Finite orbits for rational functions

Abstract

Let $K$ be a number field and $\phi\in K(z)$ a rational function. Let $S$ be the set of all archimedean places of $K$ and all non-archimedean places associated to the prime ideals of bad reduction for $\phi$. We prove an upper bound for length of finite orbits of $\phi$ in $\mathbb{P}_1(K)$ depending only on the cardinality of $S$.