Coincidence of Critical Thresholds for Collatz-Type Maps: Why the Divisor Prime Must Be Two

1. Introduction

1.1. The Problem

The Collatz, or $3x+1$, map admits a standard acceleration. On odd integers set

$$T\left(n\right)={\displaystyle \frac{3n+1}{2^{v_2(3n+1)}}},$$

where $v_q$ denotes the q-adic valuation; this is the Syracuse map, and it carries odd integers to odd integers. It sits inside an evident family: for integers $a,b$ and a prime q with $q\nmid a$, put

$$T_{a,b,q}\left(n\right)={\displaystyle \frac{an+b}{q^{v_q(an+b)}}}.$$

(1)

The classical case is $(a,b,q)=(3,1,2)$.

Two very different numbers are attached to (1). The first is archimedean and classical. Since a single step multiplies n by roughly $a/q^v$, the mean logarithmic drift is

$$\delta (a,q)=loga-\mathbb{E}\left[v\right]logq,$$

(2)

and the sign of $\delta$ decides, heuristically, whether typical orbits shrink or grow. For $(3,1,2)$ one has $\mathbb{E}\left[v\right]=2$ and $\delta =log3-2log2=log(3/4)<0$: the familiar reason to expect the Collatz conjecture to be true.

The second is non-archimedean and, in the form used here, recent. Unrolling (1) and reducing modulo $a^k$ annihilates the dependence on the initial value; what survives is the law of a random variable on the a-adic integers ${\mathbb{Z}}_a$, and that law is the invariant measure of a countable affine iterated function system all of whose maps are similarities of ratio $1/a$ (Section 3). Such a measure has a similarity dimension

$$error(a,q)={\displaystyle \frac{H\left({\nu }_q\right)}{loga}},$$

(3)

where ${\nu }_q$ is the law of the valuation $v=v_q(an+b)$ and H is Shannon entropy; error is an upper bound for the Hausdorff dimension of the measure (Proposition 4), and $error=1$ is the threshold separating the regime in which the measure can be absolutely continuous from that in which it is necessarily singular.

Each quantity singles out a critical multiplier. There is a unique $a_{\mathrm{drift}}>1$ with $\delta =0$, and a unique $a_{dim}>1$ with $error=1$. For the Collatz prime $q=2$ both equal 4. It is natural to ask whether this is an accident of the value $q=2$ or a general feature of the family.

1.2. Results

It is not general, and the failure is exact.

Theorem 1.

Let q be a prime and let ${\nu }_q$ be the law of the valuation $v=v_q(an+b)$ for n ranging, with natural density, over the arithmetic progression on which $q\mid an+b$. Then:

(i)

${\nu }_q\left(m\right)=(q-1)q^{-m}$ for $m\ge 1$, and

$$-log{\nu }_q\left(m\right)=mlogq-log(q-1)\mathrm{for}\mathrm{every}m\ge 1;$$

(ii)

$H\left({\nu }_q\right)=\mathbb{E}\left[v\right]logq-log(q-1)$;

(iii)

$a_{\mathrm{drift}}=(q-1)a_{dim}$.

In particular $a_{\mathrm{drift}}=a_{dim}$ if and only if $q=2$.

The identity (i) is pointwise, not merely in expectation, and it is the engine of everything else: it says that the archimedean cost $mlogq$ of dividing by $q^m$ and the information content $-log{\nu }_q\left(m\right)$ of observing that division differ by a constant, namely the free energy of the valuation law. When that constant vanishes, cost and information are the same functional, and the two critical multipliers must agree.

The condition $q=2$ turns out to be a single condition seen from four sides.

Theorem 2.

For a prime q the following are equivalent.

(a)

$q=2$.

(b)

$\left|{(\mathbb{Z}/q\mathbb{Z})}^{\times }\right|=1$.

(c)

The pressure $P\left(1\right)=log{\sum }_{m\ge 1}{e}^{-mlogq}$ of the archimedean cost function at inverse temperature one vanishes.

(d)

$H\left({\nu }_q\right)=\mathbb{E}\left[v\right]logq$; equivalently $a_{dim}=a_{\mathrm{drift}}$.

(e)

For some (equivalently, any) $a,b$ with $q\nmid a$ and $an+b\equiv 0\left(\mathrm{mod}q\right)$ solvable at a unit, the map $T_{a,b,q}$ is defined ateveryn prime to q, i.e. a division occurs at every step.

Two complements are proved. The first quantifies what is lost for $q\ge 3$.

Theorem 3.

Let $q\ge 3$ be prime. Then $a_{dim}<a_{\mathrm{drift}}$, and for every real $a\in (a_{dim},a_{\mathrm{drift}})$ the map (1) has strictly negative drift while its invariant measure has similarity dimension strictly less than 1, hence is singular with respect to Haar measure on ${\mathbb{Z}}_a$. For $q=2$ this band is empty.

The second says that the natural operator attached to the system carries no spectral information, which we record because the opposite is easy to assume.

Theorem 4.

Let L be the Markov operator of the system on ${\mathbb{Z}}_a$, let $V_n$ be the space of functions constant on balls of radius $a^{-n}$, and for $\alpha >0$ let $C^{\alpha }\left({\mathbb{Z}}_a\right)$ be the corresponding Hölder space. Then

(i)

$L\left(V_n\right)\subseteq V_{n-1}$, and $L^{n}f=(\int fd\mu )\mathbf{1}$ for every $f\in V_n$;

(ii)

${\parallel Lf\parallel }_{\infty }\le {\parallel f\parallel }_{\infty }$ and ${\left[Lf\right]}_{\alpha }\le a^{-\alpha }{\left[f\right]}_{\alpha }$, so L is quasi-compact on $C^{\alpha }$ with essential spectral radius at most $a^{-\alpha }$;

(iii)

the only eigenvalue of L on $C^{\alpha }$ of modulus exceeding $a^{-\alpha }$ is 1, which is simple with eigenspace the constants.

Thus L does possess a spectral gap on Hölder spaces, but a degenerate one: apart from the peripheral eigenvalue the spectrum is entirely essential and sits at the contraction rate, so no eigenvalue carries dynamical information, and at any fixed resolution convergence to equilibrium is exact after finitely many steps rather than exponentially fast.

Theorem 5

(Rényi spectrum). For $t>0$, $t\ne 1$, set

$$D_t^S(a,q)={\displaystyle \frac{tlog(q-1)-log(q^t-1)}{(1-t)loga}},D_1^S:=error.$$

Then $D_t^S$ is continuous at $t=1$, and the Rényi dimensions of ${\mu }_{a,b,q}$ satisfy $D_t\le min\left(1,D_t^S\right)$ for all $t>0$.

The similarity dimension is thus the value at $t=1$ of a one-parameter family available in closed form; this is what the free-energy computation of Section 2 buys beyond notation.

The next two results are, in our view, the substance of the paper.

