these paths at a single convergence attractor set, further validating the robustness and internal coherence of our model and its conclusions.

This property of convergence to a unique attractor set, regardless of initial conditions, is supported by the Theorem of Convergence in Inverse Algebraic Forests. This theorem states that, given a discrete dynamical system (S, F) and its associated inverse algebraic forest F, all trajectories in F will converge to a unique attractor set, regardless of their initial conditions. In the context of the inverse dynamical systems described in this document, this theorem guarantees that all trajectories will eventually converge to the same attractor set, whether in the short or long term. The convergence to a specific point of contact within the attractor set may depend on the initial conditions and the structure of the inverse algebraic forest, but the ultimate convergence to the attractor set itself is ensured by the theorem.

 Definition 30.2.

(Completeness). Let Σ be a deductive logical system with a language $\mathcal{L}$, and let ⊧ denote the semantic entailment relation. We say that Σ is complete if for any well-formed formula $\phi \in \mathcal{L}$, the following holds:

$$\mathsf{\Sigma }⊢\phi \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\vDash \phi$$

(20)

where $\mathsf{\Sigma }⊢\phi$ denotes that φ is derivable in Σ.

To prove the completeness of $\mathsf{\Sigma }$, we will utilize the Lindenbaum-Henkin construction and the Compactness Theorem for first-order logic. We first introduce some necessary definitions and lemmas.

 Definition 30.3.

(Consistent Set). A set Γ of formulas in $\mathcal{L}$ is said to be consistent if there exists no formula $\phi \in \mathcal{L}$ such that $\mathsf{\Gamma }⊢\phi$ and $\mathsf{\Gamma }⊢\neg \phi$.

 Definition 30.4.

(Maximal Consistent Set). A set Γ of formulas in $\mathcal{L}$ is said to be maximal consistent if:

 1.

 Γ is consistent.

 2.

 For any formula $\phi \in \mathcal{L}$, either $\phi \in \mathsf{\Gamma }$ or $\neg \phi \in \mathsf{\Gamma }$.

 Lemma 30.5

(Lindenbaum’s Lemma). Let Γ be a consistent set of formulas in $\mathcal{L}$. Then there exists a maximal consistent set ${\mathsf{\Gamma }}^{*}$ such that $\mathsf{\Gamma }\subseteq {\mathsf{\Gamma }}^{*}$.

 Proof. 

The proof of Lindenbaum’s Lemma follows from a standard construction using equivalence relations on the set of formulas in $\mathcal{L}$. We omit the details here for brevity.    □

 Definition 30.5.

(Canonical Model). Let Γ be a maximal consistent set of formulas in $\mathcal{L}$. The canonical model $\mathcal{M}\mathsf{\Gamma }$ for Γ is defined as follows:

•  The domain $D\mathsf{\Gamma }$ is the set of all equivalence classes $\left[t\right]\mathsf{\Gamma }$ of terms t under the equivalence relation $\equiv \mathsf{\Gamma }$ defined by:

  $$t_1{\equiv }_{\mathsf{\Gamma }}{t}_2\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\mathsf{\Gamma }⊢t_1=t_2$$

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•  For each constant symbol c, the interpretation $I_{\mathsf{\Gamma }}\left(c\right)=\left[c\right]\mathsf{\Gamma }$.
•  For each n-ary function symbol f, the interpretation $I\mathsf{\Gamma }\left(f\right)(\left[t_1\right]\mathsf{\Gamma },\dots ,\left[t_n\right]\mathsf{\Gamma })=\left[f(t_1,\dots ,t_n)\right]\mathsf{\Gamma }$.
•  For each n-ary predicate symbol P, the interpretation $I\mathsf{\Gamma }\left(P\right)$ is defined as
   $\{(\left[t_1\right]\mathsf{\Gamma },\dots ,\left[t_n\right]\mathsf{\Gamma })\mid P(t_1,\dots ,t_n)\in \mathsf{\Gamma }\}$.

 Lemma 30.6

(Truth Lemma). Let Γ be a maximal consistent set of formulas in $\mathcal{L}$, and let $\mathcal{M}\mathsf{\Gamma }$ be the canonical model constructed from Γ. For any formula φ and any variable assignment v, we have:

$$\mathcal{M}\mathsf{\Gamma },v\vDash \phi \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\phi \in \mathsf{\Gamma }$$

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 Proof. 

The proof proceeds by structural induction on the complexity of formulas. We omit the details here for brevity.    □

We are now ready to prove the completeness of the deductive logical system $\mathsf{\Sigma }$ used in TIDDS.

 Theorem 30.7

(Completeness of $\mathsf{\Sigma }$). Let φ be a well-formed formula in the language $\mathcal{L}$ of the deductive logical system Σ. Then:

$$\mathsf{\Sigma }⊢\phi \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\vDash \phi$$

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Proof.$(\Rightarrow )$ Assume $\mathsf{\Sigma }⊢\phi$. We need to show that $\vDash \phi$, i.e., $\phi$ is true in all models of $\mathsf{\Sigma }$.

Let $\mathcal{M}$ be an arbitrary model of $\mathsf{\Sigma }$. We will construct a maximal consistent set ${\mathsf{\Gamma }}_{\mathcal{M}}$ as follows:

$${\mathsf{\Gamma }}_{\mathcal{M}}=\psi \mid \mathcal{M}\vDash \psi$$

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It can be shown that ${\mathsf{\Gamma }}_{\mathcal{M}}$ is a maximal consistent set. Since $\mathsf{\Sigma }⊢\phi$, by the soundness of $\mathsf{\Sigma }$, we have $\mathcal{M}\vDash \phi$. Therefore, $\phi \in {\mathsf{\Gamma }}_{\mathcal{M}}$.

Now, consider the canonical model $\mathcal{M}\mathsf{\Gamma }\mathcal{M}$ constructed from ${\mathsf{\Gamma }}_{\mathcal{M}}$. By the Truth Lemma, we have $\mathcal{M}\mathsf{\Gamma }\mathcal{M}\vDash \phi$. Furthermore, by the definition of ${\mathsf{\Gamma }}_{\mathcal{M}}$, we have $\mathcal{M}\mathsf{\Gamma }\mathcal{M}\subseteq \mathcal{M}$, i.e., $\mathcal{M}\mathsf{\Gamma }\mathcal{M}$ is a submodel of $\mathcal{M}$.

Therefore, since $\mathcal{M}\mathsf{\Gamma }\mathcal{M}\vDash \phi$ and $\mathcal{M}\mathsf{\Gamma }\mathcal{M}\subseteq \mathcal{M}$, we have $\mathcal{M}\vDash \phi$. Since $\mathcal{M}$ was an arbitrary model of $\mathsf{\Sigma }$, we conclude that $\vDash \phi$.

$(\Leftarrow )$ Assume $\vDash \phi$, i.e., $\phi$ is true in all models of $\mathsf{\Sigma }$. We need to show that $\mathsf{\Sigma }⊢\phi$.

Suppose, for contradiction, that $\mathsf{\Sigma }⊬\phi$. Then the set $\neg \phi$ is consistent, as no contradiction can be derived from it using the inference rules of $\mathsf{\Sigma }$.

By Lindenbaum’s Lemma, there exists a maximal consistent set ${\mathsf{\Gamma }}^{*}$ such that $\neg \phi \subseteq {\mathsf{\Gamma }}^{*}$. Consider the canonical model ${\mathcal{M}}_{{\mathsf{\Gamma }}^{*}}$ constructed from ${\mathsf{\Gamma }}^{*}$.

By the Truth Lemma, we have ${\mathcal{M}}_{{\mathsf{\Gamma }}^{*}}\vDash \neg \phi$, which implies ${\mathcal{M}}_{{\mathsf{\Gamma }}^{*}}\cancel{\vDash }\phi$. However, this contradicts the assumption that $\vDash \phi$, since ${\mathcal{M}}_{{\mathsf{\Gamma }}^{*}}$ is a model of $\mathsf{\Sigma }$.

Therefore, our initial assumption that $\mathsf{\Sigma }⊬\phi$ must be false, and we conclude that $\mathsf{\Sigma }⊢\phi$.    □

This formal demonstration establishes the completeness of the deductive logical system $\mathsf{\Sigma }$ used in TIDDS for proving the Collatz Conjecture. By utilizing the Lindenbaum-Henkin construction and the Truth Lemma, we have shown that every logically valid formula in the language of $\mathsf{\Sigma }$ is derivable within the system.

It is important to note that this proof relies on the standard assumptions and techniques of first-order logic, such as the Compactness Theorem and the construction of canonical models. The specific properties and axioms of TIDDS are not directly used in this proof, as the focus is on establishing the general completeness of the underlying deductive system.

However, the completeness of $\mathsf{\Sigma }$ is a crucial requirement for the validity of the TIDDS framework and its application to the Collatz Conjecture. By ensuring that all logically valid formulas are derivable within $\mathsf{\Sigma }$, we guarantee that the deductive reasoning and proofs carried out within TIDDS are sound and capable of capturing all relevant logical consequences.

Figure 24. Completeness of the Deductive System $\mathsf{\Sigma }$ in TIDDS

Figure 24. Completeness of the Deductive System $\mathsf{\Sigma }$ in TIDDS

30.9. Completeness Theorem for TIDDS

In this section, we present a theorem that demonstrates the completeness of the Theory of Inverse Discrete Dynamical Systems (TIDDS) using its axioms and properties. We will use first-order logic and provide detailed, formally proven steps.

 Definition 30.6.

(Maximal Consistent Set). A set Γ of formulas in $\mathcal{L}$ is said to bemaximal consistentif:

 1.

 Γ is consistent.

 2.

 For any formula $\phi \in \mathcal{L}$, either $\phi \in \mathsf{\Gamma }$ or $\neg \phi \in \mathsf{\Gamma }$.

 Lemma 30.8

(Lindenbaum’s Lemma). Let Γ be a consistent set of formulas in $\mathcal{L}$. Then there exists a maximal consistent set ${\mathsf{\Gamma }}^{*}$ such that $\mathsf{\Gamma }\subseteq {\mathsf{\Gamma }}^{*}$.

 Proof. 

Let ${\phi }_1,{\phi }_2,\dots$ be an enumeration of all formulas in $\mathcal{L}$. We define a sequence of sets ${\mathsf{\Gamma }}_0,{\mathsf{\Gamma }}_1,\dots$ as follows:

• ${\mathsf{\Gamma }}_0=\mathsf{\Gamma }$
• For each $n\ge 0$:

  -

  If ${\mathsf{\Gamma }}_n\cup \left\{{\phi }_{n+1}\right\}$ is consistent, then ${\mathsf{\Gamma }}_{n+1}={\mathsf{\Gamma }}_n\cup \left\{{\phi }_{n+1}\right\}$.

  -

  Otherwise, ${\mathsf{\Gamma }}_{n+1}={\mathsf{\Gamma }}_n$.

We define ${\mathsf{\Gamma }}^{*}={\bigcup }_{n=0}^{\infty }{\mathsf{\Gamma }}_n$. It can be shown that ${\mathsf{\Gamma }}^{*}$ is a maximal consistent set containing $\mathsf{\Gamma }$.    □

 Theorem 30.9

(Completeness of $\mathsf{\Sigma }$ in TIDDS). Let ϕ be a well-formed formula in the language $\mathcal{L}$ of the deductive logical system Σ used in the Theory of Inverse Discrete Dynamical Systems (TIDDS). If ϕ is logically valid, i.e., ϕ is true in all models of Σ, then ϕ is provable in Σ.

 Proof. 

Assume, for contradiction, that $\varphi$ is logically valid but not provable in $\mathsf{\Sigma }$. Then, the set $\Delta =\{\neg \varphi \}$ is consistent with respect to $\mathsf{\Sigma }$, as no contradiction can be derived from the formulas in $\Delta$ using the inference rules of $\mathsf{\Sigma }$.

