Counting the number of $2$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI
Authors
Brian Kintu
Abstract
In this follow-up paper, we again inspect a surprising relationship between the set of $2$-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 $c, z \in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $d>2$ is an integer. As before, we again wish to study counting problems that are inspired by advances on $2$-torsion point-counting in arithmetic statistics and $2$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic $p$-adic integral points of any $\varphi_{p^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is bounded or zero or unbounded as $c\to \infty$; and then prove that for any prime $p\geq 5$ and for any $\ell\in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic $p$-adic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is $1$ or $2$ or $0$ as $c\to \infty$. Motivated by periodic $\mathbb{F}_{p}(t)$-point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and for any $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic points of any $\varphi_{p^{\ell}, c}$ modulo prime $π$ is bounded or zero or unbounded as $c$ varies; and then prove that for any prime $p\geq 5$ and for any $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime $π$ is $1$ or $2$ or $0$ as $c$ varies. Finally, we apply density, field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and then obtain counting and statistical results on irreducible polynomials, number (function) fields, and Artin $L$-functions that arise naturally in our polynomial discrete dynamical settings.
