Periodic points of polynomials over finite fields

Abstract:

Fix an odd prime $p$. If $r$ is a positive integer and $f$ is a polynomial with coefficients in $\mathbb {F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of $\mathbb {P}^1\left (\mathbb {F}_{p^r}\right )$ 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 $22/\left (\log {\log {p^r}}\right )$. 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.