Title: The image size of iterated rational maps over finite fields,
Abstract: Let $\varphi:\mathbb{F}_q\to\mathbb{F}_q$ be a rational map on a fixed finite
field. We give explicit asymptotic formulas for the size of image sets
$\varphi^n(\mathbb{F}_q)$ 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 $\mathbb{F}_q$ in terms of $q$ for certain
rational maps.