WANDERING DOMAINS AND NONTRIVIAL REDUCTION IN NON-ARCHIMEDEAN DYNAMICS

摘要

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 φ ε K(z) and Rivera-Letelier's notion of nontrivial reduction. First, if φ has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of φ 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.