Arithmetic Convergence of Double-iterated Polynomials

TL;DR

This paper investigates the convergence properties of sequences generated by iterating polynomials with integer coefficients and establishes conditions under which these sequences converge in the profinite integers, also exploring $b'$-adic approximations of related equations.

Contribution

It characterizes when such polynomial-generated sequences converge in $\u221e$ and links this to permutation properties modulo primes, also analyzing $b'$-adic approximations of fixed points.

Findings

Sequences converge in $\u221e$ iff $f$ is not a permutation mod any prime.

Limit of sequences is independent of initial value $b$ under certain conditions.

Provides $b'$-adic approximations for solutions to $f^y(a) = y$.

Abstract

Let $f$ be a polynomial with integer coefficients such that $f(n)$ positive for any positive integer $n$. We consider diverging sequences $\{ y_n\}$ given by $y_0 = b$ and $y_{n+1} = f^{y_n}(a)$ with positive integers $a$ and $b$. We show such a sequence converges in $\widehat{\mathbb{Z}}$ and the limit is independent of $b$, if and only if $f$ does not become a permutation of length $p$ on $\mathbb{Z}/p\mathbb{Z}$ for any prime number $p$. We also show that $b'$-adic asymptotic approximations of the equation $f^y(a) = y$ holds in $\mathbb{N}$ for some bases $b'$.

Equations

$$a^x{\equiv }_{b}x(i.e.a^x\equiv x\mathrm{mod}b).$$

$$a^{x_n}{\equiv }_{b^n}{x}_n\mathrm{and}{x}_n=c_n{b}^{n-1}+x_{n-1}(0\le c_n<b)$$

$$7^3$$

$$7^{43}$$

$$7^{343}$$

$$7^{\cdots 3643331265511565172343}$$

$$\underset{n\to \infty }{\lim }7\uparrow \uparrow n=\cdots 3643331265511565172343\mathrm{in}{\mathbb{Z}}_2\times {\mathbb{Z}}_5.$$

$$f^{x_n}(a){\equiv }_{b^n}{x}_n\mathrm{and}{x}_n=c_n{b}^{n-1}+x_{n-1}(0\le c_n<b).$$

$$f^3(0)$$

$$f^{43}(0)$$

$$f^{243}(0)$$

$$f^{\cdots 636048243}(0)$$

$$a\dot{+}b=\{\begin{array}{ll}a+b\in [m-1]& (\mathrm{if}a,b\in [m-1]\mathrm{and}a+b<m),\\ \overline{a+b}\in {\mathbf{Z}}_n& (\mathrm{otherwise})\end{array}$$

$${\pi }_{m,n}(a)=\{\begin{array}{ll}a\in [m-1]& (a<m),\\ \overline{a}\in {\mathbf{Z}}_n& (a\ge m).\end{array}$$

$$a\widehat{+}b=\{\begin{array}{ll}a+b\in \mathbb{N}& (a,b\in \mathbb{N})\\ a+b\in \hat{\mathbb{Z}}& (\mathrm{otherwise}).\end{array}$$

$$\underleftarrow{\lim }{\mathbf{Z}}_{m,n}=\underset{n}{\underleftarrow{\lim }}\underset{m}{\underleftarrow{\lim }}{\mathbf{Z}}_{m,n}.$$

$$a\dot{+}b=\{\begin{array}{ll}a+b\in \mathbb{N}& (a,b\in \mathbb{N})\\ \overline{a+b}\in {\mathbf{Z}}_n& (\mathrm{otherwise}).\end{array}$$

$$\underset{m}{\underleftarrow{\lim }}{\mathbf{Z}}_{m,n}\ni a=(a_m{)}_{m\in \mathbb{N}}\mapsto \{\begin{array}{ll}{a}_m\in \mathbb{N}& (\mathrm{if}{a}_m\in [m-1]\mathrm{for}\mathrm{any}m)\\ a_1\in {\mathbf{Z}}_n& (\mathrm{otherwise}).\end{array}$$

$$\mathbb{N}\cup {\mathbf{Z}}_n\ni a\mapsto \{\begin{array}{ll}({\pi }_{m,n}(a){)}_{m\in \mathbb{N}}& (\mathrm{if}a\in \mathbb{N})\\ (a{)}_{m\in \mathbb{N}}& (\mathrm{if}a\in {\mathbf{Z}}_n).\end{array}$$

$${\sigma }^{k^{\prime }+l}(a)={\sigma }^{k^{\prime }+x{l}_1+y{l}_2}(a)={\sigma }^{k^{\prime }}(a)(k^{\prime }\gg 0),$$

$${\sigma }^{k+N{l}_1}(a)={\sigma }^{k+N{l}_1+l}(a)={\sigma }^l({\sigma }^{k+N{l}_1}(a))(N\gg 0).$$

$$K+\alpha (L)=\max k_a+\alpha (\mathrm{lcm}{l}_a)=\max k_a+\max \alpha (l_a)\le \max k_a+\max l_a.$$

$$T_b$$

$$C_c$$

$$\begin{array}{ccc}\hat{\mathbb{Z}}& \stackrel{f}{\to }& \hat{\mathbb{Z}}\\ \downarrow & & \downarrow \\ {\mathbf{Z}}_{P(n)}& \stackrel{f_n}{\to }& {\mathbf{Z}}_n.\end{array}$$

$$f^{u-k}(s){\equiv }_m{f}^{v-k}(s){\equiv }_m{f}^{v-k}(t).$$

$$f^k(f^{u-k}(s)){\equiv }_n{f}^k(f^{v-k}(t)),$$

$$n_i,n_j\ge k+K\mathrm{and}{n}_i{\equiv }_L{n}_j(i,j\gg 0).$$

$$f{\uparrow }_x(n_i){\equiv }_{n}f{\uparrow }_x(n_j)(i,j\gg 0)$$

$$\underset{i\to \infty }{\lim }f{\uparrow }_s(n_i){\equiv }_n\underset{i\to \infty }{\lim }f{\uparrow }_s(n_i^{\prime })$$

$$\lambda (n)=\text{ the period of the reduction }{f}_{\mu (n)}:{\mathbf{Z}}_{\mu (n)}\to {\mathbf{Z}}_{\mu (n)}.$$

$$f^{n_i-k}(x){\equiv }_{\mu (n)}{f}^{n_i-k}(y)$$

$$f{\uparrow }_x(n_i)=f^{n_i}(x){\equiv }_n{f}^{n_i}(y)=f{\uparrow }_y(n_i)(i\gg 0).$$

$$\alpha ({\lambda }_f(p))=\alpha (\mathrm{lcm}\{l_x\})=\max \{\alpha (l_x)\}\le \max \{l_x\}<p,$$

Full text

††footnotetext: u661233h@ecs.osaka-u.ac.jp

1 Introduction

In [J-Y], J. Jiménez-Urroz and Yebra proved the following result.

