Counting the number of 1ₙ-preperiodic Zₚ- and Fₚ[t]-points of a discrete dynamical system with applications from arithmetic statistics, VIII

In this follow-up paper, we again inspect an astonishing relationship between the set of (strictly) 1ₙ-preperiodic points of a polynomial map φd, c defined by φd, c(z) = zᵈ + c for all c, z ∈ Zₚ or c, z ∈ Fₚ[t] and the coefficient c, where $d>2$ is an integer and n∈ 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 (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) n∈ Z≥ 1, the average number of distinct 1ₙ-preperiodic p-adic integral points of any φ_p^, c modulo pZₚ is unbounded or zero as c→ ∞; and also prove that for any prime p≥ 5 and for any fixed ∈ Z≥ 1, the average number of distinct 1ₙ-preperiodic p-adic integral points of any φ_(p-1)^, c modulo pZₚ is unbounded or zero as c→ ∞. Motivated by Benedetto and Doyle-Poonen on function field-valued preperiodic point-counting in arithmetic dynamics, we then also prove that for any prime p≥ 3 and for any fixed ∈ Z≥ 1 and fixed (eventual period) n∈ Z≥ 1, the average number of distinct 1ₙ-preperiodic points of any φ_p^, c modulo prime π is unbounded or zero as c varies; and also prove that for any prime p≥ 5 and for any fixed ∈ Z≥ 1, the average number of distinct 1ₙ-preperiodic points of any φ_(p-1)^, c modulo prime π is unbounded or zero as c varies. We then apply density, polynomial-and field-counting, and equidistribution results from arithmetic statistics, and obtain counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Finally, we then 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 the real world.