Counting the number of $m$-periodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, V

In this follow-up paper, we again inspect a surprising relationship between the set of $m$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n\geq 2$, $d>2$ is an integer and $m\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we again study counting problems which are inspired by advances on $m$-torsion point-counting in arithmetic statistics and $m$-periodic 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 (period) $m\in \mathbb{Z}_{\geq 2}$, the average number of distinct $m$-periodic integral points of any $\varphi_{p^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c$ tends to infinity. Motivated further by $K$-rational periodic point-counting work of Benedetto along with conjectural work of Hutz on $m$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ for any prime $p\geq 5$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$ in arithmetic dynamics, we then also prove that for any fixed (period) $m\in \mathbb{Z}_{\geq 2}$, the average number of distinct $m$-periodic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime $p\mathcal{O}_{K}$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply here density, polynomial-counting, field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further counting and statistical results on the irreducible monic polynomials, Artin-Mazur zeta functions, algebraic number fields, and lastly on Artin $L$-functions arising naturally in our polynomial discrete dynamical settings.