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 ${\phi }_{d,c}$ defined by ${\phi }_{d,c}(z)=z^d+c$ 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 $p\ge 3$ and for any $\ell \in {\mathbb{Z}}_{\ge 1}$, the average number of distinct fixed points of any ${\phi }_{p^{\ell },c}$ 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.

25 pages and any comments are very welcome!