Counting the number of $1_{m}$-preperiodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII

Abstract

In this follow-up article of a multi-part series on (strictly) preperiodic point-counting, we inspect an astonishing relationship between the set of (strictly) $1_{m}$-preperiodic 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 $n\geq 1$, $d>2$ is an integer and $m\in \mathbb{Z}_{\geq 1}$ is any fixed (eventual period). As before, we 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) $m\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any odd degree map $\varphi_{p^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c$ tends to infinity. Inspired further by work of Doyle-Poonen, along with conjectural work of Hutz and $\textit{abc}(\textit{d})$-conditional work of Panraksa on $K$-rational preperiodic points of any even degree map $\varphi_{(p-1)^{\ell}, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $m \in \mathbb{Z}_{ \geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c\to \infty$. Finally, we then apply density, polynomial- and number field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further a stream of counting and statistical results on arithmetic objects that arise naturally in our polynomial discrete dynamical settings.