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

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ or $\in \mathbb{Z}_{p}$ or $\in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $K$ is any number field of degree $n > 1$, $p>2$ is any prime, $\mathbb{Z}_{p}$ (resp., $\mathbb{F}_{p}[t]$) is the ring of all $p$-adic integers (resp., the ring of all polynomials over a finite field $\mathbb{F}_{p}$) and $d>2$ is an integer. As before, we again wish to study counting problems which are inspired by advances in arithmetic statistics, and also by Narkiewicz on totally complex $K$-periodic points along with Adam-Fares on $\mathbb{Q}_{p}$-periodic points 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 fixed points of any $\varphi_{p^{\ell}, c}$ modulo prime $p\mathcal{O}_{K}$ (modulo $p\mathbb{Z}_{p}$) is bounded or zero or unbounded as $c\to \infty$ . Motivated further by $\mathbb{F}_{p}(t)$-periodic point-counting result of Benedetto in arithmetic dynamics, we then also find that the average number of fixed points in $\mathbb{F}_{p}[t]$-setting behaves in the same way as in $\mathcal{O}_{K}$-setting. Finally, we then apply here counting and statistical results from arithmetic statistics, and as a result obtain counting and statistical results on irreducible monic ($p$-adic) integer polynomials, number fields and subfields of global function fields arising naturally in our polynomial discrete dynamical settings.