Some remarks on the Poincaré-Birkhoff theorem
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- by Patrice Le Calvez and Jian Wang PDF
- Proc. Amer. Math. Soc. 138 (2010), 703-715 Request permission
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.References
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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
- MR Author ID: 111345
- 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2557187