数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (6): 2149-2172.doi: 10.1007/s10473-021-0621-3

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HANDEL'S FIXED POINT THEOREM: A MORSE THEORETICAL POINT OF VIEW

Patrice LE CALVEZ   

  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, Sorbonne Université, Université Paris-Diderot, CNRS, F-75005, Paris, France&Institut Universitaire de France
  • 收稿日期:2021-09-04 修回日期:2021-10-08 出版日期:2021-12-25 发布日期:2021-12-27
  • 作者简介:Patrice LE CALVEZ,E-mail:patrice.le-calvez@imj-prg.fr

HANDEL'S FIXED POINT THEOREM: A MORSE THEORETICAL POINT OF VIEW

Patrice LE CALVEZ   

  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, Sorbonne Université, Université Paris-Diderot, CNRS, F-75005, Paris, France&Institut Universitaire de France
  • Received:2021-09-04 Revised:2021-10-08 Online:2021-12-25 Published:2021-12-27

摘要: Michael Handel has proved in[10] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article fits into a series of articles by the author[13] and by Juliana Xavier[21, 22], where proofs were given, related to the classical Brouwer Theory, instead of the Homotopical Brouwer Theory used in the original article. Like in[13, 21] and[22], we will use "free brick decompositions" but will present a more conceptual Morse theoretical argument. It is based on a new preliminary lemma, that gives a nice "condition at infinity" for our problem.

关键词: brick decomposition, Brouwer theory, translation arc

Abstract: Michael Handel has proved in[10] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article fits into a series of articles by the author[13] and by Juliana Xavier[21, 22], where proofs were given, related to the classical Brouwer Theory, instead of the Homotopical Brouwer Theory used in the original article. Like in[13, 21] and[22], we will use "free brick decompositions" but will present a more conceptual Morse theoretical argument. It is based on a new preliminary lemma, that gives a nice "condition at infinity" for our problem.

Key words: brick decomposition, Brouwer theory, translation arc

中图分类号: 

  • 37B20