Theorem 6

(Purity). ${\mu }_{a,b,q}$ is either absolutely continuous or singular with respect to Haar measure λ on ${\mathbb{Z}}_a$; no mixed case occurs. Consequently

$$\mu \ll \lambda \iff \mathrm{TV}(\mu ,\lambda )<1\iff \underset{n\to \infty }{lim}\mathrm{TV}({\mu }_n,{\lambda }_n)<1.$$

The proof (Section 5) uses only the uniqueness clause of Proposition 2: the absolutely continuous part of $\mu$ is itself a fixed point of the Markov operator, so if it is nonzero its normalisation must be $\mu$. Uniqueness of a fixed point is a statement about the iterated function system and not about the law of any one random variable; we do not know how to see Theorem 6 from the recursion alone. The corollary matters because it converts total-variation data, which is computable, into a decision procedure in principle for a question—absolute continuity—which is otherwise inaccessible.

Theorem 7

(Classification of the supercritical regime). Let $a\ge 2$ and q prime with $gcd(a,q)=1$. Then $a<a_{dim}\left(q\right)$ holds for exactly two pairs:

$$(a,q)=(3,2)\mathrm{and}(a,q)=(2,3).$$

For every other admissible pair, $error{\mu }_{a,b,q}<1$ and ${\mu }_{a,b,q}$ is singular with respect to λ, for every b coprime to a.

The first pair is the Collatz map. Thus, of the entire two-parameter family, precisely two members have an invariant measure whose absolute continuity is not settled by a dimension count, and the classical $3x+1$ map is one of them.

1.3. Discussion

Theorem 1 is a statement about the family (1) and not about the Collatz conjecture, on which it has no bearing. Its interest, such as it is, lies in isolating what is arithmetically special about the classical parameters. It is often said that the Collatz conjecture is delicate because $log3/log2$ is close to, but below, 2. Theorem 1 locates the delicacy elsewhere: the constant 4 at which the classical map sits just below threshold is simultaneously the dimensional and the dynamical critical value only because the divisor prime is 2, and for every other prime the two thresholds are separated by the order of the unit group of the residue field.

We have made no attempt to survey the very large literature on generalised Collatz maps; see Lagarias [6,8] for the standard entry points, Conway [1] for the undecidability of the general family, and Tao [10] for the strongest known partial result, in which the 3-adic random variable underlying Section 3 is introduced. Iterated function systems on local fields are treated in the general framework of random affine recursions; see Kesten [5] and Goldie [2] for the archimedean theory and Hutchinson [4] for the contraction principle used in Proposition 2.

2. The Valuation Law

Throughout, q is a prime and $a,b$ are integers with $q\nmid a$. Since a is invertible modulo q, the congruence $an+b\equiv 0\left(\mathrm{mod}q\right)$ has a unique solution $n\equiv n_0\left(\mathrm{mod}q\right)$, and we write

$$\mathcal{P}=\{n\in \mathbb{Z}:n\equiv n_0\left(\mathrm{mod}q\right)\}$$

for the progression on which a division actually occurs.

Lemma 1.

For n ranging over $\mathcal{P}$ with natural density,

$${\nu }_q\left(m\right):=\mathbb{P}\left(v_q(an+b)=m\right)=(q-1)q^{-m},m\ge 1.$$

Consequently ${\nu }_q$ is a probability law on ${\mathbb{Z}}_{\ge 1}$ with $\mathbb{E}\left[v\right]=q/(q-1)$.

Proof. 

Write $n=n_0+qt$ with $t\in \mathbb{Z}$. Then $an+b=(a{n}_0+b)+aqt=q(c+at)$, where $c=(a{n}_0+b)/q\in \mathbb{Z}$. Since $q\nmid a$, the map $t\mapsto c+at$ is a bijection of $\mathbb{Z}/q^k\mathbb{Z}$ for every k, so $c+at$ is equidistributed modulo every power of q as t ranges over $\mathbb{Z}$ with natural density. For an equidistributed integer u one has $\mathbb{P}(v_q\left(u\right)=j)=(1-q^{-1})q^{-j}$ for $j\ge 0$. Hence

$${\nu }_q\left(m\right)=\mathbb{P}\left(v_q(c+at)=m-1\right)=(1-q^{-1})q^{-(m-1)}=(q-1)q^{-m}.$$

Summing,

$$\sum _{m\ge 1}(q-1)q^{-m}=(q-1)·{\displaystyle \frac{q^{-1}}{1-q^{-1}}}=1,\mathbb{E}\left[v\right]=\sum _{m\ge 1}\mathbb{P}(v\ge m)=\sum _{m\ge 1}{q}^{-(m-1)}={\displaystyle \frac{q}{q-1}}.$$

□

Lemma 2

(Pointwise cost–information identity). For every $m\ge 1$,

$$-log{\nu }_q\left(m\right)=mlogq-log(q-1).$$

Proof. 

Immediate from Lemma 1: $-log\left((q-1)q^{-m}\right)=mlogq-log(q-1)$. □

Lemma 2 has a thermodynamic reading. Regard $E\left(m\right)=mlogq$ as the energy of the event $\{v=m\}$: it is exactly the archimedean cost, $-log|q^{-m}|$, of the division performed at that step. The Gibbs state at inverse temperature $\beta =1$ for this energy is $Z^{-1}{e}^{-E\left(m\right)}=Z^{-1}{q}^{-m}$ with partition function $Z={\sum }_{m\ge 1}{q}^{-m}=1/(q-1)$; that is exactly ${\nu }_q$. The associated pressure is

$$P\left(1\right)=logZ=-log(q-1).$$

(4)

Proposition 1.

$H\left({\nu }_q\right)=\mathbb{E}\left[v\right]logq-log(q-1)$, and therefore

$$H\left({\nu }_q\right)=\mathbb{E}\left[v\right]logq\iff q=2.$$

Proof. 

Since $\mathbb{E}\left[v\right]=q/(q-1)<\infty$, Lemma 2 may be integrated against ${\nu }_q$:

$$H\left({\nu }_q\right)=\mathbb{E}\left[-log{\nu }_q\left(v\right)\right]=\mathbb{E}\left[vlogq-log(q-1)\right]=\mathbb{E}\left[v\right]logq-log(q-1).$$

The difference $H\left({\nu }_q\right)-\mathbb{E}\left[v\right]logq$ equals $-log(q-1)$, which vanishes if and only if $q-1=1$. □

Remark 1.

Explicitly $H\left({\nu }_q\right)=log{\displaystyle \frac{q}{q-1}}+{\displaystyle \frac{logq}{q-1}}$. For $q=2$ this is $2log2=log4$; for $q=3$ it is $log{\displaystyle \frac{3}{2}}+{\displaystyle \frac{log3}{2}}=0.954771\dots$, against $\mathbb{E}\left[v\right]logq={\displaystyle \frac{3}{2}}log3=1.647918\dots$, the difference being $log2$ as Proposition 1 requires.

