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

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of $n$-periodic points of a polynomial map ${\phi }_{d,c}$ defined by ${\phi }_{d,c}(z)=z^d+c$ 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 $n\in {\mathbb{Z}}_{\ge 2}$ 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 $p\ge 3$ and any fixed $\ell \in {\mathbb{Z}}_{\ge 1}$, the average number of distinct $n$-periodic $p$-adic integral points of any ${\phi }_{p^{\ell },c}$ modulo $p{\mathbb{Z}}_p$ is unbounded or zero as $c\to \infty$; and also prove that for any prime $p\ge 5$, the average number of distinct $n$-periodic $p$-adic integral points of any ${\phi }_{(p-1{)}^{\ell },c}$ 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 $p\ge 3$ and any fixed $\ell \in {\mathbb{Z}}_{\ge 1}$, the average number of distinct $n$-periodic points of any ${\phi }_{p^{\ell },c}$ modulo prime $\pi$ is unbounded or zero as $c$ varies; and also prove that for any prime $p\ge 5$, the average number of distinct $n$-periodic points of any ${\phi }_{(p-1{)}^{\ell },c}$ 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.

40 pages, and any comments are very welcome! arXiv admin note: text overlap with arXiv:2505.24565, arXiv:2508.16393, arXiv:2507.08601