Counting the number of $2_{n}$-preperiodic integral points of a discrete dynamical system with applications from arithmetic statistics, IX

In this follow-up paper, we inspect a surprising relationship between the set of (strictly) $2_{n}$-preperiodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ 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 for any fixed (eventual period) $n\in \mathbb{Z}_{\geq 1}$, the average number of distinct $2_{n}$-preperiodic integral points of any $\varphi_{p, c}$ modulo $p$ is unbounded or zero as $c\to \infty$. Inspired further by work of Doyle-Poonen along with conjectural work of Hutz on rational preperiodic points of any $\varphi_{p-1, c}$ 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 $2_{n}$-preperiodic integral points of any $\varphi_{p-1, c}$ modulo $p$ is unbounded or zero as $c\to \infty$. We then apply results from arithmetic statistics, and then get counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Finally, we historically meditate on why human-created systems with unbounded capabilities that defy our universal sense of order have often prompted sustainability-related questions concerned with having such systems in real world.