Proportion of periodic points in reduction of polynomials

TL;DR

This paper investigates the asymptotic behavior of the proportion of periodic points in the reductions of polynomials over number fields, completing the classification for all polynomial cases.

Contribution

It provides a complete classification of the limit inferior of periodic points proportion for polynomial reductions over number fields.

Findings

Limit is zero for polynomials conjugate to Chebyshev polynomials.

Complete classification of polynomial cases regarding periodic points proportion.

Addresses remaining cases not covered by previous results.

Abstract

In 2014, Juul, Kurlberg, Madhu and Tucker asked the following: given $K$ a number field and $f$ a rational function with coefficients in $K$, if $f_\mathfrak{p}$ denotes the reduction of $f$ modulo a prime ideal $\mathfrak{p}$ in the ring of integers of $K$, what is the limit inferior of the proportion of periodic points of $f_\mathfrak{p}$ when the norm of $\mathfrak{p}$ goes to infinity? Recent results of Fari\~na-Asategui and the author show that when $f$ is a polynomial of degree $d \geq 2$ non-linearly conjugate over $\mathbb{C}$ to a Chebyshev polynomial then the limit is zero. In this article, we address the remaining cases to give a complete classification of the problem in the case of polynomials.