  Let $\varphi:\mathbb{P}^1(\mathbb F_q)\to\mathbb{P}^1(\mathbb F_q)$ be a
rational map of degree $d>1$ on a fixed finite field. We give asymptotic
formulas for the size of image sets $\varphi^n(\mathbb{P}^1(\mathbb F_q))$ 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.
