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Wandering domains and nontrivial reduction in non-archimedean dynamics

Robert L. Benedetto
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Abstract

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 f in K(z) and Rivera-Letelier's notion of nontrivial reduction. First, if f has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of f has wandering components by any of the usual definitions of such components. Second, we show that if k has characteristic zero and K is discretely valued, then the converse holds; that is, the existence of a wandering domain implies that some iterate has nontrivial reduction in some coordinate.

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Robert L. Benedetto. 2004-12-06. Wandering domains and nontrivial reduction in non-archimedean dynamics. https://arxiv.org/abs/math/0312034

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