Counting the number of $n$-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV

Abstract

In this follow-up paper, we 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 $c, z \in \mathbb{Z}$ and the coefficient $c$, where $d>2$ is an integer and $n\geq 2$ is any fixed integer. As before, we again wish to study counting problems which are inspired by the exciting advances of Bhargava-Shankar-Tsimerman and their collaborators on $n$-torsion point-counting in arithmetic statistics, and also by Hutz's conjecture along with Panraksa's work on $n$-periodic rational point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed (period) $n\in \mathbb{Z}_{\geq 2}$, the average number of distinct $n$-periodic integral points of any $\varphi_{p, c}$ modulo $p$ is unbounded or zero as $c$ tends to infinity. Inspired further by a conjecture of Hutz on any $\varphi_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (period) $n\in \mathbb{Z}_{\geq 2}$, the average number of distinct $n$-periodic integral points of any $\varphi_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density, polynomial-counting, number field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining a stream of counting and statistical results on irreducible polynomials, number fields, and Artin $L$-functions that arise naturally in our polynomial discrete dynamical settings.