The image size of iterated rational maps over finite fields

Abstract

Let ϕ : Fq → Fq be a rational map on a fixed finite field. We give explicit asymptotic formulas for the size of image sets ϕ n (Fq) as a function of n. This is done by using properties of the Galois groups of iterated maps, whose connection to the question of the size of image sets is established via Chebotarev's Density Theorem. We then apply these results to provide explicit bounds on the proportion of periodic points in Fq in terms of q for certain rational maps.