Theorem 1.1**.**

([J-Y, Theorem 1]) For any positive integers $a$ and $b$, there exists a positive integer $x$ such that

[TABLE]

Moreover, if $b$ is valid (i.e. for every pair of prime numbers $p$ and $q$ such that $p\mid b$ and $q\mid p-1$, we have $q\mid b$), then there exists a sequence of positive integers $\{x_{n}\}$ such that

[TABLE]

for every $n$.

For example, they gave the case $a=7$ and $b=10$ as the following:

[TABLE]

The proof of Theorem 1.1 was constructive, that is, done by giving an algorithm to obtain $x$. In [S-S], D. B. Shapiro and S. D. Shapiro proved the former part of Theorem 1.1 independently, affording a more explicit form of the same algorithm. They showed that

Theorem 1.2**.**

([S-S, Corollary 2.11]) For any positive integers $a$ and $b$, the sequence $a$, $a^{a}$, $a^{a^{a}}$, $a\uparrow\uparrow 4$, …, $a\uparrow\uparrow n$,… become stable modulo $b$ for $n\gg 0$, where $a\uparrow\uparrow n$ is Knuth’s up-arrow notation introduced in [K]. Equivalently, the sequence converges in the ring $\mathbb{Z}_{p}$ of $p$-adic integers for every prime number $p$.

This implies $x=a\uparrow\uparrow n$ for $n\gg 0$ satisfies the congruence (1). In addition, this allows to state the above example as

[TABLE]

We regard the above two theorems as results on the polynomial $ax\in\mathbb{Z}[x]$ from the viewpoint of dynamical systems. Let $X$ be a set and $x$ an element of $X$. For a map $f:X\to X$, we denote $f^{n}(x)=f\circ f\circ\cdots\circ f(x)$ by $f\uparrow_{x}(n)$ and thus obtain a map $f\uparrow_{x}:\mathbb{N}\to X$ where $\mathbb{N}$ is the set of the positive integers. If $X=\mathbb{N}$, then for every positive integer $a$, we obtain $f\uparrow_{a}:\mathbb{N}\to\mathbb{N}$ and therefore we can construct $f\uparrow_{a}\uparrow_{b}$. If $f(x)=ax$, then we have $f\uparrow_{1}(n)=a^{n}=a\uparrow n$ and $f\uparrow_{1}\uparrow_{1}(n)=(f\uparrow_{1})\uparrow_{1}(n)=a\uparrow\uparrow n$.

The purpose of this paper is to generalize Theorems 1.1 and 1.2 to more general polynomials in $\mathbb{Z}[x]$. We shall prove the following:

Theorem 1.3**.**

Let $f$ be a polynomial of one variable with integer coefficients and we assume $f(\mathbb{N})\subseteq\mathbb{N}$. Then the following conditions are equivalent to each other:

1. (i)

For any prime number $p$, the reduction map $f_{p}:\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$ is not a cyclic permutation of length $p$. 2. (ii)

For any $a,b\in\mathbb{N}$, if $\lim_{n\to\infty}f\uparrow_{a}\uparrow_{b}(n)=\infty$ in $\mathbb{R}$, then the limit $\lim_{n\to\infty}f\uparrow_{a}\uparrow_{b}(n)$ exists in $\mathbb{Z}_{p}$ for every prime number $p$ and is independent of $b$.

We call a polynomial $f\in\mathbb{Z}[x]$ tower-stable if $f$ satisfies condition (i) in the above theorem (this definition will be generalized in Definition 3.11). Note that for every $a\in\mathbb{N}$, $f(x)=ax$ is tower-stable because $f(0)=0$. Therefore Theorem 1.3 implies Theorem 1.2.

Definition 1.4**.**

Let $f$ be a tower-stable polynomial. A positive integer $b$ is said to be $f$-$valid$ if $b$ is square-free, valid and for every pair of prime numbers $p$ and $q$ such that $p\mid b$ and $q\mid\lambda_{f}(p)$, it holds $q\mid b$, where $\lambda_{f}(p)$ is the period of the reduction map $f_{p}:\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$.

With this refinement of valid numbers, we state the following generalization of the latter part of Theorem 1.1.

Theorem 1.5**.**

Let $f$ be a tower-stable polynomial. If $b$ is $f$-valid, there exists a sequence of positive integers $\{x_{n}\}$ such that

[TABLE]

For example, we choose a polynomial $f(x)=x^{2}+x+3$ and $b=10$. $f$ is tower-stable because $f(0)=f(-1)=3$ makes no reductions injective. We see $b=10$ is $f$-valid. It is easy to see $10$ is valid: if $p\mid 10$ and $q\mid(p-1)$ then $(p,q)=(5,2)$ and $q\mid 10$. Moreover, by some direct computation, we can see $\lambda_{f}(5)=2$ and $\lambda_{f}(2)=1$, this leads that $10$ is $f$-valid.

[TABLE]

The outline of the paper is as follows. In Section 2, We extend maps on $\mathbb{N}$ to some maps on $\widehat{\mathbb{Z}}$, the profinite complation of $\mathbb{Z}$. Using these extensions, we discuss dynamical systems on $\widehat{\mathbb{Z}}$ in Section 3 and prove profinite version Theorem 3.13 of Theorem 1.3, then obtain Theorem 1.3 as a corollary. In Section 4, we give a more precise evaluation of order of convergence to show Theorem 1.5.

Acknowledgement

Takao Watanabe encouraged me to try to write a paper and gave some valid comments for drafts. Takehiko Yasuda and Seidai Yasuda proofread this paper and gave warm encouragement. Without their contributions, this paper cannot be materialized. I would like to express my greatest appreciation to them.

1.1 Notation

[TABLE]

2 Profinite Completion of $\mathbb{N}$

Let $S$ be a finite quotient semigroup of $(\mathbb{N},+)$ and $\pi:\mathbb{N}\to S$ be the quotient map. The semigroup $S$ is generated by $\pi(1)$. Since $S$ is finite, the set $P_{S}:=\{(k,l)\in\mathbb{N}^{2}\mid\pi(k)=\pi(k+l)\}$ is nonempty. Let $(m,n)=(m_{S},n_{S})$ be a pair of positive integers such that $n:=\min\{l\mid$ there exists $k$ such that $(k,l)\in P_{S}\}$ and $m:=\min\{k\mid(k,n)\in P_{S}\}$. Then $S$ is uniquely determined by $m$ and $n$ up to isomorphism and identified with the semigroup ${\bf Z}_{m,n}:=([m-1]\cup{\bf Z}_{n},\dot{+})$ where the operator $\dot{+}$ is defined as

[TABLE]

and the quotient map $\pi_{m,n}:\mathbb{N}\to{\bf Z}_{m,n}$ is given by

[TABLE]

If $m\leq k$ and $n\mid l$, there is natural homomorphism $\phi_{(k,l),(m,n)}:{\bf Z}_{k,l}\to{\bf Z}_{m,n}$. Thus $(\{{\bf Z}_{m,n}\},\{\phi_{(k,l),(m,n)}\})$ is a projective system and we define $\widehat{\mathbb{N}}$, the profinite completion of $\mathbb{N}$ as the projective limit of $({\bf Z}_{m,n},\phi_{(k,l),(m,n)})$. The semigroup $\widehat{\mathbb{N}}$ has the following description with $\widehat{\mathbb{Z}}:=\varprojlim_{n}{\bf Z}_{n}$:

Proposition 2.1**.**

We have $\widehat{\mathbb{N}}\cong(\mathbb{N}\cup\widehat{\mathbb{Z}}\,,\widehat{+})$ where

[TABLE]

Proof.

