2003/12/31 by Robert L. Benedetto
Mathematics · #math.NT #math.DS #msc:11S80 #msc:37F10 #msc:54H20
published as Ill. J. Math. 49 (2005), no. 1, pp. 167--193 · 22 pages; to appear in Ill. J. Math.; added appendix and some more examples; a few other minor changes
arxiv created 2004/12/06 · arxiv updated 2009/12/01
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.