The image size of iterated rational maps over finite fields

Abstract

Let $\phi :{\mathbb{P}}^1({\mathbb{F}}_q)\to {\mathbb{P}}^1({\mathbb{F}}_q)$ be a rational map of degree $d>1$ on a fixed finite field. We give asymptotic formulas for the size of image sets ${\phi }^n({\mathbb{P}}^1({\mathbb{F}}_q))$ as a function of $n$. This is done using properties of Galois groups of iterated maps, whose connection to the size of image sets is established via the Chebotarev Density Theorem. We apply our results in the following setting. For a rational map defined over a number field, consider the reduction of the map modulo each prime of the number field. We use our results to give explicit bounds on the proportion of periodic points in the residue fields.