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

TL;DR

This paper investigates the distribution of $n$-periodic integral points in polynomial dynamical systems, revealing unbounded or zero average counts for certain primes and fixed periods, and connects these findings to broader arithmetic statistics and distribution theories.

Contribution

It establishes new average counting results for $n$-periodic points of polynomial maps modulo primes, linking dynamical systems with arithmetic statistics and distribution conjectures.

Findings

Average number of $n$-periodic points is unbounded or zero as $c$ grows for primes $p eq 2$.

Average number of $n$-periodic points is 0, 1, or 2 for primes $p-1$, as $c$ tends to infinity.

Results connect polynomial dynamics with density, counting, and distribution theories in number theory.

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.