Finite orbits for rational functions

Abstract

Let $K$ be a number field and $\varphi \in K(z)$ 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 $\varphi$. We prove an upper bound for length of finite orbits of $\varphi$ in ${\mathbb{P}}_1(K)$ depending only on the cardinality of $S$.

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