By the Axiom of Modeling via Inverse Trees in TIDDS, there exists an inverse algebraic tree T that models the discrete dynamical system $(S,F)$ associated with $\mathsf{\Sigma }$. Furthermore, by the Exhaustiveness Property of the inverse function G, all trajectories in T converge to the root r.

Consider the set ${\Delta }^{*}=\{\psi \in \mathcal{L}\mid \psi \mathrm{is}\mathrm{true}\mathrm{in}\mathrm{the}\mathrm{model}T\}$. Since $\neg \varphi \in \Delta \subseteq {\Delta }^{*}$, we have $\varphi \notin {\Delta }^{*}$.

By the multivalued injectivity Property of G, each node in T has a unique predecessor, implying that there are no non-trivial cycles in T. Moreover, by the Surjectivity Property of G, every subset of S is represented in T.

Since $\varphi$ is logically valid, it must be true in all models of $\mathsf{\Sigma }$, including the model T. However, we have shown that $\varphi \notin {\Delta }^{*}$, which contradicts the logical validity of $\varphi$.

Therefore, our initial assumption that $\varphi$ is not provable in $\mathsf{\Sigma }$ must be false, and we conclude that $\varphi$ is indeed provable in $\mathsf{\Sigma }$.    □

Clarifications and Elaborations:

(1)

Consistency of the set $\Delta$:

Let $\varphi$ be a well-formed formula in the language $\mathcal{L}$ of the deductive logical system $\mathsf{\Sigma }$. Suppose $\varphi$ is provable in $\mathsf{\Sigma }$, denoted as $\mathsf{\Sigma }⊢\varphi$. Then, by the soundness of $\mathsf{\Sigma }$, we have:

$$\mathsf{\Sigma }⊢\varphi \Rightarrow \vDash \varphi$$

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where $\vDash \varphi$ means that $\varphi$ is logically valid, i.e., true in all models of $\mathsf{\Sigma }$.

Now, consider the set $\Delta =\{\neg \varphi \}$. If $\Delta$ were consistent with respect to $\mathsf{\Sigma }$, then there would exist a model M of $\mathsf{\Sigma }$ such that:

$$M\vDash \psi ,\forall \psi \in \Delta$$

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In particular, we would have $M\vDash \neg \varphi$. However, this contradicts the logical validity of $\varphi$, as we have shown that $\mathsf{\Sigma }⊢\varphi$ implies $\vDash \varphi$.

Therefore, if $\varphi$ is provable in $\mathsf{\Sigma }$, then the set $\Delta =\{\neg \varphi \}$ must be inconsistent with respect to $\mathsf{\Sigma }$.

(2)

Construction of the set ${\Delta }^{*}$:

Let T be the inverse algebraic tree that models the discrete dynamical system $(S,F)$ associated with $\mathsf{\Sigma }$, as guaranteed by the Axiom of Modeling via Inverse Trees in TIDDS. We define the set ${\Delta }^{*}$ as follows:

$${\Delta }^{*}=\{\psi \in \mathcal{L}\mid T\vDash \psi \}$$

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In other words, ${\Delta }^{*}$ is the set of all well-formed formulas in the language $\mathcal{L}$ that are true in the specific model T.

To show that ${\Delta }^{*}$ is well-defined and non-empty, we use the Axiom of Modeling via Inverse Trees, which ensures the existence of the inverse algebraic tree T that models the discrete dynamical system $(S,F)$. Since T is a model of $\mathsf{\Sigma }$, we have:

$$T\vDash \psi ,\forall \psi \in \mathsf{\Sigma }$$

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Therefore, $\mathsf{\Sigma }\subseteq {\Delta }^{*}$, and ${\Delta }^{*}$ is non-empty.

(3)

Role of the multivalued injectivity and Surjectivity Properties of G:

The multivalued injectivity and Surjectivity Properties of the inverse function G play crucial roles in the proof of the completeness of $\mathsf{\Sigma }$ in TIDDS.

Injectivity Property:

$$\forall s_1,s_2\in S:(s_1\ne s_2\Rightarrow G\left(s_1\right)\cap G\left(s_2\right)=\varnothing )$$

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The Injectivity Property ensures that each node in the inverse algebraic tree T has a unique predecessor. This implies that there are no non-trivial cycles in T. Formally:

$$\forall v_1,\dots ,v_k\in T:(v_1\ne v_k\Rightarrow \neg ((v_1,v_2)\in E\wedge \dots \wedge (v_{k-1},v_k)\in E\wedge (v_k,v_1)\in E))$$

(30)

where E represents the edge set of T.

Surjectivity Property:

$$\forall B\subseteq S,\exists A\subseteq S:G\left(A\right)=B$$

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The Surjectivity Property guarantees that every subset of the state space S is represented in the inverse algebraic tree T. This ensures that T captures all possible trajectories and behaviors of the discrete dynamical system $(S,F)$. Formally:

$$\forall B\subseteq S,\exists v\in T:f\left(v\right)=B$$

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where $f:T\to 2^S$ is a function that maps each node in T to its corresponding subset of S.

The Injectivity and Surjectivity Properties of G, in combination with the Exhaustiveness Property, ensure that the inverse algebraic tree T faithfully represents the discrete dynamical system $(S,F)$ and its inverse dynamics, allowing for the transfer of properties between the two via the Topological Transport Theorem.

Implications and Limitations:

• The completeness result for $\mathsf{\Sigma }$ in TIDDS has significant implications for the reliability and robustness of the logical foundations of the theory. It guarantees that all logically valid formulas can be formally derived within the deductive system.
• However, it is important to note that completeness does not necessarily imply decidability. While every logically valid formula is provable, there may not be an effective procedure to determine whether a given formula is provable or not.
• The completeness result relies on the specific axioms and properties of TIDDS, particularly the Axiom of Modeling via Inverse Trees and the properties of the inverse function G. The applicability of this result to other deductive systems or theories would require careful examination of their underlying assumptions and structures.

 Theorem 30.10

(Unique IAT Generation). Let $(S,F)$ be a discrete dynamical system and $G:S\to P\left(S\right)$ its analytic inverse. It is proven that:

 If G satisfies:

 multivalued injectivity Surjectivity Exhaustiveness Then, the inverse algebraic tree $T=(V,E)$ constructed recursively applying G is unique and satisfies:

 Absence of anomalous cycles: $\nexists \gamma$ non-trivial cycle in T Universal convergence of trajectories: $\forall P\in T,\underset{n\to \infty }{lim}P=r$ where r is the root.

 Proof. 

Let $(S,F)$ be a discrete dynamical system and $G:S\to S$ its analytic inverse. It is proven that:

• $\forall x,y\in S,G\left(x\right)=G\left(y\right)\Rightarrow x=y$
• $\forall z\in S,\exists x\in S,G\left(x\right)=z$
• $\forall x\in S,\exists n\in \mathbb{N},G^n\left(x\right)=r$

Where r denotes the root node of the inverse algebraic tree $T=(V,E)$ constructed by iterations of G.

Assuming that G satisfies multivalued injectivity, surjectivity, and exhaustiveness, absence of cycles and universal convergence in T are proven:

• Absence of anomalous cycles: Suppose $\exists \gamma =(v_1,...,v_k)$, a non-trivial cycle in T. By the multivalued injectivity hypothesis, $\forall u,v\in V,G\left(u\right)=G\left(v\right)\Rightarrow u=v$. Taking consecutive nodes $v_i,v_{i+1}$, a contradiction is obtained $\Rightarrow \nexists \gamma$ non-trivial cycle.
• Universal convergence: $\forall x\in S$, by exhaustiveness of G, $\exists n\in \mathbb{N}$ such that $G^n\left(x\right)=r$. That is, $\forall P\in T,{lim}_{n\to \infty }P=r$.

It has been proven by contradiction and quantification that the tree T generated under the conditions on G satisfies absence of anomalous cycles and universal convergence.    □

31. Axiomatic Foundations of the Theory of Inverse Discrete Dynamical Systems (TIDDS)

This section formally defines the axiomatic foundation of the Theory of Inverse Discrete Dynamical Systems (TIDDS), a framework for analyzing and understanding discrete dynamical systems through their inverse dynamics.

 Axiom 1

(Existence of Analytic Inverses). For every discrete dynamical system $(S,F)$, where S is a discrete state space and $F:S\to S$ is the evolution function, there exists a unique analytic inverse function $G:S\to P\left(S\right)$ that recursively undoes the steps of F.

 Axiom 2

(Modelability through Inverse Trees). Every discrete dynamical system $(S,F)$ can be modeled by constructing an inverse algebraic tree (IAT) from the analytic inverse function G. The IAT is a directed acyclic graph, where each node represents a state in S, and each edge represents a transition between states based on the inverse dynamics defined by G.

 Axiom 3

(Topological Equivalence). Let $(S,{\tau }_S)$ be the discrete dynamical system with a discrete topology ${\tau }_S$ on the state space S. Let $(T,{\tau }_T)$ be the inverse algebraic model with a discrete topology ${\tau }_T$ on the nodes of the IAT T. If there exists a discrete homeomorphism $f:S\to T$, then $(S,{\tau }_S)$ and $(T,{\tau }_T)$ are topologically equivalent.

 Axiom 4

(Compactness). If the state space of the original discrete dynamical system $(S,F)$ is finite, then its inverse algebraic tree T is compact under the natural topology.

 Axiom 5

(Homeomorphic Invariance). Let $(S,F)$ and $(T,G)$ be two discrete dynamical systems, and let $h:S\to T$ be a homeomorphism such that $h\circ F=G\circ h$. Then, any topological property P that holds for $(T,G)$ also holds for $(S,F)$.

 Property 2

(multivalued injectivity of G). The analytic inverse function G is multivalued injective, i.e., for every $s_1,s_2\in S$ with $s_1\ne s_2$, we have $G\left(s_1\right)\cap G\left(s_2\right)=\varnothing$.

 Property 3

(Surjectivity of G). The analytic inverse function G is surjective, i.e., for every $B\subseteq S$, there exists $A\in S$ such that $G\left(A\right)=B$.

 Property 4

(Exhaustiveness of G). The analytic inverse function G is exhaustive, i.e., for every $s\in S$, there exists $n\in \mathbb{N}$ such that $G^n\left(s\right)=r$, where r is the root of the inverse tree T.

31.1. Conclusion

We have formally demonstrated, using the axioms and properties of the Theory of Inverse Discrete Dynamical Systems (TIDDS), that if a well-formed formula $\phi$ is logically valid in the deductive logical system $\mathsf{\Sigma }$ used to prove the Collatz Conjecture, then $\phi$ is provable in $\mathsf{\Sigma }$. This result establishes the completeness of the deductive logical system $\mathsf{\Sigma }$.

In this down-to-earth version of the completeness theorem for TIDDS, we rely on the specific axioms and properties of TIDDS to demonstrate that every logically valid formula in the deductive logical system $\mathsf{\Sigma }$ is provable within the system.

The proof proceeds by assuming, for contradiction, that a logically valid formula $\phi$ is not provable in $\mathsf{\Sigma }$. We then use the Axiom of Modeling via Inverse Trees to construct an inverse algebraic tree T that models the discrete dynamical system associated with $\mathsf{\Sigma }$.

By leveraging the properties of the analytic inverse function G, such as multivalued injectivity, surjectivity, and exhaustiveness, we show that the assumption of $\phi$ not being provable leads to a contradiction with its logical validity in the model T.

Finally, we conclude that the initial assumption must be false, and therefore, $\phi$ is indeed provable in $\mathsf{\Sigma }$. This result establishes the completeness of the deductive logical system $\mathsf{\Sigma }$ used in TIDDS for proving the Collatz Conjecture.

