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Some remarks on the Poincaré-Birkhoff theorem


Authors: Patrice Le Calvez and Jian Wang
Journal: Proc. Amer. Math. Soc. 138 (2010), 703-715
MSC (2000): Primary 37E30, 37E40, 37J10
DOI: https://doi.org/10.1090/S0002-9939-09-10105-3
Published electronically: October 7, 2009
MathSciNet review: 2557187
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Abstract | References | Similar Articles | Additional Information

Abstract: We define the notion of a positive path of a homeomorphism of a topological space. It seems to be a natural object to understand Birkhoff's arguments in his proof of the celebrated Poincaré-Birkhoff theorem. We write the proof of this theorem, by using positive paths, and the proof of its generalization due to P. Carter. We will also explain the links with the free disk chains introduced in the subject by J. Franks. We will finish the paper by studying the local versions where the upper curve is not invariant and will explain why this curve or its image must be a graph to get such a generalization.


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Additional Information

Patrice Le Calvez
Affiliation: Institut de Mathématiques de Jussieu, Unité Mixte de Recherche 7586, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 175 rue du Chevaleret, 75013 Paris, France
Email: lecalvez@math.jussieu.fr

Jian Wang
Affiliation: Department of Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China
Address at time of publication: Laboratoire Analyse Géométrie et Applications, Unité Mixte de Recherche 7539, Centre National de la Recherche Scientifique, Université Paris 13, 93430 Villetaneuse, France
Email: wjian05@mails.tsinghua.edu.cn, wangjian@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9939-09-10105-3
Keywords: Fixed point, boundary twist condition, positive path.
Received by editor(s): May 28, 2009
Received by editor(s) in revised form: July 2, 2009
Published electronically: October 7, 2009
Additional Notes: The authors have been supported by ANR (Symplexe, ANR-06-BLAN-0030-01) and the project 111-2-01.
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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