Un Théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe

Abstract

Let $f$ be a local homeomorphism of the plane with a fixed point $z$ which is a locally maximal invariant set and which is neither a sink nor a source. We prove that there are two integers $q\ge 1$ and $r\ge 1$ such that the sequence $i(f^k,z)$ of the indices at $z$ of the iterates of $f$ satisfy $i(f^k,z)=1-rq$ if $k$ is a multiple of $q$ and $i(f^k,z) = 1$ otherwise. As a corollary we deduce that there is no minimal homeomorphism on the infinite annulus or more generally on the two-dimensional sphere minus a finite set of points. We also construct for a local homeomorhism $f$ as above a topological invariant which is a cyclically ordered set with an automorphism on it; this allows us in particular to define a rotation number for $f$ (rational of denominator $q$).

DOI
https://doi.org/10.2307/2952463

Authors

Patrice Le Calvez

Jean-Christophe Yoccoz