Instability property of homeomorphisms on surfaces

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.