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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

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.

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

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