Devaney Chaos for the Function Envelope of a Non-Autonomous Discrete Dynamical System

1. Introduction

A nonautonomous dynamical system, abbreviated as an NDS, is a pair $(X,f_{\infty })$, where X is a compact metric space endowed with a metric d, and $f_{\infty }={\left\{f_n\right\}}_{n=1}^{\infty }$ is a sequence of continuous self-maps on X. Nonautonomous dynamical systems were systematically studied in the framework of topological dynamics by Kolyada and Snoha [1]. For integers $1\le m\le n$, put $f_{m,n}=f_n\circ f_{n-1}\circ \cdots \circ f_m,$ and set $f_{m,m-1}={id}_X.$ Then the positive orbit of $x_1$ is defined by

$$x_{n+1}=f_{1,n}\left(x_1\right),n\in {\mathbb{N}}^{+}.$$

If $f_n=f$ for every $n\ge 1$, then $(X,f_{\infty })$ reduces to the classical autonomous dynamical system $(X,f)$.

A dynamical system $(X,f)$ is said to be Devaney chaotic [2] if it satisfies the following conditions:

1.

$(X,f)$ is topologically transitive;

2.

the set of periodic points of f is dense in X;

3.

$(X,f)$ has sensitive dependence on initial conditions.

Banks et al. [3] proved that, in the definition of Devaney chaos, the sensitivity condition is redundant.

Functional envelope systems provide a natural framework for studying the relationship between the dynamics on an underlying space and the dynamics on an infinite-dimensional function space. A classical discrete dynamical system describes the evolution of a single initial point under successive iterations of continuous mappings. In contrast, a functional envelope system takes functions as its states and describes, through composition operators, the simultaneous evolution of a family of initial states, spatial distributions, or global state profiles.

More precisely, for every $n\in {\mathbb{N}}^{+}$, define $H_n\left(\phi \right)=f_n\circ \phi .$ The sequence $H_{\infty }={\left\{H_n\right\}}_{n=1}^{\infty }$ generates the $L^1$-functional envelope system. Its n-step cumulative map is given by $H_{1,n}\left(\phi \right)=f_{1,n}\circ \phi .$ Here, a function $\phi \in L^1(I,I)$ no longer represents a single state in the underlying space. Instead, it may be regarded as a family of states parametrized by points of I. Consequently, the functional envelope system describes the collective evolution of a large family of initial states under the same nonautonomous dynamical process.

The functional envelope construction lifts a dynamical problem on a finite- or low-dimensional underlying space to an infinite-dimensional function space. This makes it possible to investigate whether dynamical properties such as transitivity, weak mixing, sensitivity, and periodic structure are preserved under this lifting procedure, and whether the chaotic behavior of the functional envelope system can, conversely, characterize the dynamics of the underlying system.

In the autonomous setting, Chen et al. [4] proved that weak mixing of an autonomous interval system implies Devaney chaos of its $L^1$-functional envelope system. Several other dynamical properties of functional envelope systems associated with autonomous systems have also been studied; see, for example, [5,6,7,8,9]. For nonautonomous discrete dynamical systems, Chen et al. [10] mainly investigated transitivity, sensitivity, and Li–Yorke chaos of the corresponding functional envelope systems.

It is therefore natural and important to determine whether density of periodic points can be transferred between an underlying nonautonomous system and its functional envelope system. In particular, Sánchez et al. [11] showed that, for a nonautonomous discrete dynamical system, density of periodic points in the original system does not necessarily imply density of periodic points in the corresponding hyperspace system. Thus, identifying a mechanism that preserves the density of periodic points in a functional envelope system is a crucial step toward establishing a theory of Devaney chaos on function spaces. There also exist several research results concerning non-autonomous dynamical systems [13,14,15,16].

In this paper, we prove that density of periodic points in a nonautonomous interval system is equivalent to density of periodic points in its $L^1$-functional envelope system. Combining this equivalence with a lifting theorem for weak mixing of all orders, we further establish Devaney chaos of the functional envelope system.

Our main result is stated as follows.

Theorem 1. 

Let $(I,f_{\infty })$ be a nonautonomous dynamical system on the interval $I=[0,1]$. Suppose that

1.

