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Handel’s fixed point theorem revisited

Published online by Cambridge University Press:  06 August 2012

JULIANA XAVIER
JULIANA XAVIER*
Affiliation:
I.M.E.R.L, Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay (email: jxavier@fing.edu.uy)
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Abstract

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Michael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 5 , October 2013 , pp. 1584 - 1610
Copyright
Copyright © 2012 Cambridge University Press 

References

[1]Brouwer, L. E. J.. Beweis des ebenen Translationssatzes. Math. Ann. 72 (1912), 37–54.Google Scholar
[2]Brown, M.. A new proof of Brouwer’s lemma on translation arcs. Houston J. Math. 10 (1984), 35–41.Google Scholar
[3]Brown, M. and Kister, J.. Invariance of complementary domains of a fixed point set. Proc. Am. Math. Soc. 91 (1984), 503–504.Google Scholar
[4]Fathi, A.. An orbit closing proof of Brouwer’s lemma on translation arcs. Enseign. Math. 2(33) (1987), 315–322.Google Scholar
[5]Franks, J.. Generalizations of the Poincaré–Birkhoff theorem. Ann. of Math. (2) 128 (1998), 139–151.Google Scholar
[6]Guillou, L.. Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff. Topology 33 (1994), 331–351.Google Scholar
[7]Handel, M.. A fixed-point theorem for planar homeomorphisms. Topology 38 (1999), 235–264.Google Scholar
[8]Le Calvez, P.. Periodic orbits of Hamiltonian homeomorphisms of surfaces. Duke Math. J. 133(1) (2006), 125–184.Google Scholar
[9]Le Calvez, P.. Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol. 10 (2006), 2299–2349.CrossRefGoogle Scholar
[10]Le Roux, F.. Homéomorphismes de surfaces: théorèmes de la fleur de Leau-Fatou et de la variété stable. Astérisque 292 (2004).Google Scholar
[11]Sauzet, A.. Application des décompositions libres à l’étude des homéomorphismes de surface. Thèse, l’Université Paris 13, 2001.Google Scholar