We can decompose every map $\phi_{(k,l),(m,n)}$ as $\phi_{(k,l),(m,n)}=\phi_{(k,n),(m,n)}\circ\phi_{(k,l),(k,n)}$. Therefore the definition of projective limit allows us to compute $\widehat{\mathbb{N}}$ as

[TABLE]

First, we show $\varprojlim_{m}{\bf Z}_{m,n}\cong(\mathbb{N}\cup{\bf Z}_{n},\dot{+})$, where

[TABLE]

Indeed we can construct maps which are the inverses of each other: one is given by

[TABLE]

The other is given by:

[TABLE]

Then it remains to show $\varprojlim_{n}(\mathbb{N}\cup{\bf Z}_{n},\dot{+})=(\mathbb{N}\cup\widehat{\mathbb{Z}},\widehat{+})$, which follows from $\varprojlim_{n}{\bf Z}_{n}=\widehat{\mathbb{Z}}$. ∎

Every projection $\pi_{m,n}:\mathbb{N}\to{\bf Z}_{m,n}$ defines multiplication on ${\bf Z}_{m,n}$. We regard ${\bf Z}_{m,n}$ as a topological semiring with discrete topology. Then, $\widehat{\mathbb{N}}$ becomes a compact Hausdorff semiring by the projective limit topology. Note that a sequence $n_{1},n_{2},...$ of natural numbers converges to $s\in\widehat{\mathbb{Z}}$ in $\widehat{\mathbb{N}}$ if and only if $n_{i}$ tends to infinity in $\mathbb{R}$ and the sequence $\widehat{n_{i}}$ converges to $s$ in $\widehat{\mathbb{Z}}$ (with respect to the standard topology of $\widehat{\mathbb{Z}}$), where $\widehat{\cdot}:\mathbb{N}\to\widehat{\mathbb{Z}}$ is the natural embedding.

Proposition 2.2**.**

Let $X$ be a metric space and $f:\mathbb{N}\to X$ a map. Then the following conditions are equivalent:

1. (i)

The map can be extended to a continuous map $\widehat{f}:\widehat{\mathbb{N}}\to X$. 2. (ii)

For every $s\in\widehat{\mathbb{Z}}$ and every sequence of positive integers $\{n_{i}\}_{i\in\mathbb{N}}$ that satisfies $n_{i}\to+\infty$ in $\mathbb{R}$ and $\widehat{n_{i}}\to s$ in $\widehat{\mathbb{Z}}$ as $i\to\infty$, the limit $\lim_{i\to\infty}f(n_{i})$ exists in $X$ and is independent of $n_{i}$ (i.e. the limit $\lim_{i\to\infty}f(n_{i})$ depends only on $s$).

Proof.

(i) $\Rightarrow$ (ii) follows by the above remark about the convergence of sequence of positive integers on $\widehat{\mathbb{N}}$; we show (ii) $\Rightarrow$ (i). For every $s\in\widehat{\mathbb{Z}}$, we take $\{n_{i}\}_{i\in\mathbb{N}}$ as in the condition of (ii) and set $\widehat{f}(s):=\lim_{i\to\infty}f(n_{i})$. Then $\widehat{f}:\widehat{\mathbb{N}}\to X$ is well-defined by assumption, we show $\widehat{f}$ is continuous. Since $\widehat{\mathbb{N}}$ is given by a projective limit of a countable system of finite sets, the topology of $\widehat{\mathbb{N}}$ is first countable. Thus it is enough to see $\widehat{f}$ is sequentially continuous. Let $\{s_{i}\}_{i\in\mathbb{N}}\subset\widehat{\mathbb{Z}}$ be a sequence such that $\lim_{i\to\infty}s_{i}=s\in\widehat{\mathbb{Z}}$. Let us denote the metric on $X$ by $d$ and fix an arbitrary $\varepsilon>0$. We take sequences of positive integers $\{n_{ij}\}_{i,j\in\mathbb{N}}$ such that for every $i$, $n_{ij}\to s_{i}$ in $\widehat{\mathbb{N}}$ as $j\to\infty$. By (ii), we have $d(f(n_{ij}),f(s_{i}))<\frac{\varepsilon}{2}\ (j\gg 0)$. Thus we can take a sequence of positive integers $\{j_{i}\}_{i\in\mathbb{N}}$ such that $d(f(n_{ij_{i}}),f(s_{i}))<\frac{\varepsilon}{2}$, $n_{ij_{i}}>i$ and $n_{ij_{i}}\equiv s_{i}\mod i!$. Here $n_{ij_{i}}\to+\infty$ in $\mathbb{R}$ and $n_{ij_{i}}\to s$ in $\widehat{\mathbb{Z}}$ as $i\to\infty$, again by (ii) we have $d(f(n_{ij_{i}}),f(s))<\frac{\varepsilon}{2}\ (i\gg 0)$, thus $d(f(s_{i}),f(s))<\varepsilon\ (i\gg 0)$. ∎

Corollary 2.3**.**

Let $X$ be a complete metric space, $f:X\to X$ be a continuous map and $x$ be a point in $X$. Then the following conditions are equivalent:

1. (i)

The map $f\uparrow_{x}:\mathbb{N}\to X$ can be extended to a continuous map $\widehat{f\uparrow_{x}}:\widehat{\mathbb{N}}\to X$. 2. (ii)

For every $s\in\widehat{\mathbb{Z}}$ and every sequence of positive integers $\{n_{i}\}_{i\in\mathbb{N}}$ that satisfies $n_{i}\to+\infty$ in $\mathbb{R}$ and $\widehat{n_{i}}\to s$ in $\widehat{\mathbb{Z}}$ as $i\to\infty$, the limit $\lim_{i\to\infty}f^{n_{i}}(x)$ exists in $X$ and is independent of $n_{i}$ (i.e. limit $\lim_{i\to\infty}f^{n_{i}}(x)$ depends only on $s$).

Definition 2.4**.**

Let $X$ be a complete metric space and $f:X\to X$ be a continuous map. A point $x$ in $X$ is said to be a profinite preperiodic point of $f$ if $x$ suffices the conditions (i) or (ii) of Corollary 2.3. A continuous map $f$ is said to be profinite preperiodic if every point $x$ in $X$ is a profinite preperiodic point of $f$.

We chose the term ”profinite preperiodic” since $x$ is a preperiodic point of $f$ if and only if $f\uparrow_{x}$ factors through some finite semigroup ${\bf Z}_{m,n}$.

