## Illinois Journal of Mathematics

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

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

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/ijm/1258138313

**Mathematical Reviews number (MathSciNet)**

MR2157374

**Zentralblatt MATH identifier**

1137.11354

**Subjects**

Primary: 11S85: Other nonanalytic theory

Secondary: 37B99: None of the above, but in this section 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets

#### 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