Counting the number of $1_n$-preperiodic integral points of a discrete dynamical system with applications from arithmetic statistics, VII

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 ${\phi }_{d,c}$ defined by ${\phi }_{d,c}(z)=z^d+c$ for all $c,z\in \mathbb{Z}$ and the coefficient $c$, where $d>2$ is an integer and $n\in {\mathbb{Z}}_{\ge 1}$ 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 fixed (eventual period) $n\in {\mathbb{Z}}_{\ge 1}$, the average number of distinct $1_n$-preperiodic integral points of any odd degree map ${\phi }_{p,c}$ modulo $p$ is unbounded or zero as $c\to \mathrm{\infty }$. Inspired by work of Doyle-Poonen, along with conjectural work of Hutz and $\mathit{\text{abc}}(\mathit{\text{d}})$-conditional work of Panraksa on preperiodic points of any even degree map ${\phi }_{p-1,c}$ for any prime $p\ge 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $n\in {\mathbb{Z}}_{\ge 1}$, the average number of distinct $1_n$-preperiodic integral points of any ${\phi }_{p-1,c}$ modulo $p$ is unbounded or zero as $c\to \mathrm{\infty }$. 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.