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Wandering domains and nontrivial reduction in non-archimedean dynamics. (English) Zbl 1137.11354
Summary: Let $$K$$ be a non-archimedean field with residue field $$k$$, and suppose that $$k$$ is not an algebraic extension of a finite field. We prove two results concerning wandering domains of rational functions $$\phi\in K(z)$$ and Rivera-Letelier’s notion of nontrivial reduction. First, if $$\phi$$ has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of $$\phi$$ has wandering components by any of the usual definitions of “components of the Fatou set”. Second, we show that if $$k$$ has characteristic zero and $$K$$ is discretely valued, then the existence of a wandering domain implies that some iterate has nontrivial reduction in some coordinate.

##### MSC:
 37P20 Dynamical systems over non-Archimedean local ground fields 37P40 Non-Archimedean Fatou and Julia sets 11S85 Other nonanalytic theory
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