3. The associated system on ${\mathbb{Z}}_a$

Fix in addition an integer $a\ge 2$ with $gcd(a,q)=1$ and $gcd(a,b)=1$. Let ${\mathbb{Z}}_a=\underleftarrow{lim}\mathbb{Z}/a^n\mathbb{Z}$ be the ring of a-adic integers, equipped with $d(x,y)=a^{-sup\{n:a^n\mid x-y\}}$.

Remark 2.

The notation ${\mathbb{Z}}_a$ for $\underleftarrow{lim}\mathbb{Z}/a^n\mathbb{Z}$ with a not necessarily prime is not universal; when $a=\prod p^{e_p}$ one has ${\mathbb{Z}}_a\cong {\prod }_{p\mid a}{\mathbb{Z}}_p$ as rings, but the metric below is the one induced by powers of a, not by any single p, and it is with respect to that metric that the maps $f_s$ are similarities of ratio $1/a$. Readers interested only in the classical case may take $a=3$ throughout.

Lemma 3.

d is an ultrametric on ${\mathbb{Z}}_a$, the space $({\mathbb{Z}}_a,d)$ is compact, balls of radius $a^{-n}$ are the residue classes modulo $a^n$ and have Haar measure $a^{-n}$, and multiplication by a is a similarity of ratio exactly $1/a$.

Proof. 

If $a^n\mid x-y$ and $a^n\mid y-z$ then $a^n\mid x-z$, giving the ultrametric inequality; compactness is the inverse limit of finite sets. The ball of radius $a^{-n}$ about x is $\{y:a^n\mid x-y\}$, a class modulo $a^n$, and normalised Haar measure assigns each of the $a^n$ classes the mass $a^{-n}$. Finally $a^n\mid x-y$ if and only if $a^{n+1}\mid a(x-y)$, so $d(ax,ay)=a^{-1}d(x,y)$. □

Definition 1.

TheSyracuse systemof $(a,b,q)$ is the pair $\left({\left\{f_s\right\}}_{s\ge 1},{\nu }_q\right)$ with

$$f_s:{\mathbb{Z}}_a\to {\mathbb{Z}}_a,f_s\left(x\right)=q^{-s}(b+ax).$$

This is well defined because $gcd(q,a)=1$ makes q a unit of ${\mathbb{Z}}_a$.

Proposition 2.

Every $f_s$ is a similarity of ratio $1/a$. There is a unique Borel probability measure $\mu ={\mu }_{a,b,q}$ on ${\mathbb{Z}}_a$ with

$$\mu =\sum _{s\ge 1}{\nu }_q\left(s\right){\left(f_s\right)}_{*}\mu ,$$

(5)

and μ is supported on the units ${\mathbb{Z}}_a^{\times }$.

Proof. 

Since $q^{-s}$ is a unit, $d(f_{s}x,f_{s}y)=\left|q^{-s}\right|d(ax,ay)=a^{-1}d(x,y)$ by Lemma 3.

Let $\mathcal{M}\rho ={\sum }_s{\nu }_q\left(s\right){\left(f_s\right)}_{*}\rho$ act on the space $\mathcal{P}\left({\mathbb{Z}}_a\right)$ of Borel probability measures, metrised by the Monge–Kantorovich distance $W_1$. Given $\rho ,{\rho }^{\prime }$ and an optimal coupling $(X,Y)$, draw $s\sim {\nu }_q$ independently and couple $\mathcal{M}\rho$ with $\mathcal{M}{\rho }^{\prime }$ through $(f_{s}X,f_{s}Y)$; then

$$W_1(\mathcal{M}\rho ,\mathcal{M}{\rho }^{\prime })\le \mathbb{E}\left[d(f_{s}X,f_{s}Y)\right]=\frac{1}{a}\mathbb{E}\left[d(X,Y)\right]=\frac{1}{a}{W}_1(\rho ,{\rho }^{\prime }).$$

As ${\mathbb{Z}}_a$ is compact, $(\mathcal{P}\left({\mathbb{Z}}_a\right),W_1)$ is a complete metric space, and the Banach fixed point theorem supplies a unique solution of (5).

Finally $f_s\left(x\right)\equiv q^{-s}b(moda)$, and $gcd(b,a)=gcd(q,a)=1$, so every $f_s$ takes values in ${\mathbb{Z}}_a^{\times }$; hence so does $\mu$. □

The system is not merely modelled on (1); it is (1), branch by branch.

Proposition 3.

If n lies in the domain of $T_{a,b,q}$ and $r=v_q(an+b)$, then $T_{a,b,q}\left(n\right)=f_r\left(n\right)$. Consequently, writing $r_i=v_q\left(a{T}^i\left(n\right)+b\right)$,

$$T^k\left(n\right)=f_{r_{k-1}}\circ \dots \circ f_{r_0}\left(n\right),$$

and since this composition has linear coefficient $a^k{q}^{-R_k}$ with $R_k=r_0+\dots +r_{k-1}$, the residue $T^k\left(n\right)mod{a}^k$ does not depend on n.

Proof. 

By definition $q^{r}T\left(n\right)=an+b$, i.e. $T\left(n\right)=q^{-r}(b+an)=f_r\left(n\right)$; iterate. A composition of k maps of the form $x\mapsto q^{-s}(b+ax)$ is affine with linear coefficient $a^k{q}^{-R_k}$, and $a^k\equiv 0(mod{a}^k)$. □

Remark 3.

The forward dynamics composes on theleft, whereas the invariant measure of an iterated function system arises from composition on theright. The two agree in law, because the driving sequence is i.i.d. and hence exchangeable; this is why μ and not some other measure is the relevant object. For $q=2$ the driving sequence really is i.i.d.: for every word $(r_0,\dots ,r_{k-1})\in {\mathbb{Z}}_{\ge 1}^k$ the set of admissible n realising it is a single residue class modulo $q^{R_k+1}$, of density $q^{-R_k}$, by induction on k using the invertibility of a modulo powers of q. This is Terras’ parity-vector theorem [12] in valuation coordinates.

Proposition 4.

$error\mu \le min\left(1,error\right)$, where $error=H\left({\nu }_q\right)/loga$.

Proof. 

The bound by 1 is Lemma 3, since ${\mathbb{Z}}_a$ has Hausdorff dimension 1. For the other, let $\Sigma ={\mathbb{Z}}_{\ge 1}^{\mathbb{N}}$ carry the product measure ${\nu }_q^{\otimes \mathbb{N}}$ and let $\pi :\Sigma \to {\mathbb{Z}}_a$ be the coding map $\pi \left(w\right)={lim}_k{f}_{w_1}\circ \dots \circ f_{w_k}\left(x_0\right)$, which exists and is independent of $x_0$ by uniform contraction, and satisfies ${\pi }_{*}{\nu }_q^{\otimes \mathbb{N}}=\mu$. Each $f_{w_1\dots w_k}$ has ratio $a^{-k}$, so $f_{w_1\dots w_k}\left({\mathbb{Z}}_a\right)$ is precisely the ball $B(\pi \left(w\right),a^{-k})$; hence

$$\mu \left(B(\pi \left(w\right),a^{-k})\right)\ge {\nu }_q\left(w_1\right)\dots {\nu }_q\left(w_k\right).$$

By the Shannon–McMillan–Breiman theorem, for ${\nu }_q^{\otimes \mathbb{N}}$-almost every w one has $-\frac{1}{k}log{\prod }_{i\le k}{\nu }_q\left(w_i\right)\to H\left({\nu }_q\right)$. Therefore for $\mu$-almost every x,

$$\underset{k\to \infty }{lim\; inf}{\displaystyle \frac{log\mu \left(B(x,a^{-k})\right)}{log{a}^{-k}}}\le {\displaystyle \frac{H\left({\nu }_q\right)}{loga}},$$

and the Billingsley lemma gives $error\mu \le H\left({\nu }_q\right)/loga$. □

4. Periodic Points and Rational Cycles

Recall (Lagarias [7]) that a rational Syracuse k-cycle with valuation word $(r_0,\dots ,r_{k-1})$ is the solution of $n(q^R-a^k)=c$, where $R=\sum r_i$, $R_i=r_0+\dots +r_{i-1}$ and $c={\sum }_{i=0}^{k-1}{a}^{k-1-i}b{q}^{R_i}$.

Lemma 4.

For $w=(s_1,\dots ,s_k)$ put $S=\sum s_j$ and $S_i=s_1+\dots +s_i$. Then $f_w=f_{s_1}\circ \dots \circ f_{s_k}$ is affine with linear coefficient $a^k{q}^{-S}$, and has the unique fixed point

$$x_w={\displaystyle \frac{{\sum }_{i=1}^k{a}^{i-1}b{q}^{S-S_i}}{q^S-a^k}}\in {\mathbb{Z}}_a.$$

Proof. 

Composing $x\mapsto q^{-s}(b+ax)$ gives linear coefficient $a^k{q}^{-S}$ and constant term $\beta ={\sum }_{i=1}^k{a}^{i-1}b{q}^{-S_i}$, so $x_w=\beta /(1-a^k{q}^{-S})$. Existence and uniqueness hold because $|a^k{q}^{-S}{|}_a=a^{-k}<1$. □

Remark 4.

Lemma 4 is not new: for a general “Hydra map” it is equation (2.62) of Siegel [9], in the form $x=H_{\mathbf{j}}\left(0\right)/\left(1-H_{\mathbf{j}}^{\prime }\left(0\right)\right)$. Siegel observes that this expression is awkward on account of its denominator and therefore studies the numerator alone, ${\chi }_H\left(\mathbf{j}\right):=H_{\mathbf{j}}\left(0\right)$, which he calls thenumenof H (his Definition 2.14). Extended to infinite strings, ${\chi }_H$ is exactly the coding map π of Proposition 4, so that

$${\mu }_{a,b,q}={\left({\chi }_H\right)}_{*}\left(\mathrm{Haar}\mathrm{measure}\mathrm{on}{\mathbb{Z}}_q\right).$$

This is the cleanest description of the measure studied here, and we adopt it: μ is the distribution of Siegel’s numen under Haar measure. Its Fourier transform is Siegel’s characteristic function ${\phi }_H$ (his (2.172)), which he identifies with the characteristic function of Tao’s Syracuse random variable, and for which he proves the renormalisation identity ${\phi }_H\left(t\right)=p^{-1}{\sum }_j{e}^{-2\pi i{\{b_{j}t/d_j\}}_q}{\phi }_H(a_{j}t/d_j)$ (his Proposition 2.18). We claim no novelty for any of this.

Theorem 8.

$x_w$ equals the rational Syracuse k-cycle with valuation word $\overleftarrow{w}=(s_k,s_{k-1},\dots ,s_1)$.

Proof. 

Put $r_j=s_{k-j}$ for $0\le j\le k-1$. Then $R=S$ and $R_i={\sum }_{j<i}{s}_{k-j}=S-S_{k-i}$, so with $m=k-i$,

$$c={\displaystyle \sum _{i=0}^{k-1}}{a}^{k-1-i}b{q}^{S-S_{k-i}}={\displaystyle \sum _{m=1}^k}{a}^{m-1}b{q}^{S-S_m},$$

which is the numerator in Lemma 4. □

The reversal is not cosmetic: the forward dynamics composes on the left while the invariant measure arises from composition on the right (Remark 3), and Theorem 8 is the arithmetic shadow of that fact. We now record the correspondence in the form it deserves.

Theorem 9

(Correspondence). Let $\sigma (s_1,\dots ,s_k)=(s_2,\dots ,s_k,s_1)$ denote cyclic rotation, and call wprimitiveif $w\ne u^j$ for any $j\ge 2$. Then:

(i)

$x_{\sigma w}=f_{s_2}\circ \dots \circ f_{s_k}\left(x_w\right)=T^{k-1}\left(x_w\right)$, so the k rotations of w produce the k elements of a single cycle;

(ii)

$x_{u^j}=x_u$ for every $j\ge 1$;

(iii)

the assignment $w\mapsto {\left\{x_{{\sigma }^{i}w}\right\}}_{i<k}$ induces a bijection

$$\left\{\mathit{primitive}\mathit{words}\mathit{of}\mathit{length}k\right\}/\langle \sigma \rangle ⟷\left\{\mathit{rational}\mathit{Syracuse}k-\mathit{cycles}\mathit{of}{T}_{a,b,q}\right\}.$$

Proof. (i) Write $f_w=f_{s_1}\circ g$ with $g=f_{s_2}\circ \dots \circ f_{s_k}$, so $f_{\sigma w}=g\circ f_{s_1}$. From $f_{s_1}\left(g\left(x_w\right)\right)=x_w$, applying g gives $f_{\sigma w}\left(g\left(x_w\right)\right)=g\left(x_w\right)$; uniqueness in Lemma 4 yields $x_{\sigma w}=g\left(x_w\right)$. By Theorem 8 the valuation word of $x_w$ is $(s_k,\dots ,s_1)$, so $T\left(x_w\right)=f_{s_k}\left(x_w\right)$ and inductively $T^{k-1}\left(x_w\right)=f_{s_2}\circ \dots \circ f_{s_k}\left(x_w\right)=g\left(x_w\right)$.

(ii) $f_{u^j}={\left(f_u\right)}^j$ fixes $x_u$; uniqueness gives $x_{u^j}=x_u$.

(iii) Surjectivity: a rational k-cycle has a valuation word; reverse it and pass to its primitive root, using Theorem 8 and (ii). Injectivity: a cycle determines its valuation word up to cyclic rotation of the base point, and reversal carries rotation classes to rotation classes, since $\overleftarrow{\sigma w}={\sigma }^{k-1}\overleftarrow{w}$; hence w is determined up to $\sigma$, and primitivity removes the ambiguity of (ii). □

Corollary 1.