$(I,f_{\infty })$ is weakly mixing of order 3;

2.

the set of periodic points of $(I,f_{\infty })$ is dense in I.

Then the functional envelope system $(L^1(I,I),H_{\infty })$ is Devaney chaotic.

2. Preliminaries

Throughout this paper, ${\mathbb{N}}^{+}$ and $\mathbb{N}$ represent positive integer sets and non-negative integer sets, respectively, and let $I=[0,1]$.

A nonautonomous discrete dynamical system on I is denoted by $(I,f_{\infty }),f_{\infty }={\left\{f_n\right\}}_{n=1}^{\infty },$ where each map $f_n:I⟶I$ is continuous. For integers $1\le m\le n$, define $f_{m,n}=f_n\circ f_{n-1}\circ \cdots \circ f_m.$ In particular, $f_{1,n}=f_n\circ f_{n-1}\circ \cdots \circ f_1.$ We also put $f_{1,0}={id}_I.$

Let ${\mathcal{L}}^1(I,I)=\left\{g:I\to I:g\mathrm{is}\mathrm{Lebesgue}\mathrm{measurable}\right\}.$ Define an equivalence relation ∼ on ${\mathcal{L}}^1(I,I)$ by

$$\phi \sim \psi ⟺{\int }_I|\phi \left(t\right)-\psi \left(t\right)|d\mu \left(t\right)=0.$$

The quotient space is denoted by $L^1(I,I)={\mathcal{L}}^1(I,I)/\sim .$ It is equipped with the metric

$$\rho (\phi ,\psi )={\int }_I|\phi \left(t\right)-\psi \left(t\right)|d\mu \left(t\right),\phi ,\psi \in L^1(I,I),$$

where $\mu$ denotes the Lebesgue measure on I.

For every $n\in {\mathbb{N}}^{+}$, define the one-step functional envelope map $H_n:L^1(I,I)⟶L^1(I,I)$ by $H_n\left(\phi \right)=f_n\circ \phi .$ The sequence $H_{\infty }={\left\{H_n\right\}}_{n=1}^{\infty }$ generates the $L^1$-functional envelope system $\left(L^1(I,I),H_{\infty }\right).$

For integers $1\le m\le n$, define the cumulative map $H_{m,n}=H_n\circ H_{n-1}\circ \cdots \circ H_m.$ We also put $H_{1,0}={id}_{L^1(I,I)}.$ Then, for every $\phi \in L^1(I,I)$, $H_{m,n}\left(\phi \right)=f_{m,n}\circ \phi .$ In particular,

$$\begin{array}{cc}\hfill H_{1,n}\left(\phi \right)& =H_n\circ H_{n-1}\circ \cdots \circ H_1\left(\phi \right)\hfill \\ & =f_n\circ f_{n-1}\circ \cdots \circ f_1\circ \phi \hfill \\ & =f_{1,n}\circ \phi .\hfill \end{array}$$

It is readily verified that $\left(L^1(I,I),\rho \right)$ is a separable complete metric space.

A point $a\in I$ is called a periodic point of $(I,f_{\infty })$ if there exists $q\in {\mathbb{N}}^{+}$ such that

$$f_{1,m+q}\left(a\right)=f_{1,m}\left(a\right)\mathrm{for}\mathrm{every}m\in {\mathbb{N}}_0.$$

The least positive integer q satisfying the above equality is called the period of a. The set of all periodic points of $(I,f_{\infty })$ is denoted by $Per\left(f_{\infty }\right).$

The system $(I,f_{\infty })$ is said to be topologically transitive if, for any two nonempty open subsets $U,V\subset I$, there exists $n\in {\mathbb{N}}^{+}$ such that $f_{1,n}\left(U\right)\cap V\ne ⌀.$

For nonempty open subsets $U,V\subset I$, define the hitting-time set by

$$N_{f_{\infty }}(U,V)=\left\{n\in {\mathbb{N}}^{+}:f_{1,n}\left(U\right)\cap V\ne ⌀\right\}.$$

The system $(I,f_{\infty })$ is said to be weakly mixing of order m if, for any nonempty open subsets $U_1,\dots ,U_m,V_1,\dots ,V_m\subset I,$ one has

$${\displaystyle \bigcap _{i=1}^m}{N}_{f_{\infty }}(U_i,V_i)\ne ⌀.$$

It is said to be weakly mixing of all orders if it is weakly mixing of order m for every integer $m\ge 2$.