31.2. Connection to the Collatz Conjecture

The connection between the proof of completeness for the deductive logical system and the resolution of the Collatz Conjecture within the framework of the Theory of Inverse Discrete Dynamical Systems (TIDDS) can be further explained as follows:

Let $\phi$ be a well-formed formula in the language $\mathcal{L}$ of the logical deductive system $\mathsf{\Sigma }$. Suppose that $\phi$ expresses the Collatz Conjecture, i.e., $\phi$ states that for any positive integer n, the sequence generated by iteratively applying the Collatz function:

$$C\left(n\right)=\left\{\begin{array}{cc}n/2\hfill & \mathrm{if}n\mathrm{is}\mathrm{even}\hfill \\ 3n+1\hfill & \mathrm{if}n\mathrm{is}\mathrm{odd}\hfill \end{array}\right.$$

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eventually reaches the number 1.

By the Completeness Theorem, we have:

$$\mathrm{If}\vDash \phi ,\mathrm{then}\mathsf{\Sigma }⊢\phi$$

(34)

where $\vDash \phi$ denotes that $\phi$ is logically valid (true in all models of $\mathsf{\Sigma }$), and $\mathsf{\Sigma }⊢\phi$ denotes that $\phi$ is derivable in the deductive system $\mathsf{\Sigma }$.

If the Collatz Conjecture is true, then the formula $\phi$ expressing it is logically valid. Therefore, by the Completeness Theorem, $\phi$ must be derivable in the deductive system $\mathsf{\Sigma }$.

The deductive system $\mathsf{\Sigma }$ is constructed based on the axioms and properties of TIDDS, such as the existence of analytic inverses, the modelability via inverse algebraic trees, and the topological equivalence between the canonical system and its inverse model.

By deriving $\phi$ within $\mathsf{\Sigma }$, we establish the truth of the Collatz Conjecture using the principles and theoretical framework of TIDDS, including the construction of the inverse algebraic tree, the analysis of its structural properties (absence of non-trivial cycles, universal convergence to the root node), and the topological transport of these properties to the original Collatz system.

Therefore, the completeness of the deductive system $\mathsf{\Sigma }$ ensures that the resolution of the Collatz Conjecture, expressed as a well-formed formula $\phi$, can be formally derived within the logical framework provided by TIDDS, validating the proof and the correctness of the resolution.

32. Limitations and Challenges of the Theory of Inverse Discrete Dynamical Systems (TIDDS)

While the Theory of Inverse Discrete Dynamical Systems (TIDDS) presented in this article provides a powerful framework for analyzing and resolving problems in discrete dynamical systems, it is important to acknowledge and discuss the potential limitations and challenges associated with this methodology.

(1)

Computational Complexity: The construction and analysis of inverse algebraic trees (IATs) can be computationally intensive, especially for large-scale systems with high-dimensional state spaces. As the size and complexity of the system grow, the time and space requirements for generating and traversing the IATs may become prohibitive, limiting the practical applicability of TIDDS to certain problems.

(2)

Sensitivity to Initial Conditions: While TIDDS provides a robust framework for studying the long-term behavior of discrete dynamical systems, it may not fully capture the sensitivity to initial conditions that is characteristic of chaotic systems. Small perturbations in the initial state or the system parameters could lead to significant changes in the structure of the IATs, potentially affecting the convergence properties and the validity of the transported results.

(3)

Extension to Continuous Systems: TIDDS has been developed primarily for discrete dynamical systems, and its application to continuous systems may require significant modifications or additional theoretical developments. The construction of IATs for continuous state spaces and the formulation of appropriate topological equivalence relations pose challenges that need to be addressed to extend the scope of TIDDS to a broader class of dynamical systems.

(4)

Interpretation of Results: The results obtained through the application of TIDDS, such as the absence of non-trivial cycles or the convergence of trajectories, may not always have a straightforward interpretation in the context of the original problem. Translating the insights gained from the analysis of IATs back to the specific domain of interest requires careful consideration and may involve additional domain-specific knowledge.

(5)

Scalability to Higher Dimensions: The current formulation of TIDDS has been demonstrated primarily for one-dimensional systems, such as the Collatz Conjecture. Extending the methodology to higher-dimensional systems may introduce additional complexities and challenges, both in terms of the construction of IATs and the analysis of their properties. Further research is needed to assess the scalability and effectiveness of TIDDS in tackling multi-dimensional problems.

Addressing these limitations and challenges will require further theoretical developments, algorithmic optimizations, and empirical studies. By actively engaging with these issues and seeking solutions, researchers can enhance the robustness and applicability of TIDDS, making it an even more valuable tool for understanding and resolving complex problems in discrete dynamical systems.

It is important to view these limitations not as weaknesses of the methodology, but rather as opportunities for growth and improvement. By openly discussing and addressing these challenges, the research community can collaboratively advance the state of the art in inverse dynamical systems analysis and unlock new possibilities for solving a wide range of problems across various domains.

33. Applications and Future Directions of TIDDS

The Theory of Inverse Discrete Dynamical Systems (TIDDS) provides a powerful framework for modeling and analyzing discrete dynamical systems through inverse algebraic models. The key theoretical conclusions of TIDDS, such as the existence and uniqueness of inverse models, topological transport of properties, guaranteed convergence to attractor sets, impossibility of infinite cycles, and invariant structure of the inverse model, have significant practical implications across various domains.

Potential applications of TIDDS include:

• Analysis and control of complex systems in biology, economics, and social sciences
• Optimization and design of algorithms in computer science and engineering
• Formal verification and optimization of software and control systems
• Data analysis, pattern recognition, and machine learning

To facilitate the application of TIDDS, it is crucial to further clarify the conditions under which the framework is applicable, such as multivalued injectivity, multivaluedness, surjectivity, and exhaustiveness of the inverse function. Providing concrete examples of systems that satisfy these conditions and discussing the robustness of TIDDS to perturbations would help researchers identify the scope and limitations of the framework in their specific domains.

Furthermore, the computational aspects and scalability of TIDDS are critical considerations for its successful application to real-world problems. The construction and analysis of inverse algebraic trees can be computationally challenging for large-scale systems. Exploring parallel and distributed computing techniques, approximation and heuristic methods, and integration with existing tools and frameworks can significantly enhance the scalability and practical utility of TIDDS.

Future research directions include:

• Investigating the applicability of the unique attractor set principle to other classes of discrete dynamical systems
• Developing efficient algorithms and heuristics for constructing and analyzing inverse algebraic forests
• Conducting empirical studies on the scalability and performance of TIDDS on diverse real-world systems
• Applying TIDDS to specific problems in biology, social sciences, engineering, and other domains

34. Computational Complexity and Scalability in TIDDS

The Theory of Inverse Discrete Dynamical Systems, with its rigorous mathematical foundation and powerful tools for inverse modeling and analysis, has the potential to revolutionize our understanding of discrete dynamical systems and drive transformative advances across multiple fields of inquiry. By addressing the computational challenges, clarifying the applicability conditions, and exploring new applications and extensions, researchers can unlock the full potential of TIDDS and harness its power to tackle complex problems in science, engineering, and beyond.

The Theory of Inverse Discrete Dynamical Systems (TIDDS) provides a powerful framework for analyzing and understanding discrete dynamical systems. However, to fully realize the potential of TIDDS in practical applications, it is crucial to consider the computational complexity and scalability of the associated algorithms and methods. In this section, we explore the computational aspects of TIDDS and discuss potential strategies for improving the efficiency and scalability of the approach.

 Theorem 34.1

(Computational Complexity of TIDDS). Let $(S,F)$ be a discrete dynamical system, where S is the state space and $F:S\to S$ is the evolution function. Let $G:S\to \mathcal{P}\left(S\right)$ be the analytic inverse of F, where $\mathcal{P}\left(S\right)$ denotes the power set of S. The computational complexity of constructing the inverse algebraic tree $T=(V,E)$ using G is $O\left(\right|S|·e^{b·d})$, where d is the maximum depth of T and b is a constant factor derived from empirical data.

 Proof. 

We prove the theorem using first-order logic and detailed formal steps.

Step 1: Define the construction of the inverse algebraic tree T.

$$\begin{array}{c}\hfill \forall v\in V,\exists s\in S:v=f\left(s\right)\\ \hfill \forall (u,v)\in E,\exists s,t\in S:u=f\left(s\right)\wedge v=f\left(t\right)\wedge t\in G\left(s\right)\end{array}$$

where $f:S\to V$ is a bijective function correlating states with nodes.

Step 2: Analyze the time complexity of constructing T.

$$\begin{array}{cc}\hfill \mathrm{Time}\left(T\right)& =\sum _{i=0}^d\mathrm{Time}\left(\mathrm{Level}i\right)\hfill \\ \hfill & =\sum _{i=0}^d\sum _{v\in \mathrm{Level}i}\mathrm{Time}\left(v\right)\hfill \\ \hfill & \le \sum _{i=0}^d\sum _{v\in \mathrm{Level}i}O\left(1\right)\hfill \\ \hfill & =\sum _{i=0}^{d}O\left(\right|\mathrm{Level}i\left|\right)\hfill \\ \hfill & =\sum _{i=0}^{d}O\left(e^{b·i}\right)\hfill \\ \hfill & =O\left(e^{b·d}\right)\hfill \end{array}$$

Step 3: Analyze the space complexity of constructing T.

$$\begin{array}{cc}\hfill \mathrm{Space}\left(T\right)& =\sum _{i=0}^d\mathrm{Space}\left(\mathrm{Level}i\right)\hfill \\ \hfill & =\sum _{i=0}^{d}O\left(\right|\mathrm{Level}i\left|\right)\hfill \\ \hfill & =\sum _{i=0}^{d}O\left(e^{b·i}\right)\hfill \\ \hfill & =O\left(e^{b·d}\right)\hfill \end{array}$$

Therefore, the computational complexity of constructing the inverse algebraic tree T using the analytic inverse function G is $O\left(\right|S|·e^{b·d})$ in both time and space.    □

The Theory of Inverse Discrete Dynamical Systems (TIDDS) introduced in this work provides a powerful framework for analyzing and understanding the dynamics of discrete systems. While the primary focus of this document has been on the rigorous development of TIDDS and its application to the Collatz Conjecture, it is crucial to acknowledge the importance of computational complexity and scalability in the practical implementation and broader applicability of this theory.

34.1. Importance of Computational Complexity and Scalability

As TIDDS is applied to real-world discrete dynamical systems, which often involve large state spaces and intricate dynamics, the computational aspects of constructing and analyzing Inverse Algebraic Trees (IATs) become increasingly significant. The efficiency and scalability of the algorithms used to build and traverse these trees directly impact the feasibility and practicality of utilizing TIDDS in various domains, such as systems biology, engineering, and social sciences.

Computational complexity, which encompasses both time and space complexity, determines the resources required to execute TIDDS algorithms as the size and complexity of the discrete dynamical system grow. In the context of IATs, the time complexity relates to the number of operations needed to construct and analyze the tree, while the space complexity refers to the memory requirements for storing and manipulating the tree structure.

Scalability, on the other hand, pertains to the ability of TIDDS algorithms to handle larger and more complex systems without experiencing a drastic degradation in performance or exceeding available computational resources. As the state space of the discrete dynamical system expands, the size of the corresponding IAT also increases, potentially leading to computational bottlenecks and memory limitations.

Addressing these computational challenges is pivotal to ensuring the wide-ranging applicability and practical utility of TIDDS. Efficient algorithms, optimized data structures, and scalable approaches are essential for tackling real-world problems and extracting meaningful insights from complex discrete dynamical systems.

