Cycles for rational maps over global function fields with one prime of bad reduction

Abstract

Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The main result is an optimal bound for cycle lengths that only depends on $p$ and $D$. A bound for the cardinality of finite orbits is given as well. Our method is based on a careful analysis (for every prime of good reduction) of the $\mathfrak p$-adic distances between points belonging to the same finite orbit, in part motivated by previous work by Canci and Paladino. Valuable insight is provided by a certain family of polynomials. In this case we also gain a good deal of information about the structure and size of the set of periodic points for polynomials of given degree.