Geometry & Topology

Une nouvelle preuve du théorème de point fixe de Handel

Patrice Le Calvez

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799804

Digital Object Identifier
doi:10.2140/gt.2006.10.2299

Mathematical Reviews number (MathSciNet)
MR2284059

Zentralblatt MATH identifier
1126.37027

Subjects
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

Keywords
brick decomposition Brouwer theory fixed point translation arc

Citation

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


Export citation

References

  • L E J Brouwer, Beweis des ebenen Translationssatzes, Math. Ann. 72 (1912) 37–54
  • M Brown, A new proof of Brouwer's lemma on translation arcs, Houston J. Math. 10 (1984) 35–41
  • M Brown, J M Kister, }, Proc. Amer. Math. Soc. 91 (1984) 503–504
  • A Fathi, An orbit closing proof of Brouwer's lemma on translation arcs, Enseign. Math. $(2)$ 33 (1987) 315–322
  • A Floer, Proof of the Arnol'd conjecture for surfaces and generalizations to certain Kähler manifolds, Duke Math. J. 53 (1986) 1–32
  • M Flucher, Fixed points of measure preserving torus homeomorphisms, Manuscripta Math. 68 (1990) 271–293
  • J Franks, }, Ann. of Math. $(2)$ 128 (1988) 139–151
  • J Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math. 2 (1996) 1–19
  • J Franks, Rotation vectors and fixed points of area preserving surface diffeomorphisms, Trans. Amer. Math. Soc. 348 (1996) 2637–2662
  • L Guillou, Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, Topology 33 (1994) 331–351
  • M Handel, A fixed-point theorem for planar homeomorphisms, Topology 38 (1999) 235–264
  • M Handel, W P Thurston, New proofs of some results of Nielsen, Adv. in Math. 56 (1985) 173–191
  • T Homma, An extension of the Jordan curve theorem, Yokohama Math. J. 1 (1953) 125–129
  • P Le Calvez, Une version feuilletée équivariante du théorème de translation de Brouwer, Publ. Math. Inst. Hautes Études Sci. (2005) 1–98
  • F Le Roux, Homéomorphismes de surfaces: théorèmes de la fleur de Leau-Fatou et de la variété stable, Astérisque (2004) vi+210
  • S Matsumoto, Arnold conjecture for surface homeomorphisms, from: “Proceedings of the French–Japanese Conference: Hyperspace Topologies and Applications (La Bussière, 1997)”, Topology Appl. 104 (2000) 191–214
  • A Sauzet, Application des décompositions libres à l'étude des homéomorphismes de surface Thèse de l'Université Paris 13 (2001)
  • S Schwartzman, 0088720
  • J-C Sikorav, Points fixes d'une application symplectique homologue à l'identité, J. Differential Geom. 22 (1985) 49–79