Periodic points of rational functions over finite fields

Abstract

For $q$ a prime power and $\varphi$ a rational function with coefficients in ${\mathbb{F}}_q$, let $p(q,\varphi )$ be the proportion of ${\mathbb{P}}^1({\mathbb{F}}_q)$ that is periodic with respect to $\varphi$. 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 $p(q,\varphi )$ as $\varphi$ 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.

12 pages, main theorem strengthened, typos corrected