For a profinite preperiodic point $x$, we define $f^{s}(x)$ for $s\in\widehat{\mathbb{Z}}$ by $f^{s}(x):=\lim_{i\to\infty}f^{n_{i}}(x)$, where the integer sequence $n_{i}$ is taken as in Proposition 2.3(ii). If $f$ is profinite preperiodic, then for every $a,b\in\widehat{\mathbb{N}}$, we have $f^{a}\circ f^{b}=f^{a+b}$ and $(f^{a})^{b}=f^{ab}$.

3 Dynamical System on $\widehat{\mathbb{Z}}$

Before discussing about dynamical systems on $\widehat{\mathbb{Z}}$, we begin with reviewing basic facts on finite dynamical systems. Let $\alpha:\mathbb{N}\to\mathbb{N}$ be a function of “the largest factor prime power”, that is, $\alpha(n)=\max\{p^{a}\mid p{\ \rm is\ prime\ and\ }p^{a}\ {\rm divides}\ n\}$. By prime factorization, we can see $\alpha({\rm lcm}\{n_{i}\})=\max\{\alpha(n_{i})\}$ and $\alpha(mn)\leq\alpha(m)\alpha(n)$.

Lemma 3.1**.**

Let $n$ be a positive integer, $F$ a finite set of cardinality $n$ and $\sigma:F\to F$ a map.

1. (i)

For every $a\in F$, there exist nonnegative integers $k$ and $l$ such that $l\geq 1$, $\sigma^{k}(a)=\sigma^{k+l}(a)$ and $k+l\leq n$. 2. (ii)

For every $a\in F$, if $\sigma^{k_{i}}(a)=\sigma^{k_{i}+l_{i}}(a)$ for $i=1,2$, then $\sigma^{k}(a)=\sigma^{k+l}(a)$ holds for $(k,l)=(\min\{k_{1},k_{2}\},\gcd\{l_{1},l_{2}\})$. 3. (iii)

For every $a\in F$, there exists an unique pair of nonnegative integers $(k,l)$ such that $\sigma^{k^{\prime}}(a)=\sigma^{k^{\prime}+l^{\prime}}(a)$ if and only if $k\leq k^{\prime}$ and $l\mid l^{\prime}$. We denote this pair by $(k_{a},l_{a})$. 4. (iv)

Let $K=\max_{a\in F}\{k_{a}\}$ and $L={\rm lcm}_{a\in F}\{l_{a}\}$. Then $\sigma^{K^{\prime}}=\sigma^{K^{\prime}+L^{\prime}}$ if and only if $K\leq K^{\prime}$ and $L\mid L^{\prime}$. 5. (v)

We have $K+\alpha(L)\leq n$.

Proof.

(i) Obvious from the pigeonhole principle.

(ii) Without loss of generality, we assume $k=k_{1}$. We can take a pair of integers $(x,y)$ as $xl_{1}+yl_{2}=l$. Then we have

[TABLE]

particularly for $k^{\prime}=k+Nl_{1}$,

[TABLE]

The assumption $k=k_{1}$ implies $\sigma^{k+Nl_{1}}(a)=\sigma^{k}(a)$ for every $N$. Thus $\sigma^{k}(a)=\sigma^{l}(f^{k}(a))=\sigma^{k+l}(a)$.

(iii) Let $k:=\min\{k^{\prime}\mid$ $\sigma^{k^{\prime}}(a)=\sigma^{k^{\prime}+l^{\prime}}(a)$ for some $l^{\prime}\}$ and $l:=\gcd\{l^{\prime}\mid$ $\sigma^{k^{\prime}}(a)=\sigma^{k^{\prime}+l^{\prime}}(a)$ for some $k^{\prime}\}$. By (i) and (ii), the pair $(k,l)$ satisfies the condition.

(iv) Obvious from (iii).

(v) We have

[TABLE]

Let $b,c\in F$ be such that $\max k_{a}=k_{b}$ and $\max l_{a}=l_{c}$. Let

[TABLE]

Then $T_{b}$ and $C_{c}$ are disjoint subsets of $F$, which leads to $k_{b}+l_{c}\leq n$. ∎

Integers $k=k_{a}$ and $l=l_{a}$ are called the tail length and the cycle length of $\sigma$ on $a$ respectively. Also, $K$ and $L$ are called the preperiod length and the period of $\sigma$ respectively.

Definition 3.2**.**

Let $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ be a continuous map. A map $P:\mathbb{N}\to\mathbb{N}$ is said to be a period map of $f$ that satisfies following equivalent conditions:

1. (i)

For every $s,t\in\widehat{\mathbb{Z}}$ and $n\in\mathbb{N}$, if $s\equiv_{P(n)}t$ then $f(s)\equiv_{n}f(t)$. 2. (ii)

For every $n\in\mathbb{N}$, there exists a map $f_{n}:{\bf Z}_{P(n)}\to{\bf Z}_{n}$ that makes the following diagram commutative:

[TABLE]

Here the vertical arrows are projections.

Maps $f_{n}$ are called reductions of $f$.

Note that $\widehat{\mathbb{Z}}$ is metrizable with some metric which is invariant by translations of the form $s\mapsto s+c$. Since $\widehat{\mathbb{Z}}$ is compact, any continuous map $\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ is also uniformly continuous with such a metric, hence there exists a period map of the continuous map. A period map of $f$ is not uniquely determined by $f$.

Proposition 3.3**.**

Let $P$ be a period map of $f$ and $Q$ a period map of $g$. Then $Q\circ P$ is a period map of $f\circ g$. In particular, $P^{n}$ is a period map of $f^{n}$.

Proof.

We check (i) of the definition above. If we take $s,t\in\widehat{\mathbb{Z}}$ such that $s\equiv_{Q(P(n))}t$, then it follows $g(s)\equiv_{P(n)}g(t)$ and $f(g(s))\equiv_{n}f(g(t))$. ∎

Definition 3.4**.**

Let $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ be a continuous map.

1. (i)

$f$ is said to be $congruence$ $stable$ if there exists a period map $P$ of $f$ such that $P^{k}(n)=P^{k+1}(n)(k\gg 0)$ for each positive integer $n$. 2. (ii)

$f$ is said to be $congruence$ $preserving$ if ${\rm id}_{\mathbb{N}}$, the identity function on $\mathbb{N}$, is a period map of $f$.

Remark 3.5**.**

Every polynomial in $\widehat{\mathbb{Z}}[x]$, as a map $\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$, is congruence preserving, but not every congruence preserving map is a polynomial with $\widehat{\mathbb{Z}}$ coefficients, see [Ch] and [Ce-Gr-Gu].

Lemma 3.6**.**

Let $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ be a continuous map, $P$ a period map of $f$ and $m=P^{k}(n)=P^{k+1}(n)$ for $k,m,n\in\mathbb{N}$. Let $K$ be the preperiod length of the reduction $f_{m}:{{\bf Z}_{m}}\to{{\bf Z}_{m}}$ and $L$ be the period of $f_{m}$.

