Un Theoreme d'Indice pour les Homeomorphismes du Plan au Voisinage d'un Point Fixe

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 ≥ 1 and r ≥ 1 such that the sequence i(fk, z) of the indices at z of the iterates of f satisfy i(fk, z) = 1 - rq...

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Bibliographic Details
Published in:Annals of mathematics 1997-09, Vol.146 (2), p.241-293
Main Authors: Le Calvez, Patrice, Yoccoz, Jean-Christophe
Format: Article
Language:English
Subjects:
Dynamical Systems
Mathematics
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Summary: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 ≥ 1 and r ≥ 1 such that the sequence i(fk, z) of the indices at z of the iterates of f satisfy i(fk, z) = 1 - rq if k is a multiple of q and i(fk, 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 homeomorphism 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).
ISSN:0003-486X
DOI:10.2307/2952463