The system $(I,f_{\infty })$ is said to have sensitive dependence on initial conditions if there exists $\eta >0$ such that, for every $x\in I$ and every $\epsilon >0$, there exist $y\in B_I(x,\epsilon )\mathrm{and}n\in {\mathbb{N}}^{+}$ such that

$$\left|f_{1,n}\left(x\right)-f_{1,n}\left(y\right)\right|>\eta .$$

We now present several key properties and lemmas.

Lemma 1. 

For every $n\in {\mathbb{N}}^{+}$, the one-step map $H_n:L^1(I,I)⟶L^1(I,I)$ is well defined and continuous. Consequently, for every $n\in {\mathbb{N}}^{+}$, the cumulative map $H_{1,n}=H_n\circ H_{n-1}\circ \cdots \circ H_1$ is continuous.

Proof of Lemma 1. 

Fix $n\in {\mathbb{N}}^{+}$. We first prove that $H_n$ is well defined. Let $\phi \in L^1(I,I)$ and choose a measurable representative of its equivalence class. Since $f_n$ is continuous, the composition $f_n\circ \phi$ is measurable. Moreover,

$$0\le f_n\left(\phi \left(t\right)\right)\le 1$$

for almost every $t\in I$, and hence $f_n\circ \phi$ is Lebesgue integrable.

Suppose that $\phi =\psi$ in $L^1(I,I)$. Then $\phi \left(t\right)=\psi \left(t\right)$ for almost every $t\in I$. Consequently, $f_n\left(\phi \left(t\right)\right)=f_n\left(\psi \left(t\right)\right)$ for almost every $t\in I$. Thus, $f_n\circ \phi$ and $f_n\circ \psi$ determine the same equivalence class. Therefore, $H_n$ is well defined.

We next prove that $H_n$ is continuous. Let $\epsilon >0$. Since $f_n$ is continuous on the compact interval I, it is uniformly continuous. Hence there exists $\delta >0$ such that

$$|s-t|<\delta ⟹|f_n\left(s\right)-f_n\left(t\right)|<{\displaystyle \frac{\epsilon }{2}}$$

for all $s,t\in I$.

Set $\eta ={\displaystyle \frac{\delta \epsilon }{2}}.$ Suppose that $\rho (\phi ,\psi )<\eta .$ Define

$$A=\{t\in I:|\phi \left(t\right)-\psi \left(t\right)|<\delta \},B=I\setminus A.$$

By Chebyshev’s inequality,

$$\mu \left(B\right)\le {\displaystyle \frac{1}{\delta }}{\int }_I|\phi \left(t\right)-\psi \left(t\right)|d\mu \left(t\right)={\displaystyle \frac{\rho (\phi ,\psi )}{\delta }}<{\displaystyle \frac{\epsilon }{2}}.$$

For A, we have

$$|f_n\left(\phi \left(t\right)\right)-f_n\left(\psi \left(t\right)\right)|<{\displaystyle \frac{\epsilon }{2}}.$$

Since $f_n\left(I\right)\subset I=[0,1]$, for B we have

$$|f_n\left(\phi \left(t\right)\right)-f_n\left(\psi \left(t\right)\right)|\le 1.$$

Therefore,

$$\begin{array}{cc}\hfill \rho \left(H_n\left(\phi \right),H_n\left(\psi \right)\right)& ={\int }_I|f_n\left(\phi \left(t\right)\right)-f_n\left(\psi \left(t\right)\right)|d\mu \left(t\right)\hfill \\ & \le {\int }_A{\displaystyle \frac{\epsilon }{2}}d\mu \left(t\right)+{\int }_{B}1d\mu \left(t\right)\hfill \\ & \le {\displaystyle \frac{\epsilon }{2}}\mu \left(A\right)+\mu \left(B\right)\hfill \\ & <\epsilon .\hfill \end{array}$$

Thus $H_n$ is continuous.

