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 ${\phi }_{d,c}$ defined by ${\phi }_{d,c}(z)=z^d+c$ for all $c,z\in {\mathcal{O}}_K$ and the coefficient $c$, where $K$ is any number field of degree $n\ge 1$, $d>2$ is an integer and $m\in {\mathbb{Z}}_{\ge 1}$ 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 $p\ge 3$ and for any fixed $\ell \in {\mathbb{Z}}_{\ge 1}$ and fixed (eventual period) $m\in {\mathbb{Z}}_{\ge 1}$, the average number of distinct $1_m$-preperiodic integral points of any odd degree map ${\phi }_{p^{\ell },c}$ 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 $\text{abc}(\text{d})$-conditional work of Panraksa on $K$-rational preperiodic points of any even degree map ${\phi }_{(p-1{)}^{\ell },c}$ for any prime $p\ge 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $m\in {\mathbb{Z}}_{\ge 1}$, the average number of distinct $1_m$-preperiodic integral points of any ${\phi }_{(p-1{)}^{\ell },c}$ 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.

31 pages, as also my sincerest congratulations to Prof. Jacob Tsimerman, and any comments are very welcome!