For every $s,t\in\widehat{\mathbb{Z}}$ and $u,v\in\mathbb{N}$, if $s\equiv_{m}t$, $u,v\geq k+K$ and $u\equiv_{L}v$, then we have $f\uparrow_{s}(u)\equiv_{n}f\uparrow_{t}(v)$.

Proof.

If we have $u-k,v-k\geq K$ and $u-k\equiv_{L}v-k$, then

[TABLE]

By Proposition 3.3, $P^{k}$ is a period map of $f^{k}$, we obtain

[TABLE]

that is $f\uparrow_{s}(u)\equiv_{n}f\uparrow_{t}(v)$. ∎

Theorem 3.7**.**

Let $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ be a congruence stable map. Then $f$ is profinite preperiodic.

Proof.

We check the condition (ii) in Proposition 2.3. Let $\{n_{i}\}$ be a sequence of positive integers such that $n_{i}\to\infty$ in $\mathbb{R}$ and $\widehat{n_{i}}\to s$ in $\widehat{\mathbb{Z}}$ as $i\to\infty$ for some $s\in\widehat{\mathbb{Z}}$.

First, we will show the sequence $\{f^{n_{i}}(s)=f\uparrow_{s}(n_{i})\}_{i}$ converges, that is, becomes eventually stable modulo $n$ for an arbitrary positive integer $n$. Since $f$ is congruence stable, there is a period map $P$ of $f$ such that $P^{k}(n)=P^{k+1}(n)$ for some $k$. Therefore we take $K$ and $L$ as in Lemma 3.6. By the definition of the sequence $\{n_{i}\}$, we have

[TABLE]

This leads to

[TABLE]

by the above lemma. Thus $\{f\uparrow_{x}(n_{i})\mod n\}_{i}$ become eventually stable.

Next, let $\{n^{\prime}_{i}\}$ be another sequence that satisfies the same condition as $\{n_{i}\}$ and converges to the same limit $s$ as $\{n_{i}\}$ does. Then by a similar argument, we have

[TABLE]

for every positive integer $n$. This means $\lim_{i\to\infty}f\uparrow_{x}(n_{i})=\lim_{i\to\infty}f\uparrow_{x}(n^{\prime}_{i})$ in $\widehat{\mathbb{Z}}$. ∎

By Proposition 2.3, we obtain a continuous map $\widehat{f\uparrow_{s}}:\widehat{\mathbb{N}}\to\widehat{\mathbb{Z}}$. By Proposition 2.1, we regard the domain $\widehat{\mathbb{N}}$ as $\mathbb{N}\cup\widehat{\mathbb{Z}}$ and denote the restriction $\widehat{f\uparrow_{s}}|_{\widehat{\mathbb{Z}}}:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ by f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s}, which is also continuous.

Proposition 3.8**.**

Let $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ be congruence stable, $P$ a period map of $f$ and $\mu:\mathbb{N}\to\mathbb{N}$ a map, such that for every $n\in\mathbb{N}$ it holds $\mu(n)=P^{k}(n)=P^{k+1}(n)$ for some $k\in\mathbb{N}$. Let $\lambda:\mathbb{N}\to\mathbb{N}$ be a map such that

[TABLE]

Then $\lambda$ is a period map of f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{x} for every $x$ in $\widehat{\mathbb{Z}}$.

Moreover, for any $x,y,s\in\widehat{\mathbb{Z}}$, if $x\equiv_{\mu(n)}y$ then f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{x}(s)\equiv_{n}f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{y}(s).

Proof.

We proceed arguement of the proof of Theorem 3.7. We take $s,t\in\widehat{\mathbb{Z}}$ as $t\equiv_{L}s$. For any sequence $\{n^{\prime\prime}_{i}\}$ such that $n^{\prime\prime}_{i}\to t$, we also have $f\uparrow_{x}(n_{i})\equiv_{n}f\uparrow_{x}(n^{\prime\prime}_{j})(i,j\gg 0)$ by a similar argument to Theorem 3.7. Therefore we have f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{x}(t)\equiv_{n}f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{x}(s).

We shall show the latter part. Let $n$ be a positive integer, $n_{i}$ a sequence of positive integers such that $n_{i}\to s$ in $\widehat{\mathbb{N}}$. By the assumption $P(\mu(n))=\mu(n)$, if $n_{i}\geq k$ then we have

[TABLE]

then by the assumption $P^{k}(n)=\mu(n)$ and Proposition 3.3, we have

[TABLE]

Passing to the limits, we obtain f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{x}(s)\equiv_{n}f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{y}(s). ∎

We now discuss the case $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ is congruence preserving. We denote the period of the reduction $f_{n}:{\bf Z}_{n}\to{\bf Z}_{n}$ by $\lambda_{f}(n)$, defining $\lambda_{f}:\mathbb{N}\to\mathbb{N}$. By the above proposition, we have the following:

Proposition 3.9**.**

Let $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ be congruence preserving and $s\in\widehat{\mathbb{Z}}$. Then $\lambda_{f}$ is a period map of f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s}.

Now we shall evaluate $\lambda_{f}$.

Lemma 3.10**.**

Let $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ be congruence preserving and $p$ a prime number. The following conditions are equivalent each other:

1. (i)

$\lambda_{f}(p)\neq p$. 2. (ii)

The reduction $f_{p}:{{\bf Z}_{p}}\to{{\bf Z}_{p}}$ of $f$ is not a cyclic permutation of length $p$. 3. (iii)

For every $x\in{{\bf Z}_{p}}$, it holds $l_{x}<p$ for the cycle length $l_{x}$ of $f_{p}$ on $x$. 4. (iv)

$\alpha(\lambda_{f}(p))<p$.

Proof.

((i) $\Rightarrow$ (ii)) The contraposition is obvious.

((ii) $\Rightarrow$ (iii)) We show the contraposition. Assume that there is some $x\in{{\bf Z}_{p}}$ such that $f_{p}^{k}(x)={f_{p}}^{k+l}(x)$ implies $l\geq p$. Then $x,f_{p}(x),...,(f_{p})^{p-1}(x)$ are distinct elements of ${{\bf Z}_{p}}$ and $(f_{p})^{p}(x)=x$, that is, $f_{p}$ is a cyclic permutation of length $p$.

((iii) $\Rightarrow$ (iv)) Assume (iii). By Lemma 3.1 (iv), the period $\lambda_{f}(p)$ of $f_{p}:{{\bf Z}_{p}}\to{{\bf Z}_{p}}$ satisfies

[TABLE]

where $x$ runs over ${{\bf Z}_{p}}$.

((iv) $\Rightarrow$ (i)) The contraposition is obvious, because $p$ is a prime number. ∎

Definition 3.11**.**

A congruence preserving function $f:\widehat{\mathbb{Z}}\to\widehat{\mathbb{Z}}$ is said to be tower-stable if $f$ satisfies $\alpha(\lambda_{f}(p))<p$, the condition (iv) of Lemma 3.10, for every prime number $p$.