Finally, $H_{1,n}=H_n\circ H_{n-1}\circ \cdots \circ H_1$ is a finite composition of continuous maps, and hence it is continuous. □

Lemma 2. 

[12] Theorem 11. Suppose that $(I,f_{\infty })$ is weakly mixing of order 3. Then it is weakly mixing of order n for every $n\ge 2$.

3. Proof of the Main Results

Theorem 2. 

Let $(I,f_{\infty })$ be a nonautonomous discrete dynamical system on $I=[0,1]$, and let $\left(L^1(I,I),H_{\infty }\right)$ be its $L^1$-functional envelope system. Then the following statements are equivalent:

1. 

$Per\left(f_{\infty }\right)$ is dense in I;

2. 

$Per\left(H_{\infty }\right)$ is dense in $L^1(I,I)$.

Proof of Theorem 2. 

(1) ⟹ (2). Suppose that the set of periodic points of $(I,f_{\infty })$ is dense in I. Let $g\in L^1(I,I)$ and $\epsilon >0$ be arbitrary.

Since $C(I,I)$ is dense in $L^1(I,I)$, there exists $h\in C(I,I)$ such that $\rho (g,h)<{\displaystyle \frac{\epsilon }{2}}.$ Choose $n\in {\mathbb{N}}_{+}$ sufficiently large so that ${\displaystyle \frac{1}{n}}<{\displaystyle \frac{\epsilon }{2}}.$ Define the measurable subsets

$$J_i=\left\{t\in I:{\displaystyle \frac{i-1}{n}}\le h\left(t\right)<{\displaystyle \frac{i}{n}}\right\},i=1,\dots ,n-1,J_n=\left\{t\in I:{\displaystyle \frac{n-1}{n}}\le h\left(t\right)\le 1\right\}.$$

Then $J_1,\dots ,J_n$ form a measurable partition of I.

By the density of the periodic points of $(I,f_{\infty })$, for each $i=1,\dots ,n$, we may choose a periodic point $x_i\in \left({\displaystyle \frac{i-1}{n}},{\displaystyle \frac{i}{n}}\right).$ Let $q_i\in {\mathbb{N}}_{+}$ be a period of $x_i$, and set $q=lcm(q_1,\dots ,q_n).$ Define $\phi ={\sum }_{i=1}^n{x}_i{\chi }_{J_i}.$ Since $\phi$ is measurable and takes values in I, we have $\phi \in L^1(I,I)$.

For every $i\in \{1,\dots ,n\}$, write $q={\ell }_i{q}_i$ for some ${\ell }_i\in {\mathbb{N}}^{+}$. Since $q_i$ is a period of $x_i$, we have

$$f_{1,r+q_i}\left(x_i\right)=f_{1,r}\left(x_i\right)\mathrm{for}\mathrm{every}r\in {\mathbb{N}}_0.$$

For any m, applying this identity successively to $r=m+({\ell }_i-1)q_i,m+({\ell }_i-2)q_i,\dots ,m,$ we obtain

$$\begin{array}{cc}\hfill f_{1,m+q}\left(x_i\right)& =f_{1,m+{\ell }_i{q}_i}\left(x_i\right)=f_{1,m+({\ell }_i-1)q_i}\left(x_i\right)=\cdots =f_{1,m}\left(x_i\right).\hfill \end{array}$$

Therefore, for every $t\in I$, since $t\in J_i$ for some i, it follows that

$$\begin{array}{cc}\hfill H_{1,m+q}\left(\phi \right)\left(t\right)& =f_{1,m+q}\left(\phi \left(t\right)\right)=f_{1,m+q}\left(x_i\right)=f_{1,m}\left(x_i\right)=H_{1,m}\left(\phi \right)\left(t\right).\hfill \end{array}$$

Hence

$$H_{1,m+q}\left(\phi \right)=H_{1,m}\left(\phi \right)\mathrm{for}\mathrm{every}m\in {\mathbb{N}}_0,$$

and thus $\phi$ is a periodic point of the functional envelope system.