For $(a,b,q)=(3,1,2)$ the fixed point of $f_s$ is $1/(2^s-3)$. In particular $f_1$ fixes $-1$, the $3x+1$ cycle on the negative integers, and $f_2$ fixes 1, the trivial cycle $\{1,2,4\}$.

5. Purity, and the Absolute-Continuity Dichotomy

Proof 

(Proof of Theorem 6). Let $\mu ={\mu }_{\mathrm{ac}}+{\mu }_{\mathrm{s}}$ be the Lebesgue decomposition with respect to $\lambda$.

Each $f_s$ preserves both classes. Multiplication by a is injective on ${\mathbb{Z}}_a$ and satisfies $\lambda \left(aS\right)=a^{-1}\lambda \left(S\right)$, while translation and multiplication by the unit $q^{-s}$ preserve $\lambda$. Hence for Borel E with $\lambda \left(E\right)=0$ we get $\lambda \left(f_s^{-1}E\right)=a\lambda \left((q^{s}E-b)\cap a{\mathbb{Z}}_a\right)\le a\lambda \left(E\right)=0$, so ${\left(f_s\right)}_{*}{\mu }_{\mathrm{ac}}\ll \lambda$; and if ${\mu }_{\mathrm{s}}$ is carried by a $\lambda$-null set N then ${\left(f_s\right)}_{*}{\mu }_{\mathrm{s}}$ is carried by $f_s\left(N\right)$, with $\lambda \left(f_s\left(N\right)\right)=a^{-1}\lambda \left(N\right)=0$.

The decomposition is preserved. Therefore in

$$\mu =\sum _s{\nu }_q\left(s\right){\left(f_s\right)}_{*}{\mu }_{\mathrm{ac}}+\sum _s{\nu }_q\left(s\right){\left(f_s\right)}_{*}{\mu }_{\mathrm{s}}$$

the first sum is absolutely continuous and the second is carried by the countable union ${\bigcup }_s{f}_s\left(N\right)$, again $\lambda$-null, hence singular. Uniqueness of the Lebesgue decomposition forces

$${\mu }_{\mathrm{ac}}=\sum _s{\nu }_q\left(s\right){\left(f_s\right)}_{*}{\mu }_{\mathrm{ac}},{\mu }_{\mathrm{s}}=\sum _s{\nu }_q\left(s\right){\left(f_s\right)}_{*}{\mu }_{\mathrm{s}}.$$

Uniqueness closes the argument. Thus ${\mu }_{\mathrm{ac}}$ is a fixed point of the Markov operator $\mathcal{M}$ of Proposition 2, which preserves total mass. If $c:=\parallel {\mu }_{\mathrm{ac}}\parallel >0$ then ${\mu }_{\mathrm{ac}}/c$ is a $\mathcal{M}$-invariant probability measure, hence equals$\mu$ by the uniqueness clause of Proposition 2; so $\mu \ll \lambda$ and ${\mu }_{\mathrm{s}}=0$. Otherwise $c=0$ and $\mu ={\mu }_{\mathrm{s}}\perp \lambda$.

For the stated equivalences: $\mathrm{TV}(\mu ,\lambda )=1$ if and only if $\mu$ and $\lambda$ are mutually singular, and by the dichotomy just proved the negation is $\mu \ll \lambda$. Finally $\mathrm{TV}({\mu }_n,{\lambda }_n)$ increases to $\mathrm{TV}(\mu ,\lambda )$ by martingale convergence, the level-n algebras generating the Borel $\sigma$-algebra of ${\mathbb{Z}}_a$. □

Remark 5.

The proof uses nothing about μ beyond the fact that it is theuniquefixed point of $\mathcal{M}$. We do not know a derivation from the recursion $X\stackrel{d}{=}{q}^{-s}(b+aX)$ that avoids this; purity appears to be a genuinely system-theoretic, rather than distributional, property.

Remark 6

(a weaker criterion, and why it is redundant). The Hellinger affinity ${\rho }_n={\sum }_b\sqrt{{\mu }_n\left(b\right){\lambda }_n\left(b\right)}=a^{-n/2}{\sum }_b\sqrt{{\mu }_n\left(b\right)}$ decreases to $\rho (\mu ,\lambda )$, and $\mu \perp \lambda$ iff $\rho =0$. The case $t=\frac{1}{2}$ of Theorem 5 gives ${\rho }_n\le {\left(\Lambda \left(\frac{1}{2}\right)/\sqrt{a}\right)}^n$, so $a>\Lambda {\left(\frac{1}{2}\right)}^2$ forces singularity. This is never an improvement: $t\mapsto log\Lambda \left(t\right)$ is convex with $log\Lambda \left(1\right)=0$ and ${(log\Lambda )}^{\prime }\left(1\right)=-H\left({\nu }_q\right)$, whence $log\Lambda \left(\frac{1}{2}\right)\ge \frac{1}{2}H\left({\nu }_q\right)$ and $\Lambda {\left(\frac{1}{2}\right)}^2\ge a_{dim}$. Numerically, for $q=2$ the Hellinger threshold is $5.8284$ against $a_{dim}=4$.

6. Classification of the Supercritical Regime

Proof 

(Proof of Theorem 7). If $a>a_{dim}\left(q\right)$ then $error=H\left({\nu }_q\right)/loga<1$, so $error\mu <1$ by Proposition 4 and $\mu$ is carried by a set of Haar measure zero. It remains to find the integer pairs with $a<a_{dim}\left(q\right)$.

By Proposition 1, $H\left({\nu }_q\right)=log{\displaystyle \frac{q}{q-1}}+{\displaystyle \frac{logq}{q-1}}$. Both summands are strictly decreasing in q on $[2,\infty )$: the first has derivative ${\displaystyle \frac{1}{q}}-{\displaystyle \frac{1}{q-1}}<0$, and the second has derivative $\left[{\displaystyle \frac{q-1}{q}}-logq\right]/{(q-1)}^2$, negative because ${\displaystyle \frac{q-1}{q}}<1<logq$ for $q\ge 3$ and $\frac{1}{2}-log2<0$ at $q=2$. Hence $a_{dim}=e^{H\left({\nu }_q\right)}$ is strictly decreasing, and since $a_{dim}\left(5\right)=5^{5/4}/4=1.8692\dots <2$ we get $a_{dim}\left(q\right)<2$ for all $q\ge 5$, leaving no admissible integer $a\ge 2$.

