Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I

TL;DR

This paper investigates fixed points of polynomial maps in integer dynamics, revealing average fixed point counts modulo primes and connecting these to arithmetic statistics and number field counts.

Contribution

It establishes new average fixed point counts for specific polynomial maps and links these to broader arithmetic statistics and number field enumeration.

Findings

Average fixed points of $ ext{varphi}_{p,c}$ modulo $p$ are 3 or 0 for primes $p extgreater 2$.

Average fixed points of $ ext{varphi}_{p-1,c}$ modulo $p$ are 1, 2, or 0 for primes $p extgreater 4$.

Results connect fixed point counts to counting irreducible polynomials and number fields.

Abstract

In this first article of a multi-part series, we inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}$ and the coefficient $c$, where $d > 2$ is an integer. Inspired greatly by the elegant counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics, and also by interesting point-counting result of Narkiewicz on rational periodic points of any odd degree map $\varphi_{d, c}$ in arithmetic dynamics, we then first prove that for any prime $p\geq 3$, the average number of distinct integral fixed points of any $\varphi_{p, c}$ modulo $p$ is $3$ or $0$ as $c$ tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of $\varphi_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any $\varphi_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on the irreducible integer polynomials and number fields arising naturally in our polynomial discrete dynamical settings.