Moreover, for every $t\in J_i$, both $h\left(t\right)$ and $x_i$ belong to the interval $\left[{\displaystyle \frac{i-1}{n}},{\displaystyle \frac{i}{n}}\right],$ and hence $|x_i-h\left(t\right)|\le {\displaystyle \frac{1}{n}}.$ It follows that

$$\begin{array}{cc}\hfill \rho (\phi ,h)& ={\int }_I|\phi \left(t\right)-h\left(t\right)|d\mu \left(t\right)\hfill \\ & ={\displaystyle \sum _{i=1}^n}{\int }_{J_i}|x_i-h\left(t\right)|d\mu \left(t\right)\hfill \\ & \le {\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{n}}\mu \left(J_i\right)\hfill \\ & ={\displaystyle \frac{1}{n}}\mu \left(I\right)={\displaystyle \frac{1}{n}}.\hfill \end{array}$$

Therefore,

$$\rho (\phi ,g)\le \rho (\phi ,h)+\rho (h,g)<{\displaystyle \frac{\epsilon }{2}}+{\displaystyle \frac{\epsilon }{2}}=\epsilon .$$

Thus every open ball in $L^1(I,I)$ contains a periodic point of $(L^1(I,I),H_{\infty })$. Hence the set of periodic points of the functional envelope system is dense in $L^1(I,I)$.

(2) ⟹ (1). Suppose that the set of periodic points of $(L^1(I,I),H_{\infty })$ is dense in $L^1(I,I)$. Let $x\in I$ and $\epsilon >0$ be arbitrary, and set $r=min\left\{\epsilon ,{\displaystyle \frac{1}{2}}\right\}.$ Consider the constant function

$$g_x\left(t\right)=x,t\in I.$$

By the density of the periodic points of the functional envelope system, there exists a periodic point $\psi \in L^1(I,I)$ such that

$$\rho (\psi ,g_x)={\int }_I|\psi \left(t\right)-x|d\mu \left(t\right)<r^2.$$

Let $q\in {\mathbb{N}}_{+}$ be a period of $\psi$. Thus,

$$H_{1,m+q}\left(\psi \right)=H_{1,m}\left(\psi \right)\mathrm{in}{L}^1(I,I)$$

for every $m\in {\mathbb{N}}_0$.

Choose a measurable representative of $\psi$, still denoted by $\psi$. Define $A=\left\{t\in I:\left|\psi \right(t)-x|<r\right\}.$ By Chebyshev’s inequality,

$$\begin{array}{cc}\hfill \mu (I\setminus A)& =\mu \left(\left\{t\in I:\left|\psi \right(t)-x|\ge r\right\}\right)\hfill \\ & \le {\displaystyle \frac{1}{r}}{\int }_I|\psi \left(t\right)-x|d\mu \left(t\right)\hfill \\ & ={\displaystyle \frac{\rho (\psi ,g_x)}{r}}<r\le {\displaystyle \frac{1}{2}}.\hfill \end{array}$$

Hence $\mu \left(A\right)>0.$

For each $m\in {\mathbb{N}}_0$, the equality $H_{1,m+q}\left(\psi \right)=H_{1,m}\left(\psi \right)$ in $L^1(I,I)$ implies that there exists a measurable subset $E_m\subset I$ with $\mu \left(E_m\right)=1$ such that

$$f_{1,m+q}\left(\psi \left(t\right)\right)=f_{1,m}\left(\psi \left(t\right)\right)\mathrm{for}\mathrm{every}t\in E_m.$$

Let $E={\bigcap }_{m=0}^{\infty }{E}_m.$ Since this is a countable intersection of sets of full measure, $\mu \left(E\right)=1.$ Because $\mu \left(A\right)>0$, the intersection $A\cap E$ is nonempty. Choose $t_0\in A\cap E$, and put $y=\psi \left(t_0\right).$ Since $t_0\in A$,

$$|y-x|=\left|\psi \right(t_0)-x|<r\le \epsilon .$$

Furthermore, since $t_0\in E$, for every $m\in {\mathbb{N}}_0$ we have

$$f_{1,m+q}\left(y\right)=f_{1,m+q}\left(\psi \left(t_0\right)\right)=f_{1,m}\left(\psi \left(t_0\right)\right)=f_{1,m}\left(y\right).$$

Thus y is a periodic point of $(I,f_{\infty })$.

