The Image Size of Iterated Rational Maps over Finite Fields
The Image Size of Iterated Rational Maps over Finite Fields
Abstract 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 …