Volume 9, issue 3 (2005)

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Structure des homeomorphismes de Brouwer

Frederic Le Roux

Geometry & Topology 9 (2005) 1689–1774

arXiv: math.DS/0403406


For every Brouwer (ie planar, fixed point free, orientation preserving) homeomorphism h there exists a covering of the plane by translation domains, invariant simply-connected open subsets on which h is conjugate to an affine translation. We introduce a distance dh on the plane that counts the minimal number of translation domains connecting a pair of points. This allows us to describe a combinatorial conjugacy invariant, and to show the existence of a finite family of generalised Reeb components separating any two points x,y such that dh(x,y) > 1.


Tout homéomorphisme de Brouwer s’obtient en recollant des domaines de translation (ouverts simplement connexes, invariants, en restriction auxquels la dynamique est conjuguée à une translation). On introduit une distance dh sur le plan qui compte le nombre minimal de domaines de translation dont la réunion connecte deux points. Ceci nous permet de décrire un invariant combinatoire de conjugaison, qui décrit très grossièrement la manière dont les domaines de translation se recollent. On montre également l’existence de structures dynamiques qui généralisent la présence de composantes de Reeb dans les feuilletages non triviaux du plan.

homeomorphism, surface, fixed point, index, Reeb components, Brouwer
Mathematical Subject Classification 2000
Primary: 37E30
Secondary: 37B30
Forward citations
Received: 8 November 2004
Revised: 1 September 2005
Accepted: 18 August 2005
Published: 14 September 2005
Proposed: Benson Farb
Seconded: Leonid Polterovich, David Gabai
Frederic Le Roux
Université Paris Sud
Bat. 425
91405 Orsay Cedex