Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI

TL;DR

This paper investigates the distribution of n-periodic points of polynomial maps over p-adic and finite fields, revealing unbounded or specific bounded averages, and applies these findings to various arithmetic and dynamical objects.

Contribution

It provides new results on the average number of periodic points in polynomial dynamical systems over p-adic and finite fields, connecting these to arithmetic statistics and zeta functions.

Findings

Average number of periodic points can be unbounded or zero as c varies.

For certain primes, the average number of periodic points is 1, 2, or 0.

Results are applied to counting irreducible polynomials, zeta functions, and L-functions.

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of $n$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}_{p}$ or $\in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $d>2$ is an integer and $n\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we study counting problems that are inspired by $n$-torsion point-counting in arithmetic statistics and $n$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic $p$-adic integral points of any $\varphi_{p^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is unbounded or zero as $c\to \infty$; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic $p$-adic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is $1$ or $2$ or $0$ as $c\to \infty$. Inspired further by periodic $\mathbb{F}_{p}(t)$-point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic points of any $\varphi_{p^{\ell}, c}$ modulo prime $\pi$ is unbounded or zero as $c$ varies; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $\pi$ is $1$ or $2$ or $0$ as $c$ varies. Finally, we apply density, polynomial-and field-counting, equidistribution results from arithmetic statistics, and then obtain counting and statistical results on irreducible polynomials, (Artin-Mazur) zeta functions, global fields, and on (Artin) $L$-functions arising naturally in our polynomial discrete dynamical settings.