Geometry & Topology, Vol. 9 (2005) Paper no. 39, pages 1689--1774.

Structure des homeomorphismes de Brouwer

Frederic Le Roux

Abstract. 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 d_h 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 d_h(x,y)>1.

Résumé. Tout homeomorphisme de Brouwer s'obtient en recollant des domaines de translation (ouverts simplement connexes, invariants, en restriction auxquels la dynamique est conjuguee a une translation). On introduit une distance d_h sur le plan qui compte le nombre minimal de domaines de translation dont la reunion connecte deux points. Ceci nous permet de decrire un invariant combinatoire de conjugaison, qui decrit tres grossierement la maniere dont les domaines de translation se recollent. On montre egalement l'existence de structures dynamiques qui generalisent la presence de composantes de Reeb dans les feuilletages non triviaux du plan.

Keywords. Homeomorphism, surface, fixed point, index, Reeb components, Brouwer

AMS subject classification. Primary: 37E30. Secondary: 37B30.

E-print: arXiv:math.DS/0403406

DOI: 10.2140/gt.2005.9.1689

Submitted to GT on 8 November 2004. (Revised 1 September 2005.) Paper accepted 18 August 2005. Paper published 14 September 2005.

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Frederic Le Roux
Universite Paris Sud, Bat. 425
91405 Orsay Cedex, FRANCE
Email: frederic.le-roux@math.u-psud.fr.

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