Wandering Domains and Nontrivial Reduction in Non-archimedean Dynamics

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 ∈ 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. The theory of complex dynamics in dimension one, founded by Fatou and Julia in the early twentieth century, concerns the action of a rational function φ ∈ C(z) on the Riemann sphere P(C) = C∪{∞}. Any such φ induces a natural partition of the sphere into the closed Julia set Jφ, where small errors become arbitrarily large under iteration, and the open Fatou set Fφ = P (C) \ Jφ. There is also a natural action of φ on the connected components of Fφ, taking a component U to φ(U), which is also a connected component of the Fatou set. In 1985, using quasiconformal methods, Sullivan [32] proved that φ ∈ C(z) has no wandering domains; that is, for each component U of Fφ, there are integers M ≥ 0 and N ≥ 1 such that φ(U) = φ(U). We refer the reader to [1, 13, 24] for background on complex dynamics. Recall that a non-archimedean metric on a space X is a metric d which satisfies the ultrametric triangle inequality d(x, z) ≤ max{d(x, y), d(y, z)} for all x, y, z ∈ X. In the past two decades, beginning with a study of linearization at fixed points by Herman and Yoccoz [19], there have been a number of investigations of non-archimedean dynamics; for a small sampling, see [4, 5, 6, 10, 20, 27, 28, 31]. It is natural to ask which properties of complex dynamics extend to the non-archimedean setting and which do not. We fix the following notation throughout this paper. K a complete non-archimedean field with absolute value | · | K̂ an algebraic closure of K CK the completion of K̂ OK the ring of integers {x ∈ K : |x| ≤ 1} of K k the residue field of K OCK the ring of integers {x ∈ CK : |x| ≤ 1} of CK k̂ the residue field of CK Date: July 30, 2003. 2000 Mathematics Subject Classification. Primary: 11S80; Secondary: 37F10, 54H20.