Lemma 3.12**.**

Let $f$ be a congruence preserving map and $n,n^{\prime}$ be positive integers. The followings hold:

1. (i)

${\rm lcm}(\lambda_{f}(n),\lambda_{f}(n^{\prime}))=\lambda_{f}({\rm lcm}(n,n^{\prime}))$. In particular, if $n\mid n^{\prime}$ then $\lambda_{f}(n)\mid\lambda_{f}(n^{\prime})$. 2. (ii)

$\lambda_{f}(p^{a+1})\mid{\rm lcm}\{1,2,\ldots,p\}\cdot\lambda_{f}(p^{a})$ for every prime $p$ and every positive integer $a$. 3. (iii)

$\lambda_{f}^{k}(n)=\lambda_{f}^{k+1}(n)\ (k\gg 0)$.

Moreover, if $f$ is tower-stable, the assertions (iii) can be strengthen as

• (iii’)

$\lambda_{f}^{k}(n)=1\ (k\gg 0)$.

Proof.

(i) We recall that for any positive integers $N$ and $\nu$, $f^{M}(x)\equiv_{\nu}f^{M+N}(x)\ (M\gg 0)$ for all $x\in\widehat{\mathbb{Z}}$ if and only if $\lambda_{f}(\nu)\mid N$ by Lemma 3.1(iv). Therefore we have

[TABLE]

for every positive integer $N$. This completes the proof.

(ii) We set $l:=\lambda_{f}(p^{a})$. Let us take an arbitrary $s\in\widehat{\mathbb{Z}}$. We put $k$ the preperiod length of the reduction $f_{p^{a}}:{\bf Z}_{p^{a}}\to{\bf Z}_{p^{a}}$ and $t=f^{k}(s)$. Let $\pi:\widehat{\mathbb{Z}}\twoheadrightarrow{\bf Z}_{p^{a}}$ and $\pi_{a}:{\bf Z}_{p^{a+1}}\twoheadrightarrow{\bf Z}_{p^{a}}$ be the cannonical surjections. We define

[TABLE]

For every $\overline{t^{\prime}}\in T$, from $t=f^{k}(s)$ it follows that

[TABLE]

that is $f^{l}(T)\subset T$. Therefore by Lemma 3.1 (i), we can take $k_{t}$ and $l_{t}$ as

[TABLE]

thus in particular,

[TABLE]

since $k_{t}\leq p$ and $l_{t}\mid{\rm lcm}\{1,2,...,p\}={\rm lcm}[p]$. Thus Lemma 3.1 (iv) leads

[TABLE]

(iii) We prove this by induction on $\alpha(n)$. The case $\alpha(n)=1$ is obvious. Let us assume that the assertion holds for every $n$ of $\alpha(n)<p^{a}$ and now set $\alpha(n)=p^{a}$. By (ii), we have $\alpha(\lambda_{f}(p^{a}))\leq p\cdot\alpha(\lambda_{f}(p^{a-1}))$ thus

[TABLE]

and hence $\alpha(\lambda_{f}(n))\leq\alpha(n)$ by (i). The case $\alpha(\lambda_{f}(n))<\alpha(n)$ is evident from the assumption of the induction. Let us assume $\alpha(\lambda_{f}(n))=\alpha(n)=p^{a}$, that is $\alpha(\lambda_{f}(p^{a}))=p^{a}$, then put $\lambda_{f}(p^{a})=cp^{a}$. By (i) of this lemma, we have

[TABLE]

Therefore, by applying (i) inductively, we obtain

[TABLE]

Here $\alpha(c)$ and $\alpha\left(\frac{n}{p^{a}}\right)$ are both less than $p^{a}$, therefore by the assumption of the induction, $\lambda_{f}^{k}(n)$ is eventually constant.

(iii’) (ii) and $\alpha(\lambda_{f}(p))<p$ leads to $\alpha(\lambda_{f}(p^{a}))<p^{a}$ in (2). This leads to $\alpha(\lambda_{f}(N))<\alpha(N)$ for every $N>1$. Therefore, we have $\alpha(\lambda_{f}^{k}(n))=1\ (k\gg 0)$, that is, $\lambda_{f}^{k}(n)=1\ (k\gg 0)$. ∎

Theorem 3.13**.**

Let $f$ be congruence preserving and $s\in\widehat{\mathbb{Z}}$. Then the map f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s} is profinite preperiodic. Moreover, if $f$ is tower-stable, then f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s}\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{t} is a constant function and independent of $t$.

Proof.

By Proposition 3.9, $\lambda_{f}$ is a period map of f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s}. Therefore by Lemma 3.12 (iii), f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s} is congruence stable. Theorem 3.7 shows that f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s} is profinite preperiodic.

We now assume that $f$ is tower-stable. By Lemma 3.12 (iii’), we can apply Proposition 3.8 for f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s} and $\lambda_{f}$, with the constant functions $\mu\equiv 1$ and $\lambda\equiv 1$. We therefore obtain that if $t\equiv_{1}t^{\prime}$ and $u\equiv_{1}u^{\prime}$, then

[TABLE]

for every $n$. Thus f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s}\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{t}(u)=f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{s}\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{t^{\prime}}(u^{\prime}) for any $t,t^{\prime},u,u^{\prime}\in\widehat{\mathbb{Z}}$. ∎

Proof of Theorem 1.3.

((i)$\Rightarrow$(ii)) Let us denote by $\widehat{\cdot}:\mathbb{Z}\to\widehat{\mathbb{Z}},a\mapsto\widehat{a}$ the natural embedding. For a given polynomial $f(x)=\sum a_{i}x^{i}\in\mathbb{Z}[x]$, set $\widehat{f}(x):=\sum\widehat{a_{i}}x^{i}\in\widehat{\mathbb{Z}}[x]$. Let $F:\widehat{\mathbb{N}}\to\widehat{\mathbb{N}}$ be a map such that

[TABLE]

Let $n_{i}$ be a sequence of positive integers such that $n_{i}\to\infty$ in $\mathbb{R}$ and $n_{i}\to s\in\widehat{\mathbb{Z}}$ in $\widehat{\mathbb{Z}}$ as $i\to\infty$. Then we have

[TABLE]

If there exists a sequence $F(n_{i})\nrightarrow\infty$ in $\mathbb{R}$, then $F(n_{i})=F(n_{i^{\prime}})$ holds for some distinct nonnegative integers $i$ and $i^{\prime}$, that is, $a$ is a preperiodic point of $f$. Therefore the function $f\uparrow_{a}$ is bounded on $\mathbb{N}$ and (ii) automatically hold.

Therefore we can assume $F(n_{i})\rightarrow\infty$ in $\mathbb{R}$ and thus the map $F$ is continuous on $\widehat{\mathbb{N}}$. By Lemma 3.10 (ii) $\Rightarrow$ (i) and Theorem 3.13, there exists $t\in\widehat{\mathbb{Z}}$ such that F^{n}(s)=(\widehat{f}\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{\widehat{a}})^{n}(s)\to t as $n\to\infty$ for every $s\in\widehat{\mathbb{Z}}$. We note that this is equivalent to $\bigcap_{n=1}^{\infty}F^{n}(\widehat{\mathbb{Z}})=\{t\}$ because $\widehat{\mathbb{Z}}$ is compact and Hausdorff.

