Counting the number of $1_{n}$-preperiodic $\mathbb{Z}_{p}$- and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VIII

Abstract

In this follow-up paper, we again inspect an astonishing relationship between the set of (strictly) $1_{n}$-preperiodic 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 and $n\in \mathbb{Z}_{\geq 1}$ is any fixed (eventual period). As before, we again wish to study counting problems that are inspired by torsion point-counting in arithmetic statistics and (strictly) preperiodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed $\ell \in \mathbb{Z}_{\geq 1}$ and fixed (eventual period) $n\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{n}$-preperiodic $p$-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 $p\geq 5$ and for any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{n}$-preperiodic $p$-adic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is unbounded or zero as $c\to \infty$. Motivated by Benedetto and Doyle-Poonen on function field-valued preperiodic point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and for any fixed $\ell \in \mathbb{Z}_{\geq 1}$ and fixed (eventual period) $n\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{n}$-preperiodic points of any $\varphi_{p^{\ell}, c}$ modulo prime $\pi$ is unbounded or zero as $c$ varies; and also prove that for any prime $p\geq 5$ and for any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{n}$-preperiodic points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime $\pi$ is unbounded or zero as $c$ varies. We then apply density, polynomial-and field-counting, and equidistribution results from arithmetic statistics, and obtain counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Finally, we then historically meditate on why human-created systems with unbounded capabilities that defy our universal sense of order have often prompted sustainability-related questions concerned with having such systems in the real world.