Counting the number of 1m-preperiodic 𝒪K-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) 1m-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≥ 1, d>2 is an integer and m∈ ℤ≥ 1 is any fixed (eventual period). As before, we 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 ℓ ∈ ℤ≥ 1 and fixed (eventual period) m∈ ℤ≥ 1, the average number of distinct 1m-preperiodic integral points of any φpℓ, c modulo prime p𝒪K is unbounded or zero as c→ ∞. Inspired further by work of Doyle-Poonen along with conjectural work of Hutz and abc(d)-conditional work of Panraksa on K-rational preperiodic points of any φ(p−1)ℓ, c for any prime p≥ 5 in arithmetic dynamics, we then also prove that for any fixed (eventual period) m ∈ ℤ ≥ 1, the average number of distinct 1m-preperiodic integral points of any φ(p−1)ℓ, c modulo prime p𝒪K is unbounded or zero as c→ ∞. Finally, we then apply counting and distribution results from arithmetic statistics, and then get counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Together with the previous articles in the series, it might then be fair to view this article as an attempt to contribute to "arithmetic statistical dynamics".
Related
- Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves
- The Arithmetic of Spectra: Factorization, Statistics, and Symmetric Functions
- Random field approximation and local counting statistics for the weakly interacting thermal Bose gas
- Arithmetic k-Polynomial Dynamical Localization for k Power-Law Quasi-Periodic Long-Range Operators on ℓ2(ℤd)
- Dynamical relations on ℙ1, higher bifurcations, and arithmetic bigness
BibTeX
@misc{kintu2026counting,
title = {{Counting the number of $1_m$-preperiodic $\mathcalO_K$-points of a discrete dynamical system with applications from arithmetic statistics, VII}},
author = {Brian Kintu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.14468},
eprint = {2606.14468},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14468},
}
RIS
TY - UNPB
TI - Counting the number of 1_m-preperiodic 𝒪_K-points of a discrete dynamical system with applications from arithmetic statistics, VII
AU - Brian Kintu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2606.14468
UR - https://arxiv.org/abs/2606.14468
AB - 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 φ_d, c defined by φ_d, c(z) = z^d + c for all c, z ∈ 𝒪_K and the coefficient c, where K is any number field of degree n≥ 1, d>2 is an integer and m∈ ℤ_≥ 1 is any fixed (eventual period). As before, we 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 ℓ ∈ ℤ_≥ 1 and fixed (eventual period) m∈ ℤ_≥ 1, the average number of distinct 1_m-preperiodic integral points of any φ_p^ℓ, c modulo prime p𝒪_K is unbounded or zero as c→ ∞. Inspired further by work of Doyle-Poonen along with conjectural work of Hutz and abc(d)-conditional work of Panraksa on K-rational preperiodic points of any φ_(p−1)^ℓ, c for any prime p≥ 5 in arithmetic dynamics, we then also prove that for any fixed (eventual period) m ∈ ℤ_ ≥ 1, the average number of distinct 1_m-preperiodic integral points of any φ_(p−1)^ℓ, c modulo prime p𝒪_K is unbounded or zero as c→ ∞. Finally, we then apply counting and distribution results from arithmetic statistics, and then get counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Together with the previous articles in the series, it might then be fair to view this article as an attempt to contribute to "arithmetic statistical dynamics".
ER -