For $q=2$: $a_{dim}=4$ and a must be odd, so $a<4$ forces $a=3$. For $q=3$: $a_{dim}=3^{3/2}/2=2.5980\dots$ and $3\nmid a$, so $a<a_{dim}$ forces $a=2$. Neither threshold is attained, since $a_{dim}$ is irrational in both cases. □

Table 1. The two critical multipliers. The ratio is $q-1$ in every row, as Theorem 1(iii) requires.

q	$H\left({\mathit{\nu }}_{\mathit{q}}\right)$	${\mathit{a}}_{dim}$	${\mathit{a}}_{\mathbf{drift}}$	ratio
2	$1.386294$	$4.000000$	$4.000000$	1
3	$0.954771$	$2.598076$	$5.196152$	2
5	$0.625503$	$1.869186$	$7.476744$	4
7	$0.478469$	$1.613602$	$9.681613$	6
11	$0.335100$	$1.398080$	$13.980798$	10

Table 1. The two critical multipliers. The ratio is $q-1$ in every row, as Theorem 1(iii) requires.

q	$H\left({\mathit{\nu }}_{\mathit{q}}\right)$	${\mathit{a}}_{dim}$	${\mathit{a}}_{\mathbf{drift}}$	ratio
2	$1.386294$	$4.000000$	$4.000000$	1
3	$0.954771$	$2.598076$	$5.196152$	2
5	$0.625503$	$1.869186$	$7.476744$	4
7	$0.478469$	$1.613602$	$9.681613$	6
11	$0.335100$	$1.398080$	$13.980798$	10

Remark 7.

The two survivors are $n\mapsto (3n+b)/2^{v_2}$ and $n\mapsto (2n+b)/3^{v_3}$, with $error=log4/log3=1.26186$ and $error=H\left({\nu }_3\right)/log2=1.37747$ respectively. It is perhaps worth recording that the classical Collatz map is not merely one supercritical member of its family but, together with a single companion, theonlyone.

7. The Two Critical Multipliers

Definition 2.

For a prime q set

$$a_{dim}\left(q\right)=exp\left(H\left({\nu }_q\right)\right),a_{\mathrm{drift}}\left(q\right)=q^{\mathbb{E}\left[v\right]}=q^{q/(q-1)}.$$

These are the announced thresholds. By (3), $error=1$ exactly when $loga=H\left({\nu }_q\right)$, i.e. $a=a_{dim}$; and by (2), $\delta =0$ exactly when $loga=\mathbb{E}\left[v\right]logq$, i.e. $a=a_{\mathrm{drift}}$. Both depend on q alone and both exceed 1. We emphasise that $\delta$ is an expectation: $a_{\mathrm{drift}}$ is the multiplier at which the mean logarithmic step $\mathbb{E}[log(a/q^v)]$ vanishes, and it does not describe the behaviour of any individual orbit. Note also that $a_{dim}$ and $a_{\mathrm{drift}}$ are real numbers while the family is indexed by integers a; the content of Theorems 1 and 3 is which integers fall on which side.

Proof 

(Proof of Theorem 1). Parts (i) and (ii) are Lemmas 1 and 2 and Proposition 1. For (iii), exponentiate (ii):

$$a_{dim}=e^{H\left({\nu }_q\right)}=e^{\mathbb{E}\left[v\right]logq-log(q-1)}={\displaystyle \frac{q^{\mathbb{E}\left[v\right]}}{q-1}}={\displaystyle \frac{a_{\mathrm{drift}}}{q-1}}.$$

Hence $a_{\mathrm{drift}}=(q-1)a_{dim}$, and the two agree if and only if $q-1=1$. □

Proof 

(Proof of Theorem 2). (a) ⇔ (b) is ${|(\mathbb{Z}/q\mathbb{Z})}^{\times }|=q-1$. (b) ⇔ (c) is (4). (c) ⇔ (d) is Proposition 1 together with Theorem 1(iii). For (a) ⇔ (e): the congruence $an+b\equiv 0\left(\mathrm{mod}q\right)$ has exactly one solution modulo q, while the integers prime to q occupy $q-1$ residue classes. Every such n therefore satisfies $v_q(an+b)\ge 1$ if and only if $q-1=1$ and that single solution is the class of units, i.e. $q=2$ and $a+b\equiv 0\left(\mathrm{mod}2\right)$; as $q\nmid a$ forces a odd, this says b is odd, which is the classical normalisation. □

Proof 

(Proof of Theorem 3). By Theorem 1(iii) and $q\ge 3$ we have $a_{\mathrm{drift}}=(q-1)a_{dim}>a_{dim}$. If $a_{dim}<a<a_{\mathrm{drift}}$ then $loga>H\left({\nu }_q\right)$, so $error=H\left({\nu }_q\right)/loga<1$, and $loga<\mathbb{E}\left[v\right]logq$, so $\delta <0$. Singularity follows from Proposition 4: a measure of Hausdorff dimension $<1$ on ${\mathbb{Z}}_a$ is carried by a set of Haar measure zero. For $q=2$ the interval $(a_{dim},a_{\mathrm{drift}})$ is empty. □

Example 1.

Take $q=3$, so $a_{dim}=e^{0.954771\dots }=2.598076\dots$ and $a_{\mathrm{drift}}=3^{3/2}=5.196152\dots$, whose ratio is exactly $2=q-1$. The integer multiplier $a=4$ lies in the band: the map $n\mapsto (4n+b)/3^{v_3(4n+b)}$ has drift $log4-\frac{3}{2}log3=-0.261624\dots <0$, so contracts on average, while its invariant measure on ${\mathbb{Z}}_4$ has $error=0.954771\dots /log4=0.688654\dots <1$ and is singular. No such example exists for $q=2$.

8. The Transfer Operator and the Rényi Spectrum

Let L act on $C\left({\mathbb{Z}}_a\right)$ by $\left(Lf\right)\left(x\right)={\sum }_{s\ge 1}{\nu }_q\left(s\right)f\left(f_s\left(x\right)\right)$, let $V_n\subset C\left({\mathbb{Z}}_a\right)$ be the space of functions constant on balls of radius $a^{-n}$ (so $dim{V}_n=a^n$, and $V_0$ is the constants), and for $\alpha >0$ let ${\left[f\right]}_{\alpha }={sup}_{x\ne y}|f\left(x\right)-f\left(y\right)|/d{(x,y)}^{\alpha }$, $C^{\alpha }={\{f:\parallel f\parallel }_{\infty }+{\left[f\right]}_{\alpha }<\infty \}$.

Proof 

(Proof of Theorem 4). (i) Let $f\in V_n$. Then $\left(Lf\right)\left(x\right)$ depends on x only through the values $f_s\left(x\right)=q^{-s}(b+ax)$ modulo $a^n$; as $q^{-s}$ is a unit of ${\mathbb{Z}}_a$, this is determined by $b+axmod{a}^n$, hence by $xmod{a}^{n-1}$. So $Lf\in V_{n-1}$, and iterating, $L^n\left(V_n\right)\subseteq V_0$, i.e. $L^{n}f$ is a constant $c\left(f\right)$. By Proposition 2 the measure $\mu$ satisfies $\mu =L^{*}\mu$, hence $\int Lgd\mu =\int gd\mu$ for every $g\in C\left({\mathbb{Z}}_a\right)$; applying this n times, $c\left(f\right)=\int L^{n}fd\mu =\int fd\mu$.

