Periodic points of polynomials over finite fields

TL;DR

This paper investigates the proportion of periodic points of polynomials over finite fields, establishing bounds that decrease with field size and generalizing uniformity theorems for dynamical systems over various rings.

Contribution

It introduces a uniformity theorem for specializations of dynamical systems over residually finite Dedekind domains, extending previous results to broader algebraic settings.

Findings

Expected proportion of periodic points decreases as field size grows.

Provides effective bounds on image sizes of rational functions over finite fields.

Generalizes previous uniformity theorems to broader classes of rings and fields.

Abstract

Fix an odd prime $p$. If $r$ is a positive integer and $f$ a polynomial with coefficients in $\mathbb{F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_{p^r})$ that is periodic with respect to $f$. We show that as $r$ increases, the expected value of $P_{p,r}(f)$, as $f$ ranges over quadratic polynomials, is less than $22/(\log{\log{p^r}})$. This result follows from a uniformity theorem on specializations of dynamical systems of rational functions over residually finite Dedekind domains. The specialization theorem generalizes previous work by Juul et al. that holds for rings of integers of number fields. Moreover, under stronger hypotheses, we effectivize this uniformity theorem by using the machinery of heights over general global fields; this version of the theorem generalizes previous work of Juul on polynomial dynamical systems over rings of integers of number fields. From these theorems we derive effective bounds on image sizes and periodic point proportions of families of rational functions over finite fields.