# 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.

## Key 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 $b'$.