Let $b$ be a positive integer. We assume $F^{n}(b)\nrightarrow t$, that is, there exists some neighborhood $N$ of $t$ in $\widehat{\mathbb{Z}}$ and some subsequence $F^{n_{1}}(b),F^{n_{2}}(b),...$ of $F^{n}(b)$, such that $n_{i}$ is increasing and $\{F^{n_{i}}(b)\}\cap N=\emptyset$.

Since $\widehat{\mathbb{Z}}$ is compact metric space, there exists a converging subsequence $F^{n_{k_{i}}}(b)\to s\in\widehat{\mathbb{Z}}$. We write $m_{i}=n_{k_{i}}$. For every $i$, $F^{m_{i}}(F^{m_{i+j}-m_{i}}(b))=F^{m_{i+j}}(b)\to s$ as $j\to\infty$. Again $\widehat{\mathbb{Z}}$ is compact metric space, let $F^{m_{i+j_{l}}-m_{i}}(b)\to s^{\prime}$ be a converging subsequence of $\{F^{m_{i+j}-m_{i}}(b)\}_{j}$, then we have $s=F^{m_{i}}(s^{\prime})\in F^{m_{i}}(\widehat{\mathbb{Z}})$. This holds for any $i$, therefore $s\in\bigcap_{i=1}^{\infty}F^{m_{i}}(\widehat{\mathbb{Z}})=\bigcap_{n=1}^{\infty}F^{n}(\widehat{\mathbb{Z}})=\{t\}$, a contradiction. Thus, for any $b$, $F^{n}(b)=f\uparrow_{a}\uparrow_{b}(n)$ converges to the same $t\in\widehat{\mathbb{Z}}=\prod_{p}\mathbb{Z}_{p}$ and assertion (ii) follows.

((ii)$\Rightarrow$(i)) We don’t use profinite complation here.

We prove contraposition. We assume that for some prime number $p$ and $x\in\mathbb{Z}$, $f^{b}(x)\equiv_{p}f^{c}(x)$ if and only if $b\equiv_{p}c$ for any positive integers $b$ and $c$. This leads that for every $a\in\mathbb{N}$, $f\uparrow_{a}:\mathbb{N}\to\mathbb{N}$ induces the bijection $(f\uparrow_{a})_{p}:{{\bf Z}_{p}}\to{{\bf Z}_{p}}$. We regard $(f\uparrow_{a})_{p}$ as a permutation on ${{\bf Z}_{p}}$. The assumption also leads that $f$ is a non-constant polynomial, therefore we can take $a\in\mathbb{N}$ as that $\lim_{n\to\infty}f\uparrow_{a}\uparrow_{b}(n)=\infty$ in $\mathbb{R}$ holds for any $b\in\mathbb{N}$.

If $(f\uparrow_{a})_{p}$ has two or more cycles, then we take $b,b^{\prime}\in\mathbb{N}$ as that $\bar{b}$ and $\bar{b^{\prime}}$ are in distinct cycles of $(f\uparrow_{a})_{p}$ on ${{\bf Z}_{p}}$. In particular, for every $n$, we have $f\uparrow_{a}\uparrow_{b}(n)=(f\uparrow_{a})^{n}(b)\not\equiv_{p}f\uparrow_{a}\uparrow_{b^{\prime}}(n)$. This contradicts to the independence of $b$ in (ii).

If $(f\uparrow_{a})_{p}$ is a permutation of one cycle, then the cycle length is $p$ and $f\uparrow_{a}\uparrow_{b}$ becomes a permutation on ${{\bf Z}_{p}}$ again. Therefore the sequence $\{f\uparrow_{a}\uparrow_{b}(n)\mod p\}_{n}$ cannot be eventually stable. ∎

4 Period Map of Iterated Polynomial

Throughout this section, we assume $f(x)=\sum_{i}c_{i}x^{i}\in\mathbb{Z}[x]$ is a tower-stable polynomial. For every integer $a$, let $\kappa_{f,a}(n)$ (resp. $\lambda_{f,a}(n)$) be the tail (resp. cycle) length on $\overline{a}\in{\bf Z}_{n}$ of the reduction $f_{n}:{\bf Z}_{n}\to{\bf Z}_{n}$ of $f$. To show Theorem 1.5, we evaluate $\lambda_{f,a}$ and $\kappa_{f,a}$. Evaluation of $\lambda_{f,a}$ is established by Fun and Liao in [F-L], in a context of $p$-adic dynamics. But one for $\kappa_{f,a}$ is not explicitly written, although implicitly in evaluating process of $\lambda_{f,a}$, because it has not been of principal interest. Therefore we reevaluate the tail length. We use following Lemmas 4.1 and 4.2.

Lemma 4.1**.**

Let $a,c$ and $n$ be positive integers and $p$ prime number. The following holds:

1. (i)

$f(a+cp^{n})\equiv_{p^{n+1}}f(a)+cf^{\prime}(a)p^{n}$, 2. (ii)

(chain rule) $(f^{n})^{\prime}=\prod_{i=0}^{n-1}f^{\prime}\circ f^{i}$

where $g^{\prime}$ denotes the derivative of $g$ for each polynomial $g$.

Lemma 4.2**.**

Let $p$ be a prime number, $g(x)=bx+c\in\mathbb{Z}[x]$ be an linear polynomial and $s$ an integer. Then we have

[TABLE]

Proof.

See [F-L, Lemma 1]. ∎

From now, we fix a polynomial $f$ and a (positive) integer $a$. We denote $\kappa_{f,a}$ by $\kappa$, and $\lambda_{f,a}$ by $\lambda$. We define mod $p$ multiplier of $f$ on $a$ by

[TABLE]

Theorem 4.3**.**

Let $k$ be a positive integer and $p$ a prime number.

1. (i)

If $\mu_{p}\equiv_{p}0$, then it holds $\lambda(p^{k})=\lambda(p)$ and $\kappa(p^{k})\leq\kappa(p)+(k-1)\cdot\lambda(p)$. 2. (ii)

If $\mu_{p}\not\equiv_{p}0$, then it holds $\kappa(p^{k})=\kappa(p)$ and $\lambda(p^{k})\mid\lambda(p)\cdot(p-1)\cdot p^{k-1}$.

Proof.

