Counting the number of $1_{n}$-preperiodic integral 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 $1_{n}$-preperiodic points of a polynomial map $φ_{d, c}$ defined by $φ_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}$ 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 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 fixed (eventual period) $n\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{n}$-preperiodic integral points of any odd degree map $φ_{p, c}$ modulo $p$ is unbounded or zero as $c\to \infty$. Inspired by work of Doyle-Poonen, along with conjectural work of Hutz and $\textit{abc}(\textit{d})$-conditional work of Panraksa on preperiodic points of any even degree map $φ_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $n\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{n}$-preperiodic integral points of any $φ_{p-1, c}$ modulo $p$ is unbounded or zero as $c\to \infty$. Finally, we apply density, polynomial- and field-counting, and equidistribution results from arithmetic statistics, and then obtain several counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings.