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Counting the number of $m$-periodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, V

Brian Kintu

TL;DR

This work analyzes $m$-periodic points of polynomial maps $\varphi_{d,c}(z)=z^{d}+c$ over rings of integers $\mathcal{O}_{K}$ modulo inert primes, connecting explicit counts, densities, and averages to arithmetic statistics. It establishes precise dichotomies for counts $N_{c}^{(m)}(p)$ and $M_{c}^{(m)}(p)$ in families $\varphi_{p^{\ell},c}$ and $\varphi_{(p-1)^{\ell},c}$ under inert primes, then derives asymptotic densities, average behaviors, and zero/constant Artin-Mazur zeta functions for these reductions. The paper extends the analysis to the associated monogenic number fields $\mathbb{Q}_{f}$ and $\mathbb{Q}_{g}$, giving bounds on the number of fields with bounded discriminant, proving existence of fields with prescribed Galois groups and class numbers, and establishing Sato-Tate equidistribution for families of Artin $L$-functions. Overall, it creates a bridge between discrete dynamical systems over number fields and arithmetic-statistical phenomena, highlighting how dynamical counts translate into field-counting, zeta, and L-function behavior with broad implications for equidistribution and monogenicity in number theory.

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of $m$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n\geq 2$, $d>2$ is an integer and $m\in \mathbb{Z}_{\geq 2}$ is any fixed (period). As before, we again study counting problems which are inspired by advances on $m$-torsion point-counting in arithmetic statistics and $m$-periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed $\ell\in \mathbb{Z}_{ \geq 1}$ and (period) $m\in \mathbb{Z}_{\geq 2}$, the average number of distinct $m$-periodic integral points of any $\varphi_{p^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c$ tends to infinity. Motivated further by $K$-rational periodic point-counting work of Benedetto along with conjectural work of Hutz on $m$-periodic points of any $\varphi_{(p-1)^{\ell}, c}$ for any prime $p\geq 5$ and any fixed $\ell \in \mathbb{Z}_{\geq 1}$ in arithmetic dynamics, we then also prove that for any fixed (period) $m\in \mathbb{Z}_{\geq 2}$, the average number of distinct $m$-periodic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime $p\mathcal{O}_{K}$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply here density, polynomial-counting, field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further counting and statistical results on the irreducible monic polynomials, Artin-Mazur zeta functions, algebraic number fields, and lastly on Artin $L$-functions arising naturally in our polynomial discrete dynamical settings.

Counting the number of $m$-periodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, V

TL;DR

This work analyzes -periodic points of polynomial maps over rings of integers modulo inert primes, connecting explicit counts, densities, and averages to arithmetic statistics. It establishes precise dichotomies for counts and in families and under inert primes, then derives asymptotic densities, average behaviors, and zero/constant Artin-Mazur zeta functions for these reductions. The paper extends the analysis to the associated monogenic number fields and , giving bounds on the number of fields with bounded discriminant, proving existence of fields with prescribed Galois groups and class numbers, and establishing Sato-Tate equidistribution for families of Artin -functions. Overall, it creates a bridge between discrete dynamical systems over number fields and arithmetic-statistical phenomena, highlighting how dynamical counts translate into field-counting, zeta, and L-function behavior with broad implications for equidistribution and monogenicity in number theory.

Abstract

In this follow-up paper, we again inspect a surprising relationship between the set of -periodic points of a polynomial map defined by for all and the coefficient , where is any number field of degree , is an integer and is any fixed (period). As before, we again study counting problems which are inspired by advances on -torsion point-counting in arithmetic statistics and -periodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime and for any fixed and (period) , the average number of distinct -periodic integral points of any modulo prime ideal is unbounded or zero as tends to infinity. Motivated further by -rational periodic point-counting work of Benedetto along with conjectural work of Hutz on -periodic points of any for any prime and any fixed in arithmetic dynamics, we then also prove that for any fixed (period) , the average number of distinct -periodic integral points of any modulo prime is or or as . Finally, we then apply here density, polynomial-counting, field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further counting and statistical results on the irreducible monic polynomials, Artin-Mazur zeta functions, algebraic number fields, and lastly on Artin -functions arising naturally in our polynomial discrete dynamical settings.

Paper Structure

This paper contains 13 sections, 46 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. The Number of $m$-Periodic Integral Points of any Family of Polynomial Maps $\varphi_{p^{\ell},c}$
  3. On Number of $m$-Periodic Integral Points of any Family of Polynomial Maps $\varphi_{(p-1)^{\ell},c}$
  4. Dynamical Complexity of Forward Periodic Orbit Structure of any $\overline{\varphi_{p^{\ell},c}}$ and $\overline{\varphi_{(p-1)^{\ell},c}}$
  5. The Average Number of $m$-Periodic Points of any Polynomial Map $\varphi_{p^{\ell},c}$ and $\varphi_{(p-1)^{\ell},c}$
  6. On the Density of Integer Polynomials $\varphi_{p^{\ell},c}(x)\in \mathcal{O}_{K}[x]$ with the Number $N_{c}^{(m)}(p) = p$
  7. The Densities of Monic Integer Polynomials $\varphi_{(p-1)^{\ell},c}(x)\in \mathcal{O}_{K}[x]$ with $M_{c}^{(m)}(p) = 1$ or $2$
  8. The Density of Polynomials $\varphi_{p^{\ell},c}(x)$ with $N_{c}^{(m)}(p) = 0$ and $\varphi_{(p-1)^{\ell},c}(x)$ with $M_{c}^{(m)}(p) = 0$
  9. On Artin-Mazur Zeta Function of $\overline{\varphi_{p^{\ell},c}}$ with $N_{c}^{(m)}(p) = 0$ and $\overline{\varphi_{(p-1)^{\ell},c}}$ with $M_{c}^{(m)}(p) = 0$
  10. On Number of Monic Polynomials $f\in \mathbb{Z}[x]$ and $g\in \mathbb{Z}[x]$ with Primitive Galois groups
  11. On the Number of Algebraic Number fields having Bounded Absolute Discriminant
  12. On the Number of Algebraic Number fields $K_{f}$ & $L_{g}$ with Prescribed Class Number
  13. On the Equidistribution of Families of Artin $L$-Functions induced by Fields $K_{f}$ & $L_{g}$

Key Result

Theorem 1.1

Let $K\slash \mathbb{Q}$ be any number field of degree $n \geq 2$ with the ring of integers $\mathcal{O}_{K}$, and in which any fixed prime integer $p\geq 3$ is inert. Let $m\geq 2$ be any fixed integer, and $\varphi_{p, c}$ be a map defined by $\varphi_{p, c}(z) = z^p + c$ for all $c, z\in\mathcal{

Theorems & Definitions (97)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 87 more