Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI
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2025
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| author | Kintu, Brian |
| author_facet | Kintu, Brian |
| contents | In this follow-up paper, we again inspect a surprising relationship between the set of $n$-periodic points of a polynomial map $φ_{d, c}$ defined by $φ_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}_{p}$ or $\in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $d>2$ is an integer and $n\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we study counting problems that are inspired by $n$-torsion point-counting in arithmetic statistics and $n$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic $p$-adic integral points of any $φ_{p^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is unbounded or zero as $c\to \infty$; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic $p$-adic integral points of any $φ_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is $1$ or $2$ or $0$ as $c\to \infty$. Inspired further by periodic $\mathbb{F}_{p}(t)$-point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic points of any $φ_{p^{\ell}, c}$ modulo prime $π$ is unbounded or zero as $c$ varies; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic points of any $φ_{(p-1)^{\ell}, c}$ modulo $π$ is $1$ or $2$ or $0$ as $c$ varies. Finally, we apply density, polynomial-and field-counting, equidistribution results from arithmetic statistics, and then obtain counting and statistical results on irreducible polynomials, (Artin-Mazur) zeta functions, global fields, and on (Artin) $L$-functions arising naturally in our polynomial discrete dynamical settings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_00322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI Kintu, Brian Number Theory Dynamical Systems In this follow-up paper, we again inspect a surprising relationship between the set of $n$-periodic points of a polynomial map $φ_{d, c}$ defined by $φ_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}_{p}$ or $\in \mathbb{F}_{p}[t]$ and the coefficient $c$, where $d>2$ is an integer and $n\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we study counting problems that are inspired by $n$-torsion point-counting in arithmetic statistics and $n$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic $p$-adic integral points of any $φ_{p^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is unbounded or zero as $c\to \infty$; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic $p$-adic integral points of any $φ_{(p-1)^{\ell}, c}$ modulo $p\mathbb{Z}_{p}$ is $1$ or $2$ or $0$ as $c\to \infty$. Inspired further by periodic $\mathbb{F}_{p}(t)$-point-counting in arithmetic dynamics, we then also prove that for any prime $p\geq 3$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$, the average number of distinct $n$-periodic points of any $φ_{p^{\ell}, c}$ modulo prime $π$ is unbounded or zero as $c$ varies; and also prove that for any prime $p\geq 5$, the average number of distinct $n$-periodic points of any $φ_{(p-1)^{\ell}, c}$ modulo $π$ is $1$ or $2$ or $0$ as $c$ varies. Finally, we apply density, polynomial-and field-counting, equidistribution results from arithmetic statistics, and then obtain counting and statistical results on irreducible polynomials, (Artin-Mazur) zeta functions, global fields, and on (Artin) $L$-functions arising naturally in our polynomial discrete dynamical settings. |
| title | Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2511.00322 |