From Lemma 3.1 and an evident inequality $\kappa(p^{k})\geq\kappa(p)$, it is enough to show $f^{K+L}(a)\equiv_{p^{k}}f^{K}(a)$ for

[TABLE]

We show this by induction on $k$. The case $k=1$ is evident. We take a pair $(K,L)$ as above and assume that they satisfy

[TABLE]

We put $t:=f^{K}(a)$ and get

[TABLE]

Applying Lemma 4.1(i) to $f^{L}$, we obtain

[TABLE]

that is

[TABLE]

where $g(s):=bs+c$. This implies

[TABLE]

Therefore if $g^{m}(s)\equiv_{p}g^{m+n}(s)$ for some $m$ and $n$, we have

[TABLE]

Applying Lemma 4.1(ii) to $f^{L}$, we have

[TABLE]

where the first congruence follows from

[TABLE]

and $f^{j},f^{\prime}\in\mathbb{Z}[x]$ (this implies they are congruence preserving), the last congruence in (5) for the case $\mu_{p}\not\equiv_{p}0$ follows from Fermat’s little theorem. Applying Lemma 4.2 to $g(s)=bs+c$, by (5) we obtain $g^{m}(s)\equiv_{p}g^{m+n}(s)$ for

[TABLE]

By subsutituting the above pair $(m,n)$, $t=f^{K}(a)$ and (3) into (4), we complete the induction. ∎

Proposition 4.4**.**

If $\mu_{p}\equiv_{p}0$, then the $p$-adic part of f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{a}(s) is algebraic number for every $s\in\widehat{\mathbb{Z}}$.

Proof.

Let $s^{\prime}$ be a positive integer such that $s^{\prime}>\kappa(p)$ and $s^{\prime}\equiv_{\lambda(p)}s$. By the above theorem, for every $k\in\mathbb{N}$ we have

[TABLE]

Therefore the sequence $\{x_{k}:=f^{k\lambda(p)+s^{\prime}}(a)\}$ converges in $\mathbb{Z}_{p}$ as $k\to\infty$. Since $x_{k+1}=f^{\lambda(p)}(x_{k})$, $x:=\lim_{k\to\infty}x_{k}$ satisfies $f^{\lambda(p)}(x)=x$, hence $x$ is algebraic over $\mathbb{Q}$.

For every increasing sequence of positive integers $\{n_{i}\}$ such that $n_{i}\to s$ in $\widehat{\mathbb{N}}$ as $i\to\infty$, $\{f^{n_{i}}(a)\}$ is a partial sequence of $\{x_{k}\}$ eventually. Thus f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{a}(s)=\lim_{i\to\infty}f^{n_{i}}(a)=x is algebraic. ∎

Proposition 4.5**.**

Let $b$ be an $f$-valid number. Then it holds $\lambda_{f}(b^{n+1})\mid b^{n}$ for every sufficiently large $n$.

Proof.

From Lemma 3.12, Theorem 4.3 and $b$ is square-free,

[TABLE]

For $p$ with $p\mid b$, if we decompose $\lambda_{f}(p)\cdot(p-1)=\prod q_{i}^{r_{i}}$, then since $b$ is $f$-valid, we have $q_{i}\mid b$ for every $q_{i}$. Therefore if we put $N=\max_{p}\max_{q_{i}}r_{i}$, then we have $\lambda_{f}(p)\cdot(p-1)\cdot p^{n}\mid b^{n}$ for every prime $p\mid b$ and integer $n\geq N$. ∎

Lemma 4.6**.**

Let $\kappa$ and $\lambda$ be positive integers, $p$ prime, $x\in\mathbb{Z}_{p}$ be $p$-adic integer that is algebraic over $\mathbb{Q}$ and $\{x_{n}\}$ be an nondecreasing sequence of positive integers such that $x_{n}\equiv_{p^{n}}x$ as $p$-adic integers. If $x_{n}$ is not eventually constant, then we have

[TABLE]

for every sufficiently large $n$.

Proof.

Let $F$ be a polynomial with integer coefficients such that $F(x)=0$. Then $x_{n}\equiv_{p^{n}}x$ leads to

[TABLE]

We have $F(x_{n})\neq 0(n\gg 0)$, because $x_{n}$ is nondecreasing and not eventually constant. Hence

[TABLE]

this leads $x_{n}\geq O(p^{n/\deg F})>O(n)$ as $n\to\infty$. ∎

Proof of Theorem 1.5.

By Theorem 3.13, we can take the limit x=\lim_{k\to\infty}f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{a}\uparrow_{t}(k). This $x$ satisfies f\mathrel{\rotatebox[origin={c}]{90.0}{\twoheadrightarrow}}_{a}(x)=x. Write the expansion of the $b$-adic part of $x$ as

[TABLE]

and $x_{n|}$ the positive integers given by the first $n$ digits of the expansion, that is,

[TABLE]

Then the sequence $x_{n|}$ is nondecreasing. First, we assume the sequence $x_{n|}$ is not eventually constant. We show that

[TABLE]

We have $\kappa(b^{n})=\max_{p^{a}\mid b}\{\kappa(p^{an})\}$ by a similar arguement as Lemma 3.12(i). Therefore by Theorem 4.3, we have

[TABLE]

for some integers $l,k$. Moreover, if $\mu_{p}\not\equiv_{p}0$ for every $p\mid b$, (7) holds for $l=0$ and (6) follows obviously. We assume $\mu_{p}\equiv_{p}0$ for some $p$. By Proposition 4.4, the $p$-adic component of $x$ is algebraic, and $x_{n|}\equiv_{p^{n}}x$ by the definition of $x_{n|}$. Thus by Lemma 4.6, $x_{n|}>ln+k\ (n\gg 0)$. Combining this with (7) gives (6).

By Proposition 4.5, $x_{n|}\equiv_{b^{n}}x$ implies

[TABLE]

From (6) and (8),

[TABLE]

Therefore taking $x_{n}$ as

[TABLE]

satisfies

[TABLE]

the required condition.

If $x_{n|}$ is eventually constant, then let us denote the constant by $x^{\prime}$ (to distinguish from $x\in\widehat{\mathbb{Z}}$). We replace $x_{n|}$ as $x_{n|}:=x^{\prime}+b^{n}$ and apply the above argument. (In this case, the argument is easier because (7) and the defitnition of $x_{n|}$ induces (6) directly.) ∎

5 Remaining Problems

All remaining problems of tetration raised in [S-S] is generalizable to iterated polynomial. The most interesting one is for the limit

[TABLE]

Problem 1**.**

Are ${}^{\infty}f(a)$ irrational, and transcendental over $\mathbb{Q}$ except trivial counterexamples (such that ${}^{\infty}f(a)\in\mathbb{N}$, or $\mu_{p}\equiv_{p}0$: see Proposition 4.4)? Are there some nontrivial algebraic, or analytic correlations over the set $\{^{\infty}f(a)\}_{f,a}$?

Relatively many of polynomials are tower-stable. Theorem 3.13 and Lemma 3.10 (ii) lead a probability of tower-stable function $C_{tow}$ as

[TABLE]

where we denote by $\mathop{\mathrm{Cong}}(X)$ the set of all congruence preserving functions, or the set of their reductions, on $X$. $\mu$ and $\mu_{p}$s are the additive Haar probability measures.

Problem 2**.**

Is $C_{tow}$ a transcendental number?

In Chapter 4, we employed polynomial $f$. However, some congruence preserving maps on $\mathbb{N}$ are not polynomial. For instance, the map $n\mapsto\lceil e^{n}n!\rceil$ is introduced in [Ce-Gr-Gu].

Problem 3**.**

Does Theorem 1.5 hold for every congruence preserving map?