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Instability property of homeomorphisms on surfaces

Published online by Cambridge University Press:  17 March 2006

JOSÉ M. SALAZAR
Affiliation:
Departamento de Matemáticas, Universidad de Alcalá, Alcalá de Henares, Madrid 28871, Spain (e-mail: [email protected])

Abstract

Let S be a compact surface without boundary, with Euler characteristic $\chi(S)>0$, and let $K \subset S$ be a compact set such that each component of K is cellular. The aim of this paper is to prove that, if $f:S \setminus K \rightarrow S \setminus K$ is a homeomorphism which has a certain instability property in the proximity of K, then the set of periodic orbits of f is non-empty. This result give us some corollaries:

  1. there is no minimal homeomorphism $f:S^2 \setminus K \rightarrow S^2 \setminus K$ for K an arbitrary compact set;

  2. if $f:S^2 \rightarrow S^2$ is an area-preserving homeomorphism, then $K=\text{cl(Per($f$))}$ is not an isolated invariant set.

Type
Research Article
Copyright
2006 Cambridge University Press

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