Since $x\in I$ and $\epsilon >0$ were arbitrary, the periodic points of $(I,f_{\infty })$ are dense in I. □

Theorem 3. 

Let $(I,f_{\infty })$ be a nonautonomous discrete dynamical system on $I=[0,1]$, and let $\left(L^1(I,I),H_{\infty }\right)$ be its $L^1$-functional envelope system. Then the following statements are equivalent:

1. 

$(I,f_{\infty })$ is weakly mixing of all orders;

2. 

$\left(L^1(I,I),H_{\infty }\right)$ is topologically transitive;

3. 

$\left(L^1(I,I),H_{\infty }\right)$ is weakly mixing of all orders.

Proof of Theorem 3. 

The implication $\left(3\right)\Rightarrow \left(2\right)$ follows from weak mixing of order 2. Indeed, let $\mathcal{U},\mathcal{V}$ be nonempty open subsets of $L^1(I,I)$. Applying weak mixing of order 2 to the two pairs $(\mathcal{U},\mathcal{V})\mathrm{and}(L^1(I,I),L^1(I,I)),$ we obtain $N_{H_{\infty }}(\mathcal{U},\mathcal{V})\ne ⌀.$ Hence the functional envelope system is topologically transitive.

(2) ⟹ (1). This is analogous to the proof of [10].

(1) ⟹ (3). Suppose that $(I,f_{\infty })$ is weakly mixing of all orders. Let $m\in {\mathbb{N}}_{+}$, and let ${\mathcal{U}}_1,\dots ,{\mathcal{U}}_m,{\mathcal{V}}_1,\dots ,{\mathcal{V}}_m$ be arbitrary nonempty open subsets of $L^1(I,I)$.

For each $l=1,\dots ,m$, choose

$$g_l\in {\mathcal{U}}_l\mathrm{and}{h}_l\in {\mathcal{V}}_l.$$

Since the above sets are open and their number is finite, there exists $\epsilon >0$ such that

$$B_{\rho }(g_l,\epsilon )\subseteq {\mathcal{U}}_l\mathrm{and}{B}_{\rho }(h_l,\epsilon )\subseteq {\mathcal{V}}_l$$

for every $l=1,\dots ,m$.

Choose $n\ge 2$ such that ${\displaystyle \frac{1}{n}}<\epsilon .$ For $r=1,\dots ,n-1$, set

$$C_r=\left[{\displaystyle \frac{r-1}{n}},{\displaystyle \frac{r}{n}}\right),r=1,\dots ,n-1,\mathrm{and}{C}_n=\left[{\displaystyle \frac{n-1}{n}},1\right].$$

Choose measurable representatives of $g_l$ and $h_l$, still denoted by $g_l$ and $h_l$, and define

$$L_{r,l}=\{t\in I:g_l\left(t\right)\in C_r\}\mathrm{and}{J}_{s,l}=\{t\in I:h_l\left(t\right)\in C_s\}.$$

where $l=1,\dots ,m,r,s=1,\dots ,n.$ For each l, both

$$L_{1,l},\dots ,L_{n,l}\mathrm{and}{J}_{1,l},\dots ,J_{n,l}$$

are measurable partitions of I.

Let $O_r=\left({\displaystyle \frac{r-1}{n}},{\displaystyle \frac{r}{n}}\right),r=1,\dots ,n.$ Apply weak mixing of order $n^2$ to the collection of pairs $\left\{(O_r,O_s):1\le r,s\le n\right\}.$ Then there exists a common integer $k\in {\mathbb{N}}^{+}$ such that

$$f_{1,k}\left(O_r\right)\cap O_s\ne ⌀\mathrm{for}\mathrm{all}1\le r,s\le n.$$

For each pair $(r,s)$, choose $x_{r,s}\in O_r$ such that $f_{1,k}\left(x_{r,s}\right)\in O_s.$

For $l=1,\dots ,m$ and $r,s=1,\dots ,n$, define $W_{r,s}^l=L_{r,l}\cap J_{s,l}.$ For each fixed l, the sets $\{W_{r,s}^l:1\le r,s\le n\}$ form a measurable partition of I. Define

$${\phi }_l={\displaystyle \sum _{r=1}^n}{\displaystyle \sum _{s=1}^n}{x}_{r,s}{\chi }_{W_{r,s}^l}.$$

Then ${\phi }_l$ is measurable and takes values in I, and hence ${\phi }_l\in L^1(I,I)$.