(ii) ${\parallel Lf\parallel }_{\infty }\le {\parallel f\parallel }_{\infty }$ since L is an average. For the seminorm, Proposition 2 gives $d(f_{s}x,f_{s}y)=a^{-1}d(x,y)$, so

$$|Lf\left(x\right)-Lf\left(y\right)|\le \sum _{s\ge 1}{\nu }_q\left(s\right){\left[f\right]}_{\alpha }d{(f_{s}x,f_{s}y)}^{\alpha }=a^{-\alpha }{\left[f\right]}_{\alpha }d{(x,y)}^{\alpha }.$$

This is a Lasota–Yorke inequality; since ${\mathbb{Z}}_a$ is compact the embedding $C^{\alpha }\hookrightarrow C\left({\mathbb{Z}}_a\right)$ is compact by Arzelà–Ascoli, so the Ionescu-Tulcea–Marinescu theorem in the form of Hennion [3] yields quasi-compactness with essential spectral radius at most $a^{-\alpha }$.

(iii) Iterating the seminorm estimate of (ii) gives ${\left[L^{n}g\right]}_{\alpha }\le a^{-n\alpha }{\left[g\right]}_{\alpha }$ for every $n\ge 1$. Suppose now $Lg=\lambda g$ with $g\in C^{\alpha }$ and $\left|\lambda \right|>a^{-\alpha }$. Then ${\left[g\right]}_{\alpha }={\left|\lambda \right|}^{-n}{\left[L^{n}g\right]}_{\alpha }\le {\left(a^{-\alpha }/\left|\lambda \right|\right)}^n{\left[g\right]}_{\alpha }$ for every n, and since $a^{-\alpha }/\left|\lambda \right|<1$ the factor tends to 0; hence ${\left[g\right]}_{\alpha }=0$ and g is constant, so $\lambda =1$. The eigenspace is exactly the constants, so 1 is simple. □

Remark 8.

We stress the reading of Theorem 4, since the presence of a uniform contraction invites the expectation of a meaningful mixing rate. A spectral gap exists on each $C^{\alpha }$, of size $1-a^{-\alpha }$; but it is the gap forced by the contraction alone, the residual spectrum is essential, and no eigenvalue other than the trivial one occurs. In particular there is no Ruelle–Pollicott resonance to compute. All analytic content of the system lies in the behaviour as the resolution n grows, not as time grows.

Proof 

(Proof of Theorem 5). By Lemma 1, for $t>0$

$$\Lambda \left(t\right):=\sum _{s\ge 1}{\nu }_q{\left(s\right)}^t={(q-1)}^t\sum _{m\ge 1}{q}^{-mt}={\displaystyle \frac{{(q-1)}^t}{q^t-1}},$$

so that $D_t^S=log\Lambda \left(t\right)/\left((1-t)loga\right)$. Since $\Lambda \left(1\right)=1$, the numerator vanishes at $t=1$; by l’Hôpital, ${lim}_{t\to 1}{D}_t^S=-{\Lambda }^{\prime }\left(1\right)/loga=H\left({\nu }_q\right)/loga=error$, giving continuity.

For the inequality, set

$$S_n\left(t\right)=\sum _{b\in \mathbb{Z}/a^n\mathbb{Z}}{\mu }_n{\left(b\right)}^t,D_t^{\left(n\right)}={\displaystyle \frac{log{S}_n\left(t\right)}{(1-t)nloga}},$$

so that $D_t={lim}_n{D}_t^{\left(n\right)}$ when the limit exists, and ${\overline{D}}_t={lim\; sup}_n{D}_t^{\left(n\right)}$ in general.

Step 1: an exact partition. Iterating (5) n times gives $\mu ={\sum }_{\left|w\right|=n}{\nu }_q\left(w\right){\left(f_w\right)}_{*}\mu$ with ${\nu }_q\left(w\right)={\prod }_i{\nu }_q\left(w_i\right)$. Each ${\left(f_w\right)}_{*}\mu$ is carried by $f_w\left({\mathbb{Z}}_a\right)$, a ball of radius $a^{-n}$ by Proposition 2. In an ultrametric space two balls of equal radius are either equal or disjoint, so for a ball B of radius $a^{-n}$ one has $\left({\left(f_w\right)}_{*}\mu \right)\left(B\right)\in \{0,1\}$, and therefore

$${\mu }_n\left(b\right)=\sum _{w:f_w\left({\mathbb{Z}}_a\right)=B(b,a^{-n})}{\nu }_q\left(w\right),$$

the index classes partitioning ${\mathbb{Z}}_{\ge 1}^n$ as b ranges over $\mathbb{Z}/a^n\mathbb{Z}$. This identity is exact, with no error term from partial overlap; it is the one place where ultrametricity does genuine work, and it has no analogue over $\mathbb{R}$.

Step 2: sub- and superadditivity. For $x_i\ge 0$ one has ${\left({\sum }_i{x}_i\right)}^t\ge {\sum }_i{x}_i^t$ when $t>1$, with the inequality reversed when $0<t<1$. Applying this within each class and summing over b, Step 1 gives

$$S_n\left(t\right)\ge \sum _{\left|w\right|=n}{\nu }_q{\left(w\right)}^t=\Lambda {\left(t\right)}^n(t>1),S_n\left(t\right)\le \Lambda {\left(t\right)}^n(0<t<1).$$

Step 3: sign bookkeeping. If $t>1$ then $log{S}_n\left(t\right)\ge nlog\Lambda \left(t\right)$ while $(1-t)loga<0$, so dividing reverses the inequality:

$$D_t^{\left(n\right)}={\displaystyle \frac{log{S}_n\left(t\right)}{(1-t)nloga}}\le {\displaystyle \frac{log\Lambda \left(t\right)}{(1-t)loga}}=D_t^S.$$

If $0<t<1$ then $log{S}_n\left(t\right)\le nlog\Lambda \left(t\right)$ while $(1-t)loga>0$, so dividing preserves it and the same bound results. Thus $D_t^{\left(n\right)}\le D_t^S$ for everyn and every $t>0$ with $t\ne 1$; in particular ${\overline{D}}_t\le D_t^S$, no passage to a limit being required.

Finally ${\overline{D}}_t\le 1$ because ${\mathbb{Z}}_a$ has Hausdorff dimension 1 (Lemma 3), and the two bounds combine to give ${\overline{D}}_t\le min(1,D_t^S)$. □

Example 2.

For $(a,q)=(3,2)$ one has $log(q-1)=0$ and $D_t^S=log(2^t-1)/\left((t-1)log3\right)$, so $D_2^S=1$ exactly and $D_3^S=log7/(2log3)=0.885622\dots$. Exact computation of ${\mu }_n$ for $n\le 16$ gives $D_2^{\left(n\right)}=0.8184,0.8546,0.8770$ and $D_3^{\left(n\right)}=0.7649,0.7953,0.8136$ at $n=8,12,16$: increasing, and consistent with the bounds $min(1,D_t^S)$ in each case.

