## Illinois Journal of Mathematics

### Wandering domains and nontrivial reduction in non-Archimedean dynamics

Robert L. Benedetto

#### 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 $\phi\in K(z)$ 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.

#### Article information

Source
Illinois J. Math., Volume 49, Number 1 (2005), 167-193.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138313

Digital Object Identifier
doi:10.1215/ijm/1258138313

Mathematical Reviews number (MathSciNet)
MR2157374

Zentralblatt MATH identifier
1137.11354

#### Citation

Benedetto, Robert L. Wandering domains and nontrivial reduction in non-Archimedean dynamics. Illinois J. Math. 49 (2005), no. 1, 167--193. doi:10.1215/ijm/1258138313. https://projecteuclid.org/euclid.ijm/1258138313