If $t\in W_{r,s}^l$, then

$$g_l\left(t\right)\in C_r\mathrm{and}{x}_{r,s}\in O_r\subseteq C_r.$$

Consequently, $|x_{r,s}-g_l\left(t\right)|\le {\displaystyle \frac{1}{n}}.$ It follows that

$$\begin{array}{cc}\hfill \rho ({\phi }_l,g_l)& ={\int }_I|{\phi }_l\left(t\right)-g_l\left(t\right)|d\mu \left(t\right)\hfill \\ & ={\displaystyle \sum _{r=1}^n}{\displaystyle \sum _{s=1}^n}{\int }_{W_{r,s}^l}|x_{r,s}-g_l\left(t\right)|d\mu \left(t\right)\hfill \\ & \le {\displaystyle \frac{1}{n}}{\displaystyle \sum _{r=1}^n}{\displaystyle \sum _{s=1}^n}\mu \left(W_{r,s}^l\right)\hfill \\ & ={\displaystyle \frac{1}{n}}\mu \left(I\right)={\displaystyle \frac{1}{n}}<\epsilon .\hfill \end{array}$$

Thus,

$${\phi }_l\in B_{\rho }(g_l,\epsilon )\subseteq {\mathcal{U}}_l.$$

Moreover, if $t\in W_{r,s}^l$, then

$$h_l\left(t\right)\in C_s\mathrm{and}{f}_{1,k}\left(x_{r,s}\right)\in O_s\subseteq C_s.$$

Therefore,

$$|f_{1,k}\left(x_{r,s}\right)-h_l\left(t\right)|\le {\displaystyle \frac{1}{n}}.$$

Hence

$$\begin{array}{cc}\hfill \rho \left(H_{1,k}\left({\phi }_l\right),h_l\right)& ={\int }_I|f_{1,k}\left({\phi }_l\left(t\right)\right)-h_l\left(t\right)|d\mu \left(t\right)\hfill \\ & ={\displaystyle \sum _{r=1}^n}{\displaystyle \sum _{s=1}^n}{\int }_{W_{r,s}^l}|f_{1,k}\left(x_{r,s}\right)-h_l\left(t\right)|d\mu \left(t\right)\hfill \\ & \le {\displaystyle \frac{1}{n}}{\displaystyle \sum _{r=1}^n}{\displaystyle \sum _{s=1}^n}\mu \left(W_{r,s}^l\right)\hfill \\ & ={\displaystyle \frac{1}{n}}<\epsilon .\hfill \end{array}$$

Thus,

$$H_{1,k}\left({\phi }_l\right)\in B_{\rho }(h_l,\epsilon )\subseteq {\mathcal{V}}_l.$$

It follows that

$$H_{1,k}\left({\mathcal{U}}_l\right)\cap {\mathcal{V}}_l\ne ⌀\mathrm{for}\mathrm{every}l=1,\dots ,m.$$

Equivalently,

$$k\in {\displaystyle \bigcap _{l=1}^m}{N}_{H_{\infty }}({\mathcal{U}}_l,{\mathcal{V}}_l).$$

Since m and the open sets ${\mathcal{U}}_l,{\mathcal{V}}_l$ were arbitrary, the functional envelope system $(L^1(I,I),H_{1,\infty })$ is weakly mixing of all orders. □

Lemma 3. 

Let $\left(L^1(I,I),H_{\infty }\right)$ be an $L^1$-functional envelope system. If it is weakly mixing of order 2, then it has sensitive dependence on initial conditions.

Proof of Lemma 3. 

