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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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