9. The Classical Case

For $(a,b,q)=(3,1,2)$ we have ${\nu }_2\left(m\right)=2^{-m}$, $\mathbb{E}\left[v\right]=2$, $H\left({\nu }_2\right)=log4$, and hence

$$\delta =log3-log4=log\frac{3}{4}<0,error={\displaystyle \frac{log4}{log3}}=1.261859\dots >1.$$

Both inequalities are strict and both point the same way, because $a_{dim}=a_{\mathrm{drift}}=4$ by Theorem 1. Theorem 1 says that these are not two coincidences but one, and Theorem 2 that the reason is the triviality of ${(\mathbb{Z}/2\mathbb{Z})}^{\times }$.

We are careful about what $error>1$ does and does not give. It renders the upper bound of Proposition 4 vacuous; it does not establish that $\mu$ is absolutely continuous, nor even that $error\mu =1$. By Theorem 7, $(3,2)$ is one of only two integer pairs for which this is so.

What Theorem 6 adds is that the question is now binary and, in principle, decidable from the total variation alone: $\mu \ll \lambda$ if and only if ${lim}_n\mathrm{TV}({\mu }_n,{\lambda }_n^{\times })<1$. Exact computation for $n\le 15$ gives

$$0.16667,0.27778,0.34601,\dots ,0.43301,0.43584,0.43825,0.44028,$$

with increments decaying geometrically at ratio $\approx 0.84$ and extrapolating to $\approx 0.451$. Since the sequence increases, no finite computation can exclude the limit 1, and we state the dichotomy as open; but the numerical evidence points firmly towards absolute continuity, and by Theorem 6 nothing intermediate is available. A rigorous upper bound $\mathrm{TV}(\mu ,{\lambda }^{\times })<1$ would settle the matter outright, and we regard this as the most interesting question raised here.

10. Comparison with earlier work

The valuation law of Lemma 1 is classical, and we claim no novelty for it; the same is true of the affine unrolling of Proposition 3, which is Tao’s identity (1.7) in [10] and the n-path of [12]. Proposition 2 is Hutchinson’s contraction principle [4] in an ultrametric setting, where it is if anything easier than over $\mathbb{R}$, since balls of equal radius are either equal or disjoint and all overlaps are exact.

The rational-cycle formula recalled in Section 4 is Lagarias [7]; Lemma 4 is Siegel’s (2.62) (Remark 4) and Theorem 8 a reindexing of the two. We claim novelty only for the correspondence Theorem 9, which organises these by rotation classes and primitivity.

The results we believe to be new are Theorems 1, 2, 3, 4, 5, 6, 7 and 9: the exact factor $q-1$ separating the two critical multipliers, the four-way equivalence, the band of multipliers, the spectral degeneracy, the closed-form Rényi spectrum, the purity dichotomy, the classification of the supercritical regime, and the cycle correspondence. We have found no prior statement of any of these. We regard Theorems 6 and 7 as the principal contributions.

A second body of work requires careful demarcation. Siegel’s dissertation [9] develops a $(p,q)$-adic analysis for exactly the family of maps considered here (his “Hydra maps”, of which (1) is the two-branch case), and constructs the numen ${\chi }_H$ discussed in Remark 4. His programme is functional-analytic: he generalises Monna–Springer integration, proves a $(p,q)$-adic Wiener Tauberian theorem, and characterises periodic points of H through density of translates of $\widehat{{\chi }_H}$. The measure $\mu ={\left({\chi }_H\right)}_{*}\lambda$ is therefore implicit throughout his work, and its Fourier transform is one of his central objects.

What his treatment does not contain is the measure-geometric analysis undertaken here. Siegel studies ${\chi }_H$ as a function and ${\phi }_H$ as its Fourier transform; he does not consider the distribution of ${\chi }_H$ as a measure with a dimension, an entropy, or a Lebesgue type. In particular the notions of similarity dimension, Hausdorff dimension of a measure, iterated function system, invariant or stationary measure, Lebesgue decomposition, Rényi spectrum, Markov or transfer operator, and archimedean drift do not appear in his text, and neither of the two critical multipliers of Section 4 is defined there. The results we claim—Theorems 1, 2, 3, 4, 5, 6, 7 and 9—are therefore disjoint from his, and are best read as a measure-theoretic complement to a functional-analytic programme that had already identified the right object.

One further overlap requires explicit acknowledgement. Wegner [11] studies the family $3x+k$ and obtains a relation between the distributions of the Syracuse iterates for different k, deducing that the oscillation appearing in Tao’s argument is independent of k. That result concerns variation of the additive parameter with $(a,q)=(3,2)$ fixed, and it anticipates the isometry remark of Section 11 below, which we therefore do not claim. The present theorems concern variation of the divisor prime q, a direction Wegner does not consider.

11. Limitations

We close by delimiting the scope of the foregoing, since the family (1) contains the Collatz map and it would be easy to read more into Theorem 1 than it contains.

Proposition 5.

For b coprime to a, the measures ${\mu }_{a,b,q}$ and ${\mu }_{a,-b,q}$ are exchanged by the map $x\mapsto -x$, which is an isometry of ${\mathbb{Z}}_a$. More generally, ${\mu }_{a,cb,q}={(c·)}_{*}{\mu }_{a,b,q}$ for any unit c of ${\mathbb{Z}}_a$.

Proof. 

Unrolling as in Proposition 3, the constant term of the k-fold composition is homogeneous of degree one in b; replacing b by $cb$ therefore multiplies the limit by c. Multiplication by a unit preserves d. □

Corollary 2.

No isometry invariant of ${\mu }_{a,b,q}$—in particular neither error, nor $error\mu$, nor the Rényi spectrum of Theorem 5, nor the spectrum of L—distinguishes $3x+1$ from $3x-1$.

This is not a technicality. The map $3x-1$ possesses the nontrivial cycles $\{5,7\}$ and $\{17,25,37,41,55,61,91\}$, whereas $3x+1$ is conjectured to have none; equivalently, the $3x+1$ map on $\mathbb{Z}$ has at least four cycles, of which three lie among the negative integers. Any argument for the Collatz conjecture must therefore use the positivity of n, or the sign of b, in an essential step. By Corollary 2 the invariants studied here cannot do so. The quantities $a_{dim}$ and $a_{\mathrm{drift}}$ likewise depend only on $(a,q)$ and not on b.

Consequently Theorem 1 explains a feature of the classical parameters—that two a priori independent inequalities are one—but supplies no information about the orbits of any particular integer, and has no bearing on the truth of the Collatz conjecture. We regard the interest of the result, such as it is, as lying in the identification of ${|(\mathbb{Z}/q\mathbb{Z})}^{\times }|=1$ as the arithmetic source of a coincidence usually left unremarked.

12. Declarations

This work was carried out independently of any institution and received no funding from any agency in the public, commercial, or not-for-profit sectors. The author declares no competing interests. All computations reported in the paper were performed by the author in double-precision floating point unless stated otherwise; the quantities of Section 5 and 9 involve only sums of positive terms, so no cancellation occurs and the relative error is of order ${10}^{-13}$ throughout. Source code reproducing every table is available from the author on request.