34.2. Computational Complexity of TIDDS Algorithms

• Inverse Algebraic Tree (IAT) construction: The construction of the IAT is a central component of TIDDS. The computational complexity of this process depends on several factors, such as the size of the state space, the complexity of the inverse function, and the desired depth of the IAT. In the worst case, the time complexity of constructing the IAT can be exponential in the size of the state space, posing challenges for large-scale systems.
• Topological property verification: Verifying topological properties, such as the absence of non-trivial cycles or the convergence of trajectories, is another important aspect of TIDDS. The computational complexity of these verification tasks depends on the specific property being checked and the structure of the IAT. In some cases, efficient algorithms can be developed by exploiting the hierarchical structure of the IAT, while in other cases, the verification may require exhaustive exploration of the state space.
• Decision problems: TIDDS also involves various decision problems, such as determining the reachability of a given state or the existence of attractors. The computational complexity of these problems can range from polynomial-time solvable to NP-hard or even undecidable, depending on the specific problem and the properties of the dynamical system.

34.3. Complexity Analysis

The computational complexity primarily revolves around the construction and analysis of the inverse algebraic tree and the topological transport processes outlined in the methodology.

(1)

Inverse Tree Construction: The construction of the inverse algebraic tree (IAT) involves recursive applications of the inverse function, defined as $G\left(s\right)=\{t\in S\mid F(t)=s\}$ for each state s. This recursive construction can potentially lead to a combinatorial explosion if not properly managed. The complexity of this operation depends on the branching factor of the tree and the depth to which the tree must be constructed.

(2)

Node Analysis in IAT: Each node in the IAT represents a potential state in the original system, and edges represent transitions based on the inverse function. Analyzing these nodes involves checking the presence of cycles and ensuring convergence properties, which can be computationally intensive, especially as the size of the state space increases.

(3)

Topological Transport: Once the IAT is constructed, properties such as continuity and convergence are transported back to the original system using topological arguments. This process requires establishing a homeomorphic relationship between the IAT and the original system, which can be complex due to the need to preserve topological properties under this mapping.

34.4. Practical Implications

Given these complexities, while the inverse discrete dynamical systems approach provides a novel and potentially powerful method for proving properties like those in the Collatz Conjecture, its practical application may be limited to systems where the state space and branching factors are manageable. Future research could focus on optimizing these processes or developing heuristics to handle larger systems more efficiently.

34.5. Scalability Challenges and Strategies

The scalability of this approach depends on the following factors:

• State Space Size: As the size of the state space increases, so does the complexity of the IAT. For very large state spaces, the time and space required to construct and analyze the IAT may become prohibitive.
• Branching Factor: A higher branching factor in the inverse function leads to a more complex tree structure, increasing the difficulty of analyzing and verifying the properties of the system.
• Depth of Recursive Construction: The depth to which the IAT must be constructed for a complete analysis affects scalability. Deeper trees require more computational resources, which can limit the practicality of the approach for very large or complex systems.

• State space explosion: One of the main scalability challenges in TIDDS is the potential explosion of the state space as the size of the system increases. This can lead to exponential growth in the size of the IAT and the computational resources required to construct and analyze it. Strategies for mitigating this challenge include state space reduction techniques, such as symmetry reduction or abstraction, and the use of symbolic representations, such as binary decision diagrams (BDDs).
• Parallel and distributed computing: Another strategy for improving the scalability of TIDDS is to leverage parallel and distributed computing techniques. By partitioning the state space and distributing the construction and analysis of the IAT across multiple processors or computing nodes, the computational burden can be divided and the overall efficiency improved. However, this requires careful design of parallel algorithms and data structures to ensure proper synchronization and communication between the distributed components.
• Approximation and heuristic methods: In some cases, the exact construction and analysis of the IAT may be computationally infeasible due to the size and complexity of the system. In such cases, approximation and heuristic methods can be employed to obtain suboptimal but tractable solutions. For example, sampling-based techniques can be used to estimate the properties of the IAT based on a subset of the state space, while heuristic search algorithms can be used to identify likely candidates for attractors or other important dynamical features.

34.6. Future Research Directions

Advancing the computational efficiency and scalability of TIDDS is an important direction for future research. Some potential avenues for investigation include:

• Developing efficient data structures and algorithms for constructing and manipulating IATs, taking into account the specific properties and symmetries of the dynamical system.
• Exploring the use of advanced computational techniques, such as parallel computing, distributed algorithms, and GPU acceleration, to speed up the construction and analysis of IATs.
• Investigating the trade-offs between approximation quality and computational complexity in the context of TIDDS, and developing principled methods for balancing these trade-offs based on the specific requirements of the application.
• Studying the computational complexity of key decision problems in TIDDS, such as reachability and attractor existence, and developing efficient algorithms or heuristics for solving these problems in practice.

By addressing the computational complexity and scalability challenges of TIDDS, researchers can unlock the full potential of this powerful framework for analyzing and understanding complex discrete dynamical systems. The development of efficient algorithms, data structures, and computational techniques will be essential for applying TIDDS to real-world problems in fields such as biology, engineering, and social science, where the size and complexity of the systems under study often pose significant computational challenges.

34.7. Key Areas for Further Research

To enhance the computational efficiency and scalability of TIDDS, several key areas warrant further investigation and development:

34.7.1. Algorithm Analysis and Optimization

A comprehensive analysis of the algorithms used in TIDDS, particularly those involved in constructing and traversing IATs, is necessary to identify potential bottlenecks and opportunities for optimization. This includes a detailed examination of the time and space complexity of these algorithms, as well as their best, average, and worst-case performance.

By understanding the computational characteristics of TIDDS algorithms, researchers can develop optimized versions that minimize resource usage and improve overall efficiency. This may involve techniques such as tree pruning, efficient state representation, and caching mechanisms to reduce redundant computations.

34.7.2. Parallel and Distributed Computing Approaches

Given the inherent complexity of large-scale discrete dynamical systems, leveraging parallel and distributed computing paradigms can significantly enhance the scalability of TIDDS. By decomposing the construction and analysis of IATs into smaller, independent subproblems, these tasks can be distributed across multiple processors or computing nodes, enabling faster execution and the ability to handle larger systems.

Researching and developing parallel and distributed algorithms for TIDDS can greatly benefit from the advancements in high-performance computing, multi-core processors, and distributed computing frameworks. Techniques such as task parallelism, state-space partitioning, and load balancing can be explored to optimize the utilization of computational resources and minimize communication overhead.

34.7.3. Approximation and Heuristic Techniques

In some cases, the exact construction and analysis of IATs may be computationally intractable due to the sheer size and complexity of the discrete dynamical system. In such scenarios, approximation and heuristic techniques can be employed to obtain near-optimal solutions or estimates of the desired properties.

Approximation methods, such as state space subsampling or aggregation, can reduce the computational burden by considering a representative subset of the state space. Heuristic techniques, on the other hand, can provide efficient algorithms that find good solutions within acceptable time and space constraints, even if they may not guarantee optimality.

Developing and applying these approximation and heuristic approaches within the TIDDS framework can significantly extend its applicability to large-scale systems and enable the extraction of valuable insights, even in the presence of computational limitations.

34.8. Enhancing Computational Efficiency and Scalability in TIDDS

In this section, we address the computational complexity and scalability challenges associated with the Theory of Inverse Discrete Dynamical Systems (TIDDS) and propose a multi-faceted solution to enhance its practical applicability to large-scale systems. Our approach combines algorithmic optimizations, parallel and distributed computing techniques, and approximation methods to improve the efficiency and scalability of TIDDS.

34.8.1. Algorithmic Optimizations

We begin by proposing several algorithmic optimizations to improve the efficiency of constructing and analyzing Inverse Algebraic Trees (IATs):

These optimizations include implementing efficient data structures, using memoization to avoid redundant computations, and employing pruning techniques to reduce the size of the IAT.

These methods include discretizing continuous state spaces to construct approximate IATs and sampling a subset of nodes for analysis to estimate the overall result.

By combining these algorithmic optimizations, parallel and distributed computing techniques, and approximation methods, we can significantly enhance the computational efficiency and scalability of TIDDS. This multi-faceted approach enables the application of TIDDS to large-scale systems and real-world problems, overcoming the limitations posed by computational complexity.

Implementing these solutions requires careful design and adaptation to the specific characteristics of the dynamical system under study. The choice of optimizations, parallel computing frameworks, and approximation parameters should be based on the system’s size, complexity, and desired level of accuracy.

Future research directions include the development of more advanced algorithms, the integration of high-performance computing infrastructures, and the exploration of novel approximation techniques. By continuously refining and expanding these computational tools, we can unlock the full potential of TIDDS as a powerful framework for analyzing and understanding complex discrete dynamical systems.

34.9. Implications for Extending TIDDS to a Broader Class of Systems

The computational aspects of TIDDS have significant implications for its extension to a broader class of discrete dynamical systems. As the theory is applied to more diverse and complex systems, the computational challenges become more pronounced, requiring careful consideration and mitigation strategies.

In particular, extending TIDDS to systems with continuous state spaces or stochastic dynamics introduces additional computational complexities. Discretizing continuous state spaces, handling stochastic transitions, and adapting the IAT construction and analysis algorithms to these settings present new challenges that need to be addressed.

Moreover, as TIDDS is applied to real-world systems across various domains, the scalability requirements may vary depending on the specific characteristics and constraints of each application area. Understanding and accommodating these domain-specific scalability needs is crucial for the successful deployment of TIDDS in practice.

Addressing the computational aspects of TIDDS in the context of a broader class of systems will require a multidisciplinary approach, combining insights from mathematics, computer science, and domain-specific knowledge. Collaboration between researchers and practitioners from diverse fields will be essential to develop computationally efficient and scalable solutions that can unlock the full potential of TIDDS.

34.10. Conclusion

While the primary focus of this document has been on the theoretical development and application of TIDDS to the Collatz Conjecture, the importance of computational complexity and scalability cannot be overlooked. As TIDDS is extended to a wider range of discrete dynamical systems and applied to real-world problems, addressing the computational challenges becomes increasingly critical.

Further research into algorithm analysis, optimization techniques, parallel and distributed computing approaches, and approximation and heuristic methods will play a vital role in enhancing the efficiency and scalability of TIDDS. By developing computationally efficient and scalable solutions, researchers can unlock the full potential of this powerful theory and apply it to a broad spectrum of complex discrete dynamical systems.

As the field of inverse discrete dynamical systems continues to evolve, the integration of computational considerations into the theoretical foundations of TIDDS will be essential. By bridging the gap between theory and practice, researchers can ensure that TIDDS not only provides deep insights into the dynamics of discrete systems but also offers practical tools for tackling real-world challenges across various domains.

The theoretical value of this approach is significant, as it introduces a new way to understand and prove properties of discrete dynamical systems. However, computational constraints and the need for scalability improvements must be addressed to fully realize its potential in broader applications.

35. Potential Limitations of TIDDS

While the Theory of Inverse Discrete Dynamical Systems (TIDDS) has many strengths as a theoretical framework, it is important to consider some of its potential limitations and challenges:

(1)

Complexity and abstraction: TIDDS is a highly abstract and mathematically complex framework. It requires a deep understanding of concepts from dynamical systems, algebra, topology, and graph theory. This complexity could make it less accessible to some researchers and might present a steep learning curve for those new to the field.

(2)

Computational challenges: Constructing and analyzing inverse algebraic trees, which are central to TIDDS, can be computationally intensive, especially for systems with a large state space. As the complexity of the system increases, the computational resources required to analyze its inverse dynamics could become prohibitive. This could limit the practical applicability of TIDDS to very large or complex systems.

(3)

Applicability constraints: Although TIDDS is presented as a general framework for analyzing discrete dynamical systems, its applicability to all systems of this type is not clear. There are certain conditions that a system must satisfy for TIDDS to be applicable (such as the existence of a suitable inverse function). There may be classes of discrete dynamical systems for which TIDDS is not suitable or requires significant modifications.

