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Des points fixes communs pour des difféomorphismes de ${ \mathbb{S} }^{2} $ qui commutent et préservent une mesure de probabilité

Part of: Low-dimensional dynamical systems

Published online by Cambridge University Press:  08 February 2013

F. Béguin ,
P. Le Calvez ,
S. Firmo  and
T. Miernowski
F. Béguin
Affiliation:
Université Paris 13 Nord, France
P. Le Calvez
Affiliation:
Université Pierre-et-Marie-Curie, France
S. Firmo
Affiliation:
Universidade Federal Fluminense, Brésil
T. Miernowski
Affiliation:
Université de Aix-Marseille II, France
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Résumé

Nous montrons des résultats d’existence de points fixes communs pour des homéomorphismes du plan ${ \mathbb{R} }^{2} $ ou la sphère ${ \mathbb{S} }^{2} $, qui commutent deux à deux et préservent une mesure de probabilité. Par exemple, nous montrons que des ${C}^{1} $-difféomorphismes ${f}_{1} , \ldots , {f}_{n} $ de ${ \mathbb{S} }^{2} $ suffisamment proches de l’identité, qui commutent deux à deux, et qui préservent une mesure de probabilité dont le support n’est pas réduit à un point, ont au moins deux points fixes communs.

Abstract

We prove the existence of common fixed points for some homeomorphisms of the plane ${ \mathbb{R} }^{2} $ or the two-sphere ${ \mathbb{S} }^{2} $ which commute and preserve a probability measure. For example, if ${f}_{1} , \ldots , {f}_{n} $ are commuting ${C}^{1} $-diffeomorphisms of ${ \mathbb{S} }^{2} $ that are sufficiently close to the identity, and that preserve a probability measure whose support is not a single point, then they have at least two common fixed points.

Keywords

point fixemesure invariantehoméomorphisme conservatifrécurrencenombre de rotationfeuilletage topologiquenombre d’intersection

MSC classification

Secondary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37E45: Rotation numbers and vectors
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 4 , October 2013 , pp. 821 - 851
Copyright
©Cambridge University Press 2013 

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