Volume 10, issue 4 (2006)

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Une nouvelle preuve du théorème de point fixe de Handel

Patrice Le Calvez

Geometry & Topology 10 (2006) 2299–2349
 arXiv: 0903.0369
Abstract

M Handel has proved in [Topology 38 (1999) 235–264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel’s fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index $1$. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1–19, Trans.AMS 348 (1996) 2637–2662] S Matsumoto [Topol. Appl. 104 (2000) 191–214]) to study homeomorphisms of surfaces.

Keywords
brick decomposition, Brouwer theory, fixed point, translation arc
Mathematical Subject Classification 2000
Primary: 37B20, 37C25, 37E30
Secondary: 37E45, 37J10
Publication
Received: 1 June 2006
Accepted: 28 October 2006
Published: 8 December 2006
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Benson Farb
Authors
 Patrice Le Calvez Laboratoire Analyse Géométrie et Applications C.N.R.S.-U.M.R 7539 Institut Galilée Université Paris 13 Avenue J.-B.Clément 93430 Villetaneuse France