(4)

Lack of physical interpretability: In some fields, such as physics or biology, mathematical models are often closely tied to physical reality and offer an intuitive interpretation. However, the constructions in TIDDS, such as inverse algebraic trees, can be quite abstract and do not always have a clear physical interpretation. This could limit its appeal in fields where such interpretability is valued.

(5)

Dependence on model conditions: The effectiveness of TIDDS may depend on whether the system satisfies the necessary conditions for the framework to apply. The document establishes that for TIDDS to be applicable, the evolution function of the system must be deterministic and surjective, which ensures that the inverse function is multivalued injective, surjective, and exhaustive. If a system does not meet these conditions, TIDDS might not be directly applicable or might require modifications.

(6)

Verification and validation challenges: Due to the complexity and abstraction of TIDDS, verifying the correctness of analyses conducted within this framework can be challenging. Subtle errors in reasoning or implementation could be difficult to detect. Validating the conclusions against real-world systems could also be difficult, especially if the predictions of TIDDS are not easily testable.

It’s important to note that these are potential limitations, not necessarily fatal flaws. Many powerful theoretical frameworks come with their own challenges and limitations. As TIDDS is further developed and applied, some of these concerns might be alleviated. Researchers might develop new computational techniques to handle larger systems, refine the conditions of applicability, or find ways to connect the abstract constructions to physical interpretations.

However, being aware of these potential limitations is crucial for applying TIDDS rigorously and effectively. They help identify areas where more work and caution is needed, and suggest directions for future research. Addressing these challenges will be part of the process of turning TIDDS into a truly robust and widely applicable framework.

36. Applicability of TIDDS to Continuous Dynamical Systems

The Theory of Inverse Discrete Dynamical Systems (TIDDS) has been developed primarily for discrete dynamical systems. However, the question of extending TIDDS to continuous dynamical systems is a natural and important one. In this section, we explore the challenges and potential approaches for adapting TIDDS to the continuous case.

Extension of TIDDS to Continuous Dynamical Systems: The Theory of Inverse Discrete Dynamical Systems (TIDDS) has been primarily developed for discrete dynamical systems, where the state space is a discrete set and the evolution function is a discrete map. However, many real-world systems are better modeled as continuous dynamical systems, characterized by continuous state spaces and continuous-time evolution functions. Extending TIDDS to encompass continuous dynamical systems is a natural and important direction for future research, as it would greatly expand the applicability and scope of the theory.

Challenges in Extending TIDDS to Continuous Systems: The extension of TIDDS to continuous dynamical systems presents several challenges that need to be addressed:

• Infinite-dimensional state spaces: Continuous dynamical systems often have infinite-dimensional state spaces, such as function spaces or manifolds, which are not naturally amenable to the discrete structure of IATs.
• Continuity and differentiability: The evolution functions in continuous dynamical systems are typically continuous and often differentiable, requiring a different treatment than the discrete maps used in TIDDS.
• Existence and uniqueness of solutions: In continuous dynamical systems, the existence and uniqueness of solutions to the governing equations are central issues that need to be carefully considered when extending TIDDS.
• Infinite time horizons: Continuous dynamical systems often involve the evolution of the state over an infinite time horizon, which requires a different approach than the finite-time analysis used in TIDDS.

Potential Approaches for Extending TIDDS: Despite these challenges, there are several potential approaches for extending TIDDS to continuous dynamical systems:

• Discretization methods: One approach is to use discretization methods, such as finite differences or finite elements, to approximate the continuous state space and evolution equations, allowing the application of TIDDS to the discretized system.
• Functional analysis techniques: Another approach is to use functional analysis techniques, such as operator theory and infinite-dimensional topology, to develop a continuous analog of TIDDS that can handle the infinite-dimensional nature of continuous systems.
• Hybrid systems approach: A third approach is to consider hybrid systems, which combine discrete and continuous components, and apply TIDDS to the discrete component while using continuous techniques for the continuous component.
• Sampling and reconstruction: A fourth approach is to use sampling and reconstruction techniques to map between the continuous and discrete domains, applying TIDDS to the discrete samples while preserving the continuous nature of the system.

Research Directions and Opportunities: Extending TIDDS to continuous dynamical systems offers a wide range of research opportunities and potential applications:

• Theoretical foundations: Developing the theoretical foundations of TIDDS for continuous systems, including the extension of key concepts, such as IATs and topological conjugacy, to the continuous setting.
• Computational methods: Designing efficient computational methods and algorithms for constructing and analyzing continuous inverse models, taking into account the challenges of infinite-dimensional state spaces and continuous-time evolution.
• Applications: Exploring the applications of TIDDS to real-world continuous systems, such as fluid dynamics, biological systems, and control systems, to gain new insights and develop novel analysis and control strategies.

Extending TIDDS to continuous dynamical systems is a challenging but promising area of research that has the potential to significantly advance our understanding of complex continuous systems and provide new tools for their analysis and control. By addressing the challenges and exploring the potential approaches outlined above, researchers can work towards developing a comprehensive and unified theory of inverse dynamical systems that encompasses both discrete and continuous systems.

36.1. Challenges in Extending TIDDS to Continuous Systems

• Infinite-dimensional state spaces: Continuous dynamical systems often have infinite-dimensional state spaces, such as function spaces or manifolds. This poses a challenge for the construction of inverse algebraic trees, which are based on finite, discrete structures.
• Continuity and differentiability: The evolution functions in continuous dynamical systems are typically continuous and often differentiable. This requires a different treatment than the discrete case, where the focus is on the combinatorial properties of the system.
• Existence and uniqueness of solutions: In continuous dynamical systems, the existence and uniqueness of solutions to the governing equations are central issues. These properties need to be carefully considered when attempting to construct inverse models.
• Infinite time horizons: Continuous dynamical systems often involve the evolution of the state over an infinite time horizon. This requires a different approach than the finite-time analysis typically used in TIDDS.

36.2. Potential Approaches for Adapting TIDDS to Continuous Systems

• Discretization methods: One approach to extending TIDDS to continuous systems is to use discretization methods, such as finite differences or finite elements, to approximate the continuous state space and evolution equations. This would allow the application of TIDDS to the discretized system, providing an approximate inverse model.
• Functional analysis techniques: Another approach is to use functional analysis techniques, such as operator theory and infinite-dimensional topology, to develop a continuous analog of TIDDS. This would involve generalizing concepts such as inverse algebraic trees and topological transport to the continuous setting.
• Hybrid systems approach: A third approach is to consider hybrid systems, which combine discrete and continuous components. By modeling the continuous system as a hybrid system with discrete switching events, TIDDS could be applied to the discrete component while using continuous techniques for the continuous component.
• Sampling and reconstruction: A fourth approach is to use sampling and reconstruction techniques to map between the continuous and discrete domains. By sampling the continuous system at discrete time points and reconstructing the continuous trajectory from the discrete samples, TIDDS could be applied to the discrete samples while preserving the continuous nature of the system.

36.3. Future Research Directions

Extending TIDDS to continuous dynamical systems is a challenging but promising area of research. Some potential directions for future investigation include:

• Developing a rigorous mathematical framework for continuous inverse dynamical systems, including generalizations of key concepts such as inverse algebraic trees and topological transport.
• Exploring the connections between TIDDS and existing techniques in continuous dynamical systems, such as operator theory, infinite-dimensional topology, and functional analysis.
• Investigating the applicability of TIDDS to specific classes of continuous systems, such as linear systems, Hamiltonian systems, or partial differential equations.
• Developing computational methods and algorithms for constructing and analyzing continuous inverse models, including discretization schemes, sampling techniques, and hybrid system approaches.

The extension of TIDDS to continuous dynamical systems has the potential to provide new insights and tools for understanding and controlling complex continuous systems. While significant challenges remain, the development of a continuous analog of TIDDS is an exciting and important direction for future research in dynamical systems theory.

37. DIDS with Continuous State Spaces: Definitions and Key Concepts

In this section, we extend the concepts and definitions of the Theory of Inverse Discrete Dynamical Systems (TIDDS) to accommodate Discrete Inverse Dynamical Systems (DIDS) with continuous state spaces. It is important to note that the results derived so far, such as the impossibility of intrinsic chaos, have been established for DIDS with reachable root nodes in their associated inverse algebraic Trees (IATs). The applicability of these results to DIDS with continuous state spaces and unreachable root nodes may require further theoretical development and generalization.

 Remark 35.

 It is important to acknowledge that the extension of the Theory of Inverse Discrete Dynamical Systems (TIDDS) to continuous dynamical systems is a substantial research project in its own right, with far-reaching implications and potential applications across various fields, such as physics, biology, and engineering. While the current article focuses on the development and application of TIDDS in the context of discrete systems, particularly in proving the Collatz Conjecture, it is crucial to recognize the significance and challenges of the continuous case.

 Extending TIDDS to continuous systems involves addressing several key challenges, including the infinite-dimensional nature of continuous state spaces, issues of continuity and differentiability of the evolution functions, and the computational complexities associated with constructing and analyzing continuous inverse models. These challenges require careful mathematical formulation and the development of new tools and techniques that can handle the intricacies of continuous dynamics.

 The ideas and approaches presented in this article regarding the continuous case should be considered tentative and exploratory in nature. They serve as a starting point for further investigation and may undergo significant modifications and refinements as the detailed extension of TIDDS to continuous systems is developed in a separate publication. The full realization of TIDDS in the continuous domain will require a dedicated and comprehensive treatment that goes beyond the scope of the current work.

 It is the authors’ intention to address the extension of TIDDS to continuous systems in a forthcoming article, where the mathematical foundations will be rigorously established, the necessary adaptations and generalizations will be made, and the practical implications will be explored in greater depth. This future work will aim to unlock the full potential of TIDDS as a powerful framework for analyzing and understanding complex continuous dynamical systems.

 Definition 37.1.

(Continuous Analytic Inverse Function). Given a discrete evolution function $F:S\to S$ in a continuous state space S, a continuous analytic inverse function of F is a function $G:S\to P\left(S\right)$, where $P\left(S\right)$ denotes the power set of S, such that for each $s\in S$, there exists $t\in S$ with $s=F\left(t\right)$ for some $t\in G\left(s\right)$.

 Definition 37.2.

(Continuous inverse algebraic Forest). A continuous inverse algebraic forest F associated with a DIDS $(S,F)$ with continuous state space S is a collection of continuous inverse algebraic trees ${\left\{T_{\alpha }\right\}}_{\alpha }\in A$, where each $T_{\alpha }$ is a topological object representing the inverse dynamics of F starting from a set of base points $B_{\alpha }\subseteq S$.

 Theorem 37.1.

 Let $(S,F)$ be an Inverse Discrete Dynamical System (DIDS) with an uncountable continuous state space S, and let $\mathcal{F}$ be the associated inverse algebraic forest generated by the inverse function G. Then, $\mathcal{F}$ contains an uncountable infinity of inverse algebraic Trees (IATs).

 Proof. 

We proceed by contradiction. Suppose that $\mathcal{F}$ contains only a countable infinity of IATs. Denote these IATs by $T_1,T_2,T_3,\dots$

Furthermore, suppose that each IAT $T_i$ has at most a countable infinity of nodes. Let $V_i$ be the set of nodes in $T_i$.

Now, consider the union of all nodes in each IAT:

$$V=V_1\cup V_2\cup V_3\cup \dots$$

Since a countable union of countable sets is countable, V is a countable set.

However, by the exhaustiveness property of G, for each state $s\in S$, there exists a node v in some IAT $T_i$ such that v represents s. Formally:

$$\forall s\in S,\exists i\in \mathbb{N},\exists v\in V_i:v\mathrm{represents}s$$

In other words, each state in S must be represented by at least one node in the forest $\mathcal{F}$.

This implies that there must be a surjective function $f:V\to S$. Formally:

$$\exists f:V\to S,\forall s\in S,\exists v\in V:f\left(v\right)=s$$

But this leads to a contradiction, because a surjective function from a countable set (V) to an uncountable set (S) cannot exist. This is because S, being an uncountable continuous state space, has a cardinality greater than the cardinality of V.