Define the constant functions

$$\mathbf{0}\left(t\right)=0\mathrm{and}\mathbf{1}\left(t\right)=1,t\in I.$$

Clearly,

$$\rho (\mathbf{0},\mathbf{1})={\int }_I|\mathbf{0}\left(t\right)-\mathbf{1}\left(t\right)|d\mu \left(t\right)=1,\mathbf{0},\mathbf{1}\in L^1(I,I).$$

Set $\delta ={\displaystyle \frac{1}{4}},$ and let $V_0=B_{\rho }(\mathbf{0},\delta ),V_1=B_{\rho }(\mathbf{1},\delta ).$ If $\alpha \in V_0$ and $\beta \in V_1$, then the triangle inequality gives

$$\begin{array}{cc}\hfill \rho (\alpha ,\beta )& \ge \rho (\mathbf{0},\mathbf{1})-\rho (\alpha ,\mathbf{0})-\rho (\beta ,\mathbf{1})>1-\delta -\delta =2\delta .\hfill \end{array}$$

We prove that $\delta$ is a sensitivity constant. Let $\phi \in L^1(I,I)$ and let $\epsilon >0$. Put $U=B_{\rho }(\phi ,\epsilon ).$ Since the functional envelope system is weakly mixing of order 2, applied to the two pairs $(U,V_0)\mathrm{and}(U,V_1),$ there exists a common integer $n\in {\mathbb{N}}^{+}$ such that

$$H_{1,n}\left(U\right)\cap V_0\ne ⌀,H_{1,n}\left(U\right)\cap V_1\ne ⌀.$$

Consequently, there exist ${\phi }_0,{\phi }_1\in U$ such that

$$H_{1,n}\left({\phi }_0\right)\in V_0\mathrm{and}{H}_{1,n}\left({\phi }_1\right)\in V_1.$$

Hence

$$\rho \left(H_{1,n}\left({\phi }_0\right),H_{1,n}\left({\phi }_1\right)\right)>2\delta .$$

On the other hand, by the triangle inequality,

$$\begin{array}{cc}\hfill \rho \left(H_{1,n}\left({\phi }_0\right),H_{1,n}\left({\phi }_1\right)\right)& \le \rho \left(H_{1,n}\left({\phi }_0\right),H_{1,n}\left(\phi \right)\right)+\rho \left(H_{1,n}\left(\phi \right),H_{1,n}\left({\phi }_1\right)\right).\hfill \end{array}$$

Therefore, at least one of the following inequalities holds:

$$\rho \left(H_{1,n}\left({\phi }_0\right),H_{1,n}\left(\phi \right)\right)>\delta ,\rho \left(H_{1,n}\left({\phi }_1\right),H_{1,n}\left(\phi \right)\right)>\delta .$$

In the first case, set $\psi ={\phi }_0$, and in the second case, set $\psi ={\phi }_1$. Since ${\phi }_0,{\phi }_1\in B_{\rho }(\phi ,\epsilon ),$ we obtain $\psi \in B_{\rho }(\phi ,\epsilon )$ and

$$\rho \left(H_{1,n}\left(\phi \right),H_{1,n}\left(\psi \right)\right)>\delta .$$

Thus $\left(L^1(I,I),H_{\infty }\right)$ has sensitive dependence on initial conditions. □

Proof of Theorem 1. 

By Lemma 2, the assumption that $(I,f_{\infty })$ is weakly mixing of order 3 implies that it is weakly mixing of every order $m\ge 2$. Hence $(I,f_{\infty })$ is weakly mixing of all orders.

By Theorem 3, the functional envelope system $\left(L^1(I,I),H_{\infty }\right)$ is weakly mixing of all orders. In particular, it is topologically transitive and weakly mixing of order 2. Therefore, by Lemma 3, it has sensitive dependence on initial conditions. On the other hand, since $Per\left(f_{\infty }\right)$ is dense in I, Theorem 2 implies that $Per\left(H_{\infty }\right)$ is dense in $L^1(I,I)$.

Thus the functional envelope system $\left(L^1(I,I),H_{\infty }\right)$ is topologically transitive, has a dense set of periodic points, and has sensitive dependence on initial conditions. Therefore, it is Devaney chaotic. □