Geometry & Topology

Une nouvelle preuve du théorème de point fixe de Handel

Patrice Le Calvez

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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.

Article information

Geom. Topol., Volume 10, Number 4 (2006), 2299-2349.

Received: 1 June 2006
Accepted: 28 October 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B20: Notions of recurrence 37C25: Fixed points, periodic points, fixed-point index theory 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 37E45: Rotation numbers and vectors 37J10: Symplectic mappings, fixed points

brick decomposition Brouwer theory fixed point translation arc


Le Calvez, Patrice. Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol. 10 (2006), no. 4, 2299--2349. doi:10.2140/gt.2006.10.2299.

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