Counting the number of 2ₘ-preperiodic OK-points of a discrete dynamical system with applications from arithmetic statistics, X

In this follow-up article, we again inspect a surprising relationship between the set of (primitive strictly) 2ₘ-preperiodic points of a polynomial map φd, c defined by φd, c(z) = zᵈ + c for all c, z ∈ OK and the coefficient c, where K is any number field of degree n≥ 2, $d>2$ is an integer and m∈ Z≥ 1 is any fixed (eventual period). As before, we again wish to study counting problems that are inspired by torsion point-counting in arithmetic statistics and (primitive strictly) preperiodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime p≥ 3 and for any fixed ∈ Z≥ 1 and fixed (eventual period) m∈ Z≥ 1, the average number of distinct 2ₘ-preperiodic integral points of any φ_p^, c modulo prime ideal pOK is unbounded or zero as c→ ∞. Inspired further by work of Doyle-Poonen along with conjectural work of Hutz on K-rational preperiodic points of any φ_(p-1)^, c for any prime p≥ 5 in arithmetic dynamics, we then also prove that for any fixed ∈ Z≥ 1 and fixed (eventual period) m∈ Z≥ 1, the average number of distinct 2ₘ-preperiodic integral points of any φ_(p-1)^, c modulo prime ideal pOK is unbounded or zero as c→ ∞. We then apply counting and distribution results from arithmetic statistics, and obtain counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Finally, we historically continue meditating on why human-created systems with unbounded capabilities that defy our universal sense of order have often incited sustainability-related questions concerned with having such systems in the real world. Together with the previous and follow-up articles in our multi-part series, it might then also be fair to view this article as an attempt to contribute to "arithmetic statistical dynamics", which aspires to understand counting, distribution, and application of families of arithmetic objects from discrete dynamical systems.