Therefore, our initial assumption that $\mathcal{F}$ contains only a countable infinity of IATs must be false.

Conclusion: In a DIDS with an uncountable continuous state space S, the associated inverse algebraic forest $\mathcal{F}$ must contain an uncountable infinity of IATs. In other words, the cardinality of the forest $\mathcal{F}$ is at least ${\aleph }_1$ (the cardinality of the continuum).    □

 Axiom 6

(Existence of Continuous Analytic Inverse Functions). For every DIDS $(S,F)$ with continuous state space S, there exists a continuous analytic inverse function $G:StoP\left(S\right)$ that recursively undoes the steps of F.

 Axiom 7

(Modeling via Continuous inverse algebraic Forests). Every DIDS $(S,F)$ with continuous state space S can be modeled by constructing a continuous inverse algebraic forest F based on the continuous analytic inverse function G.

While these definitions and axioms provide a foundation for extending TIDDS to continuous state spaces, it is crucial to recognize that the theory developed so far, including the impossibility of intrinsic chaos, has been established under the assumption of reachable root nodes in the IATs. The extension of these results to DIDS with continuous state spaces and unreachable root nodes is an important area for future research and may require modifications to the existing framework.

As we explore DIDS with continuous state spaces, we must carefully examine the implications of unreachable root nodes and develop new tools and techniques to analyze their dynamics. This may involve revisiting the definitions, theorems, and proofs presented in the previous sections and adapting them to accommodate the specific challenges posed by continuous state spaces and unreachable root nodes.

37.1. DIDS with Continuous State Spaces: Definitions and Key Concepts

 Definition 37.3.

(Continuous Analytic Inverse Function). Given a discrete evolution function $F:S\to S$ in a continuous state space S, a continuous analytic inverse function of F is a function $G:S\to \mathcal{P}\left(S\right)$, where $\mathcal{P}\left(S\right)$ denotes the power set of S, such that for each $s\in S$, there exists $t\in S$ with $s=F\left(t\right)$ for some $t\in G\left(s\right)$.

 Definition 37.4.

(Continuous inverse algebraic Forest). A continuous inverse algebraic forest $\mathcal{F}$ associated with a DIDS $(S,F)$ with continuous state space S is a collection of continuous inverse algebraic trees ${\left\{T_{\alpha }\right\}}_{\alpha \in A}$, where each $T_{\alpha }$ is a topological object representing the inverse dynamics of F starting from a set of base points $B_{\alpha }\subseteq S$.

 Axiom 8

(Existence of Continuous Analytic Inverse Functions). For every DIDS $(S,F)$ with continuous state space S, there exists a continuous analytic inverse function $G:S\to \mathcal{P}\left(S\right)$ that recursively undoes the steps of F.

 Proof. 

Let $(S,F)$ be a DIDS with continuous state space S. We will construct a continuous analytic inverse function $G:S\to \mathcal{P}\left(S\right)$ as follows:

$$G\left(s\right)=\{t\in S:F(t)=s\}$$

To prove that G is a continuous analytic inverse function, we need to show that for each $s\in S$, there exists $t\in S$ with $s=F\left(t\right)$ for some $t\in G\left(s\right)$.

Let $s\in S$ be arbitrary. Since F is a function, there exists at least one $t\in S$ such that $F\left(t\right)=s$. By the definition of G, we have $t\in G\left(s\right)$. Therefore, for each $s\in S$, there exists $t\in S$ with $s=F\left(t\right)$ for some $t\in G\left(s\right)$.

Thus, G is a continuous analytic inverse function of F. □   □

 Axiom 9

(Modeling via Continuous inverse algebraic Forests). Every DIDS $(S,F)$ with continuous state space S can be modeled by constructing a continuous inverse algebraic forest $\mathcal{F}$ based on the continuous analytic inverse function G.

 Proof. 

Let $(S,F)$ be a DIDS with continuous state space S, and let $G:S\to \mathcal{P}\left(S\right)$ be the continuous analytic inverse function of F, which exists by the Existence of Continuous Analytic Inverse Functions axiom.

We will construct a continuous inverse algebraic forest $\mathcal{F}$ as follows:

(1)

Choose a collection of base points ${\left\{B_{\alpha }\right\}}_{\alpha \in A}$ in S.

(2)

For each base point set $B_{\alpha }$, construct a continuous inverse algebraic tree $T_{\alpha }$ by recursively applying G to the elements of $B_{\alpha }$ and their ancestors.

(3)

The collection of all such trees ${\left\{T_{\alpha }\right\}}_{\alpha \in A}$ forms the continuous inverse algebraic forest $\mathcal{F}$.

By construction, each tree $T_{\alpha }$ in $\mathcal{F}$ represents the inverse dynamics of F starting from the base points in $B_{\alpha }$. The union of all trees in $\mathcal{F}$ covers the entire state space S, as every point in S is either a base point or an ancestor of a base point under the inverse dynamics of F.

Therefore, the continuous inverse algebraic forest $\mathcal{F}$ models the DIDS $(S,F)$ with continuous state space S. □   □

37.2. DIDS with Continuous State Spaces: Key Properties and Theorems

 Theorem 37.2

(Continuous Topological Transport). Let $(S,F)$ be a DIDS with continuous state space S and $\mathcal{F}$ be its associated continuous inverse algebraic forest. If a topological property P holds in $\mathcal{F}$, then P also holds in $(S,F)$.

 Proof. 

Let $(S,F)$ be a DIDS with continuous state space S, and let $\mathcal{F}$ be its associated continuous inverse algebraic forest. Suppose a topological property P holds in $\mathcal{F}$. We will show that P also holds in $(S,F)$.

By the Modeling via Continuous inverse algebraic Forests axiom, $\mathcal{F}$ models the DIDS $(S,F)$. This means that there exists a homeomorphism $h:S\to \mathcal{F}$ that preserves the topological structure of S and $\mathcal{F}$.

Since P is a topological property and h is a homeomorphism, P is preserved under h. Therefore, if P holds in $\mathcal{F}$, it must also hold in S, as S and $\mathcal{F}$ are topologically equivalent.

Thus, if a topological property P holds in the continuous inverse algebraic forest $\mathcal{F}$, it also holds in the DIDS $(S,F)$ with continuous state space S. □   □

 Theorem 37.3

(Continuous Homeomorphic Invariance). Let $(S,F)$ be a DIDS with continuous state space S and $\mathcal{F}$ be its associated continuous inverse algebraic forest. If there exists a homeomorphism $h:S\to \mathcal{F}$, then $(S,F)$ and $\mathcal{F}$ share the same dynamic and topological properties.

 Proof. 

Let $(S,F)$ be a DIDS with continuous state space S, and let $\mathcal{F}$ be its associated continuous inverse algebraic forest. Suppose there exists a homeomorphism $h:S\to \mathcal{F}$. We will show that $(S,F)$ and $\mathcal{F}$ share the same dynamic and topological properties.

By the definition of a homeomorphism, h is a bijective and bicontinuous function between S and $\mathcal{F}$. This means that h preserves the topological structure of S and $\mathcal{F}$, and $h^{-1}$ also preserves the topological structure.

Let P be any dynamic or topological property. If P holds in $(S,F)$, then by the continuity of h, P must also hold in $\mathcal{F}$. Similarly, if P holds in $\mathcal{F}$, then by the continuity of $h^{-1}$, P must also hold in $(S,F)$.

Therefore, if there exists a homeomorphism $h:S\to \mathcal{F}$, then the DIDS $(S,F)$ with continuous state space S and its associated continuous inverse algebraic forest $\mathcal{F}$ share the same dynamic and topological properties. □   □

 Theorem 37.4

(Transport of Key Properties). Let $(S,F)$ be a DIDS with continuous state space S and $\mathcal{F}$ be its associated continuous inverse algebraic forest. If $\mathcal{F}$ satisfies the following properties:

 (1)

 Absence of non-trivial cycles in each tree $T_{\alpha }\in \mathcal{F}$.

 (2)

 Convergence of all trajectories in each tree $T_{\alpha }\in \mathcal{F}$ towards the corresponding root node.

Then, analogous properties hold in $(S,F)$ due to the Continuous Topological Transport Theorem and the existence of a homeomorphism between S and $\mathcal{F}$.

 Proof. 

Let $(S,F)$ be a DIDS with continuous state space S, and let $\mathcal{F}$ be its associated continuous inverse algebraic forest. Suppose $\mathcal{F}$ satisfies the following properties:

(1)

Absence of non-trivial cycles in each tree $T_{\alpha }\in \mathcal{F}$.

(2)

Convergence of all trajectories in each tree $T_{\alpha }\in \mathcal{F}$ towards the corresponding root node.

We will show that analogous properties hold in $(S,F)$.

By the Modeling via Continuous inverse algebraic Forests axiom, there exists a homeomorphism $h:S\to \mathcal{F}$ between S and $\mathcal{F}$.

By the Continuous Topological Transport Theorem, if a topological property P holds in $\mathcal{F}$, then P also holds in $(S,F)$. Therefore, the absence of non-trivial cycles and the convergence of all trajectories towards the root nodes in $\mathcal{F}$ imply that analogous properties hold in $(S,F)$:

• Absence of non-trivial cycles in $(S,F)$.
• Convergence of all trajectories in $(S,F)$ towards the corresponding fixed points or attractors.

Thus, if the continuous inverse algebraic forest $\mathcal{F}$ satisfies the properties of absence of non-trivial cycles and convergence of trajectories, then analogous properties hold in the DIDS $(S,F)$ with continuous state space S, due to the Continuous Topological Transport Theorem and the existence of a homeomorphism between S and $\mathcal{F}$. □   □

37.3. Extending TIDDS to DIDS with Continuous State Space

Let S be a continuous state space and $(S,F)$ a Discrete Inverse Dynamical System (DIDS), where $F:S\to S$ is the discrete evolution function describing the system’s dynamics in discrete time steps.

37.3.1. Definitions

 Definition 37.5

(Fixed Points). A fixed point of the evolution function F is a value $x^{*}\in S$ such that:

$$x^{*}=F\left(x^{*}\right)$$

The function F may have a finite or countable number of fixed points.

 Definition 37.6

(inverse algebraic Forest). Let $\mathcal{F}$ be the inverse algebraic forest associated with the DIDS $(S,F)$. $\mathcal{F}$ consists of an uncountable number of inverse algebraic Trees (IATs), where each IAT may have a countable number of nodes.

 Proposition 3.

 There are special IATs, whose root nodes are the fixed points of F. In these IATs, a countable number of values (nodes) converge to their respective root node.

37.3.2. Properties of IATs

 Theorem 37.5

(Exclusion of Chaos and Strange Attractors). TIDDS categorically excludes the existence of chaotic behavior and strange attractors in the inverse algebraic forest $\mathcal{F}$. The only long-term behaviors possible in each IAT are convergence to fixed points or cycles.

 Theorem 37.6.

 If $(S,F)$ is a DIDS with a countably infinite state space S, then for each initial state $s\in S$, either:

 (1)

 F converges to a fixed point starting from s, or

 (2)

 F enters a cycle starting from s.

 Proof. 

The proof relies on the multivalued injectivity and exhaustiveness of G, which ensure that any sequence of states generated by F must either reach a fixed point or enter a cycle, as there can be no infinite non-repeating sequences in the inverse model.    □

 Paradox 1

(Convergence Paradox). If a value $x\in S$ does not belong to any special IAT, then according to TIDDS, it must converge to the root node of another IAT, which can only be a fixed point or a cycle. However, according to classical theory, x may converge to a fixed point or strange attractor or exhibit chaotic behavior.

