Illinois Journal of Mathematics

Wandering domains and nontrivial reduction in non-Archimedean dynamics

Robert L. Benedetto

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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

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

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

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


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.

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