Counting the number of $2$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI

TL;DR

This work probes the existence and counting of 2-periodic points for polynomial dynamical systems of the form $\varphi_{d,c}(z)=z^d+c$ over $\mathbb{Z}_p$ and $\mathbb{F}_p[t]$, focusing on reductions modulo $p\mathbb{Z}_p$ and modulo irreducible $\pi$ in $\mathbb{F}_p[t]$. It establishes precise, sometimes uniform, counts for 2-periodic points in the $p$-adic and function-field settings when $d$ takes the form $p^\ell$ or $(p-1)^\ell$, revealing that the number of such points often collapses to the small set ${p,1,2,0}$ depending on the residue class of $c$ and the choice of $d$, with explicit bounds and exceptional cases. The paper then studies averaging across coefficients and primes, deriving zero, bounded, or unbounded average behavior, and connects these dynamical counts to arithmetic statistics through densities of polynomials, monogenic number fields, discriminant bounds, and class numbers. It further establishes equidistribution of Artin $L$-functions in the associated families (via work of Nico and Shankar–Södergren–Templier) and analyzes intermediate subfields in both number-field and function-field contexts, yielding concrete bounds and density results. Overall, the results create a bridge between $2$-periodic dynamics in $p$-adic and function-field settings and statistical properties of associated number fields and $L$-functions, highlighting universal patterns and explicit probabilistic densities across families.

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of $2$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}_{p}$ or $c, z \in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $d>2$ is an integer. As before, we again wish to study counting problems that are inspired by advances on $2$-torsion point-counting in arithmetic statistics and $2$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic $p$-adic integral points of any $\varphi_{p^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is bounded or zero or unbounded as $c\to \infty$; and then prove that for any prime $p\geq 5$ and for any $\ell\in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic $p$-adic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is $1$ or $2$ or $0$ as $c\to \infty$. Motivated by periodic $\mathbb{F}_{p}(t)$-point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and for any $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic points of any $\varphi_{p^{\ell}, c}$ modulo prime $π$ is bounded or zero or unbounded as $c$ varies; and then prove that for any prime $p\geq 5$ and for any $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $2$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime $π$ is $1$ or $2$ or $0$ as $c$ varies. Finally, we apply density, field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and then obtain counting and statistical results on irreducible polynomials, number (function) fields, and Artin $L$-functions that arise naturally in our polynomial discrete dynamical settings.

Theorems & Definitions (95)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Conjecture 1.6
• Corollary 1.7
• Conjecture 1.8
• Conjecture 1.9
• Theorem 1.10