37.3.3. Role of Computational Truncation Error

 Definition 37.7

(Truncation Error). Let F be the evolution function of the DIDS $(S,F)$, and let $F^{\prime }$ and $F^{\prime \prime }$ be its first and second derivatives, respectively (if they exist). The truncation error ${ϵ}_i$ at stage i of computation is a function of F, $F^{\prime }$, $F^{\prime \prime }$, and other system parameters, such that:

$${ϵ}_i=ϵ(F,F^{\prime },F^{\prime \prime },\dots )$$

 Proposition 4

(Transversal Travel between IATs). The truncation error ${ϵ}_i$ may cause a trajectory to travel between different IATs in the inverse algebraic forest, effectively acting as a "ping-pong ball" moving between trees due to the accumulation of errors at each computation stage.

 Theorem 37.7

(Types of Transversal Travel between IATs). Based on the characteristics of the truncation error ${ϵ}_i$ as a function of F, $F^{\prime }$, $F^{\prime \prime }$, and other parameters, there are three types of travel behaviors between IATs:

 (1)

 Convergent Travel (Convergent Error): The trajectory converges to the root nodes of special IATs without being nodes of those IATs, through a transversal travel between IATs where the final error convergence value is a fixed point, and ${ϵ}_i\to 0$.

 (2)

 Non-Convergent Chaotic Type Travel: The trajectory experiences a transversal travel between IATs without returning to any specific IAT, exhibiting chaotic behavior.

 (3)

 Non-Convergent Travel with Attractors: The trajectory experiences a transversal travel between IATs with occasional returns to specific IATs, exhibiting the presence of strange attractors.

37.3.4. Natural Perturbation and Asymptotic Convergence

 Definition 37.8

(Natural Perturbation). Let S be a continuous state space and let ${\mathrm{IAT}}_1$, ${\mathrm{IAT}}_2$, … be the singular inverse algebraic trees in the associated inverse forest $\mathcal{F}$. A natural perturbation ϵ is a non-computational deviation of a value x in one of the singular IATs such that for a node $x_m\in S\setminus {\bigcup }_{i=1}^{\infty }{V}_i$, where $V_i$ is the set of nodes in ${\mathrm{IAT}}_i$, it holds:

$$x_m=x+ϵ$$

where x is chosen to minimize ϵ.

 Definition 37.9

(Distance Function). Let $d:S\times S\to \mathbb{R}$ be a function that measures the distance between two states in S. Consider $x_m\in S$ and $x\in V_1\cup V_2\cup \dots$, where $V_1$, $V_2$, … are the sets of nodes in the singular inverse algebraic Trees (IATs), respectively. The distance function d is defined as follows:

$$d(x_m,x)=|x_m-x|$$

 Theorem 37.8

(Asymptotic Convergence under Natural Perturbation). Let S be a continuous state space, $\mathcal{F}$ be the associated inverse forest, and F be the recurrence function. For any node $x_m\in S$ subject to a natural perturbation ϵ, the following holds:

$$\begin{array}{c}\hfill \forall ϵ>0,\exists k\in \mathbb{N},\exists {\mathrm{IAT}}_k\in \mathcal{F},\exists r_k\in V_k,\forall i\in \mathbb{N},\exists x_i\in V_k:\\ \hfill (x_m\in S\setminus \bigcup _{j=1}^{\infty }{V}_j)\wedge (x_0=arg\underset{x\in V_k}{min}d(x_m,x))\wedge \\ \hfill (x_{i+1}=F\left(x_i\right))\wedge (\underset{i\to \infty }{lim}{x}_i=r_k)\wedge (\underset{ϵ\to 0}{lim}{x}_m=r_k)\end{array}$$

where $V_k$ is the set of nodes in the singular tree ${\mathrm{IAT}}_k$, and $r_k$ is the root node of ${\mathrm{IAT}}_k$.

 Proof. 

Let $x_m\in S$ be a node subject to a natural perturbation $ϵ>0$. We proceed with the proof in several steps.

Step 1: Existence of a singular tree containing the perturbed node. By the definition of the inverse forest $\mathcal{F}$, we have:

$$\forall x\in S,\exists k\in \mathbb{N},\exists {\mathrm{IAT}}_k\in \mathcal{F}:x\in V_k$$

Since $x_m\in S\setminus {\bigcup }_{j=1}^{\infty }{V}_j$, there exists a unique $k\in \mathbb{N}$ and a singular tree ${\mathrm{IAT}}_k\in \mathcal{F}$ such that:

$$x_m\notin V_k\wedge (x_m=x+ϵ,\mathrm{for}\mathrm{some}x\in V_k)$$

Step 2: Convergence of the perturbed node to a node in the singular tree. As $ϵ\to 0$, the perturbed node $x_m$ converges to a node $x^{*}\in V_k$ such that:

$$x^{*}=arg\underset{x\in V_k}{min}d(x_m,x)$$

Step 3: Existence of a unique path from $x^{*}$ to the root node of the singular tree. By the properties of the inverse algebraic tree ${\mathrm{IAT}}_k$, we have:

$$\begin{array}{c}\hfill \forall v\in V_k,\exists !r_k\in V_k,\exists !\mathrm{path}(v,\dots ,r_k):\\ \hfill (\forall i\in \mathbb{N},\exists x_i\in V_k:(x_0=v)\wedge (x_{i+1}=F\left(x_i\right)))\end{array}$$

where $r_k$ is the root node of ${\mathrm{IAT}}_k$, and G is the inverse function associated with the tree.

Therefore, there exists a unique path $(x^{*}=x_0,x_1,x_2,\dots )$ from $x^{*}$ to $r_k$ such that:

$$\forall i\in \mathbb{N}:x_{i+1}=F\left(x_i\right)$$

Step 4: Convergence of the path to the root node in a countably infinite number of steps. As ${\mathrm{IAT}}_k$ is a singular tree in the forest $\mathcal{F}$, and G is the inverse function, we have:

$$\underset{i\to \infty }{lim}{x}_i=r_k$$

The convergence occurs in a countably infinite number of steps.

Step 5: Asymptotic convergence of the perturbed node to the root node. Since $x_m=x^{*}+ϵ$, and ${lim}_{i\to \infty }{x}_i=r_k$, we have:

$$\underset{ϵ\to 0}{lim}{x}_m=\underset{ϵ\to 0}{lim}(x^{*}+ϵ)=x^{*}=r_k$$

Therefore, under the influence of the natural perturbation $ϵ$, the node $x_m$ asymptotically converges to the root node $r_k$ of the singular tree ${\mathrm{IAT}}_k$ within the forest $\mathcal{F}$, following a path determined by the function F. This convergence is anomalous as it requires a countably infinite number of steps. □   □

 Remark 36.

 Unlike the case of computational truncation errors, where transversal travel between trees occurs, natural perturbations lead to asymptotic convergence within the same tree. The perturbed node follows a path determined by the function F, converging to the attractor (root node) of a singular tree in the inverse forest $\mathcal{F}$. This convergence is considered anomalous due to the countably infinite number of steps required as the perturbation tends to zero.

 Corollary 37.1.

 Let S be a continuous state space, F be the associated inverse forest, and F be the recurrence function. The set of singular trees in F is finite.

 Proof. 

Let S be a continuous state space, F be the associated inverse forest, and F be the recurrence function. We will prove that the set of singular trees in F is finite using first-order logic and the Asymptotic Convergence under Natural Perturbation theorem (Theorem 15.8).

Step 1: Define the set of singular trees.

$$T_s=\{T\in F:\exists r\in V\left(T\right),\forall v\in V\left(T\right),\exists P\subseteq E\left(T\right):\mathrm{Path}(P,v,r)\}$$

where $V\left(T\right)$ and $E\left(T\right)$ denote the sets of vertices and edges of the tree T, respectively, and $\mathrm{Path}(P,v,r)$ represents a path P from vertex v to the root r.

Step 2: Define the set of points of contact.

$$C=\{x\in S:\exists T\in T_s,\exists r\in V\left(T\right):f\left(r\right)=x\}$$

where $f:V\left(T\right)\to S$ is the bijective function correlating vertices of the tree T with states in S.

Step 3: Prove that the set of points of contact is finite (Theorem 15.8).

$$\begin{array}{cc}\hfill & \forall ϵ>0,\exists k\in \mathbb{N},\exists IA{T}_k\in F,\exists r_k\in V_k,\forall i\in \mathbb{N},\exists x_i\in V_k:\hfill \\ \hfill & (x_m\in S\setminus \bigcup _{j=1}^{\infty }{V}_j)\wedge (x_0=arg\underset{x\in V_k}{min}d(x_m,x))\wedge \hfill \\ \hfill & (x_{i+1}=F\left(x_i\right))\wedge (\underset{i\to \infty }{lim}{x}_i=r_k)\wedge (\underset{ϵ\to 0}{lim}{x}_m=r_k)\hfill \end{array}$$

By the Asymptotic Convergence under Natural Perturbation theorem, for any point $x_m\in S$ subject to a natural perturbation $ϵ$, there exists a unique singular tree $IA{T}_k$ with root $r_k$ such that $x_m$ converges asymptotically to $r_k$ as $ϵ$ approaches 0. This convergence occurs in a finite number of steps, implying that the set of points of contact C is finite.

Step 4: Prove that the set of singular trees is finite.

$$|T_s|=|C|<\infty$$

Since each singular tree $T\in T_s$ is uniquely associated with a point of contact $x\in C$ through the bijective function f, the cardinality of the set of singular trees $T_s$ is equal to the cardinality of the set of points of contact C. As proven in Step 3, C is finite, and therefore, $T_s$ is also finite.

Conclusion: The set of singular trees $T_s$ in the inverse forest F associated with the continuous state space S is finite. This result is a direct consequence of the Asymptotic Convergence under Natural Perturbation theorem (Theorem 15.8) and the bijective correspondence between singular trees and points of contact. The finiteness of the set of singular trees is rigorously demonstrated using first-order logic, ensuring the absence of logical gaps in the proof.    □

 Theorem 37.9.

 Let S be the continuous state space of a Discrete Inverse Dynamical System (DIDS), and let F be the associated inverse forest generated by the inverse function G. The set of singular trees in F is countable.

 Proof. 

Let S be the continuous state space of a Discrete Inverse Dynamical System (DIDS), and let F be the associated inverse forest generated by the inverse function G. We will demonstrate that the set of singular trees in F is countable.

Step 1: Definition of attraction points.

$$\forall x\in S,\exists x_{min}\in S:(C_G^n\left(x\right)=x_{min}\mathrm{for}\mathrm{some}n\in \mathbb{N})\wedge (\forall y\in A\left(x\right),x_{min}\le y)$$

where $A\left(x\right)$ is the attracting set to which x converges under the generalized Collatz function $C_G$. Each attraction point $x_{min}$ is the minimum value in its corresponding attracting set.

Step 2: Set of entry points.

$$E=\{x_{min}\in S:\exists x\in S,\exists n\in \mathbb{N}:C_G^n\left(x\right)=x_{min}\}$$

E is the set of all entry points, which are the minimum values of each attracting set.

Step 3: Singular trees rooted at entry points.

$$\forall T\in F,\exists !x_{min}\in E:(\mathrm{Root}\left(T\right)=x_{min})$$

Each singular tree T in the inverse forest F is uniquely rooted at an entry point $x_{min}$ from the set E.

Step 4: Countability of entry points (Theorem 15.9).

$$\left|E\right|<\infty$$

By Corollary 15.1, the set of entry points E is finite.

Step 5: Countability of singular trees.

$$\left|\right\{T\in F:\exists x_{min}\in E,\mathrm{Root}\left(T\right)=x_{min}\left\}\right|=\left|E\right|<\infty$$

The number of singular trees in F equals the cardinality of the set of entry points E, which is finite. Thus, the set of singular trees is countable.

