Periodic points of rational functions over finite fields

TL;DR

This paper investigates the proportion of periodic points of rational functions over finite fields, showing that for functions of degree at least 2, this proportion tends to zero as the field size increases, generalizing previous quadratic polynomial results.

Contribution

It proves that the expected proportion of periodic points tends to zero for rational functions of degree at least 2 over large finite fields, extending earlier quadratic polynomial findings.

Findings

Expected proportion of periodic points tends to zero as field size increases.

Generalization from quadratic polynomials to higher degree rational functions.

Established a uniformity theorem for specializations of dynamical systems.

Abstract

For $q$ a prime power and $\phi$ a rational function with coefficients in $\mathbb{F}_q$, let $p(q,\phi)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_q)$ that is periodic with respect to $\phi$. And if $d$ is a positive integer, let $Q_d$ be the set of prime powers coprime to $d!$ and let $\mathcal{P}(d,q)$ be the expected value of $p(q,\phi)$ as $\phi$ ranges over rational functions with coefficients in $\mathbb{F}_q$ of degree $d$. We prove that if $d$ is a positive integer no less than $2$, then $\mathcal{P}(d,q)$ tends to 0 as $q$ increases in $Q_d$. This theorem generalizes our previous work, which held only for quadratic polynomials, and only in fixed characteristic. To deduce this result, we prove a uniformity theorem on specializations of dynamical systems of rational functions with coefficients in certain finitely-generated algebras over residually finite Dedekind domains. This specialization theorem generalizes our previous work, which held only for algebras of dimension one.