Geometry & Topology
- Geom. Topol.
- Volume 10, Number 4 (2006), 2299-2349.
Une nouvelle preuve du théorème de point fixe de Handel
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 . 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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.gt/1513799804