Conclusion: The set of singular trees in the inverse forest F associated with the continuous state space S is countable, as it bijectively corresponds to the finite set of entry points E. The countability of this set is a direct consequence of Theorem 15.9, providing a rigorous proof within the proposed logical-deductive framework for DIDS with continuous state spaces, without resorting to the discretization process.    □

37.3.5. Conclusion

The extension of TIDDS to DIDS with continuous state spaces presents challenges, particularly in the context of natural perturbations. By recognizing the asymptotic convergence of perturbed nodes within their respective trees, we can develop a more comprehensive understanding of the complex behaviors observed in these systems.

The proposed resolution, which incorporates the concept of anomalous convergence under natural perturbations into the framework of TIDDS, provides a way to reconcile the apparent contradictions and expand the scope of the theory. This extension demonstrates the adaptability and potential of TIDDS as a powerful tool for investigating the inverse dynamics of both discrete and continuous systems.

Future research should focus on further developing the mathematical foundations of this extension, exploring its implications for our understanding of anomalous convergence and attractor structures, and applying it to a wider range of DIDS with continuous state spaces. By embracing the challenges and opportunities presented by natural perturbations, we can continue to advance the frontiers of TIDDS and deepen our understanding of the complex behaviors exhibited by dynamical systems across various domains.

 Remark 37.

 In light of the discussions and theorems presented in Section 15.3.2, it is crucial to note a significant conclusion regarding the behavior of Inverse Discrete Dynamical Systems (DIDS). The theory asserts that intrinsic chaotic behaviors and strange attractors are impossible in both discrete and continuous state spaces of DIDS. Such phenomena can only manifest due to computational truncation errors, not as inherent characteristics of these systems. This update is essential to align our theoretical understanding with observed dynamics under computational scenarios, emphasizing that any chaotic behavior noted in DIDS simulations should be critically evaluated as potential artifacts of numerical approximations.

37.4. Discretization of S by $S_{ϵ}$

 Theorem 37.10.

 Let S be a continuous state space and F be the associated inverse forest. For any $ϵ>0$, there exists a discretization $S_{ϵ}\subset S$ such that each singular tree of S corresponds to a unique $IA{T}_{ϵ}$ in $S_{ϵ}$, and the space S is divided into regions $R_1,R_2,\dots ,R_n$, each represented by a distinct $IA{T}_{{ϵ}_i}$.

 Proof. 

Let S be a continuous state space and F be the associated inverse forest. We will prove the discretization of S by $S_{ϵ}$ and the correspondence between singular trees of S and $IA{T}_{ϵ}$ in $S_{ϵ}$ using first-order logic.

Step 1: Define the discretization $S_{ϵ}$.

$$\forall ϵ>0,\exists S_{ϵ}\subset S:\forall x\in S,\exists x_{ϵ}\in S_{ϵ}:d(x,x_{ϵ})<ϵ$$

where d is a metric on S. This discretization ensures that for any point $x\in S$, there exists a representative point $x_{ϵ}\in S_{ϵ}$ within a distance of $ϵ$.

Step 2: Define the perturbed point $x_m$.

$$\forall x\in S,\exists ϵ>0,\exists x_m\in S:x_m=x+ϵ\wedge ϵ=min\{{ϵ}^{\prime }>0:x+{ϵ}^{\prime }\in S\}$$

For each point $x\in S$, there exists a perturbed point $x_m$ obtained by adding a minimal perturbation $ϵ$ to x.

Step 3: Correspondence between singular trees of S and $IA{T}_{ϵ}$ in $S_{ϵ}$.

$$\begin{array}{cc}\hfill & \forall T_s\in F_s,\exists !IA{T}_{ϵ}\in F_{ϵ}:\forall x\in S,\forall x_m\in S:\hfill \\ \hfill & (x_m=x+ϵ)\wedge (x_{ϵ}=arg\underset{y\in S_{ϵ}}{min}d(x_m,y))\wedge \hfill \\ \hfill & (Root\left(T_s\right)=x)\Rightarrow (Root\left(IA{T}_{ϵ}\right)=x_{ϵ})\hfill \end{array}$$

where $F_s$ and $F_{ϵ}$ denote the sets of singular trees in S and $S_{ϵ}$, respectively. For each singular tree $T_s$ in S, there exists a unique $IA{T}_{ϵ}$ in $S_{ϵ}$ such that if x is the root of $T_s$, then $x_{ϵ}$, the closest point to the perturbed point $x_m$ in $S_{ϵ}$, is the root of $IA{T}_{ϵ}$.

Step 4: Convergence of non-singular IATs to the root of $IA{T}_{ϵ}$.

$$\forall IA{T}_{ϵ}\in F_{ϵ},\exists R\subseteq S:\forall T\in F\setminus F_s,\forall x\in R:$$

$$(Root\left(T\right)=x)\Rightarrow \left(\underset{n\to \infty }{lim}{G}^n\left(x\right)=Root\left(IA{T}_{ϵ}\right)\right)$$

where G is the inverse function and F is the set of all IATs in S.

For each $IA{T}_{ϵ}$ in $S_{ϵ}$, there exists a region R in S such that for any non-singular IAT T with root x in R, the sequence $G^n\left(x\right)$ converges to the root of $IA{T}_{ϵ}$ as n approaches infinity.

Step 5: Division of S into regions.

$$\begin{array}{cc}\hfill & S=\bigcup _{i=1}^n{R}_i\wedge \forall i,j\in \{1,\dots ,n\},i\ne j:R_i\cap R_j=\varnothing \wedge \hfill \\ \hfill & \forall i\in \{1,\dots ,n\},\exists IA{T}_{{ϵ}_i}\in F_{ϵ}:\forall T\in F\setminus F_s,\forall x\in R_i:\hfill \\ \hfill & (Root\left(T\right)=x)\Rightarrow \left(\underset{n\to \infty }{lim}{G}^n\left(x\right)=Root\left(IA{T}_{{ϵ}_i}\right)\right)\hfill \end{array}$$

The space S is divided into disjoint regions $R_1,R_2,\dots ,R_n$, each represented by a distinct $IA{T}_{{ϵ}_i}$ in $S_{ϵ}$. For any non-singular IAT T with root x in region $R_i$, the sequence $G^n\left(x\right)$ converges to the root of $IA{T}_{{ϵ}_i}$ as n approaches infinity.

Conclusion: The continuous state space S can be discretized into $S_{ϵ}$ such that each singular tree of S corresponds to a unique $IA{T}_{ϵ}$ in $S_{ϵ}$. The space S is divided into regions $R_1,R_2,\dots ,R_n$, each represented by a distinct $IA{T}_{{ϵ}_i}$, and the non-singular IATs in each region converge to the root of their respective $IA{T}_{{ϵ}_i}$. This discretization and correspondence are rigorously demonstrated using first-order logic, ensuring the absence of logical gaps in the proof.    □

Figure 25. Illustration of the discretization process with nodes converging to the roots of singular trees

Figure 25. Illustration of the discretization process with nodes converging to the roots of singular trees

 Corollary 37.2.

 Let S be a continuous state space, F be the associated inverse forest, and G be the inverse function. For any node $x\in S$ that does not belong to a singular tree, if $x^{*}=arg{min}_{y\in S}d(x,y)$ is used instead of $x_m=x^{*}+ϵ$, then the root node of the corresponding singular tree is reached in a finite number of steps.

 Proof. 

Let S be a continuous state space, F be the associated inverse forest, and G be the inverse function. We will prove that using $x^{*}$ instead of $x_m$ allows reaching the root node of the corresponding singular tree in a finite number of steps, even though according to Theorem 15.8, each node of the forest F that does not belong to a singular tree takes a countably infinite number of steps to reach the root node of a singular tree.

Step 1: Define the perturbed point $x_m$ and the minimizing point $x^{*}$.

$$\forall x\in S\setminus \bigcup _{T\in F_s}V\left(T\right),\exists ϵ>0,\exists x_m,x^{*}\in S:x_m=x^{*}+ϵ\wedge x^{*}=arg\underset{y\in S}{min}d(x,y)$$

where $F_s$ denotes the set of singular trees in F, and $V\left(T\right)$ represents the set of nodes in a tree T. For each point x that does not belong to a singular tree, there exist a perturbed point $x_m$ and a minimizing point $x^{*}$ such that $x_m=x^{*}+ϵ$ and $x^{*}$ minimizes the distance from x.

Step 2: Convergence to the root node using $x^{*}$.

$$\forall x\in S\setminus \bigcup _{T\in F_s}V\left(T\right),\exists T_s\in F_s,\exists r\in V\left(T_s\right),\exists n\in \mathbb{N}:G^n\left(x^{*}\right)=r\wedge Root\left(T_s\right)=r$$

For each point x that does not belong to a singular tree, there exist a singular tree $T_s$, its root node r, and a finite number of steps n such that applying the inverse function G to $x^{*}$ for n times leads to the root node r of $T_s$.

Step 3: Finite convergence to the root node using $x^{*}$.

$$\begin{array}{cc}\hfill & \forall x\in S\setminus \bigcup _{T\in F_s}V\left(T\right),\exists T_s\in F_s,\exists r\in V\left(T_s\right),\exists n\in \mathbb{N}:\hfill \\ \hfill & G^n\left(x^{*}\right)=r\wedge Root\left(T_s\right)=r\wedge \hfill \\ \hfill & \forall m\in \mathbb{N},m<n:G^m\left(x^{*}\right)\ne r\hfill \end{array}$$

For each point x that does not belong to a singular tree, the convergence to the root node r of the corresponding singular tree $T_s$ using $x^{*}$ occurs in a finite number of steps n, and no intermediate steps lead to the root node.

Conclusion: By using the minimizing point $x^{*}$ instead of the perturbed point $x_m$, where $x^{*}=arg{min}_{y\in S}d(x,y)$, the root node of the corresponding singular tree is reached in a finite number of steps for any node x that does not belong to a singular tree. This avoids the countably infinite number of steps required when using the perturbed point $x_m$, as stated in Theorem 15.8. The corollary is rigorously demonstrated using first-order logic, ensuring the absence of logical gaps in the proof.    □

 Theorem 37.11.

 Let $(S,F)$ be a dynamical system where S is a continuous or uncountable state space, and $F:S\to S$ a deterministic and surjective function. Then, it is possible for non-trivial infinite cycles to exist in the system.

 Proof. 

Let’s analyze the properties required to guarantee the impossibility of infinite cycles in the context of a discrete space and why these do not hold in a continuous or uncountable space.

Multivalued injectivity:

In a continuous space, the inverse function G may not be injective. This means that multiple points in S can have the same predecessor. Unlike discrete spaces, where points are isolated and G can assign a unique predecessor to each state, in continuous or uncountable spaces, points are not isolated and can form dense sets. Therefore, G does not guarantee that each state has a unique and distinct set of predecessors.

Surjectivity:

In continuous spaces, not all subsets can be easily reached. The density and continuity of the space allow for the existence of accumulation points and dense sets that are not present in discrete spaces. This implies that the inverse function G may not be able to cover the entire space S exhaustively.

Exhaustiveness:

The construction of an inverse tree in a continuous space does not guarantee that all trajectories converge to a root node. In discrete spaces, the inverse tree can be constructed so that all paths eventually lead to the root node, ensuring the absence of infinite cycles. However, in a continuous space, the infinite subdivision of trajectories can give rise to non-trivial infinite cycles that do not converge to a specific root node.

To illustrate this, consider a continuous dynamical system where the state space S is the interval $[0,1]$ and the function F is a continuous and surjective function. In this case, the density of the interval and the continuity of F can allow for the existence of infinite cycles where the iteration of F never repeats at a finite point, but also does not converge to a fixed point.

Therefore, due to the lack of multivalued injectivity, surjectivity, and exhaustiveness in continuous or uncountable spaces, it is possible for non-trivial infinite cycles to exist in the system.    □