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

doi:10.1007/s10473-021-0621-3

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

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

Patrice LE CALVEZ

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

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/Abstract

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

Patrice LE CALVEZ. HANDEL'S FIXED POINT THEOREM: A MORSE THEORETICAL POINT OF VIEW[J]. 数学物理学报(英文版), 2021, 41(6): 2149-2172.

Patrice LE CALVEZ. HANDEL'S FIXED POINT THEOREM: A MORSE THEORETICAL POINT OF VIEW[J]. Acta mathematica scientia,Series B, 2021, 41(6): 2149-2172.

参考文献 22

[1]	Abate M, Mongodi S, Raissy J. Toeplitz operators and skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains. J Operator Theory, 2020, 84(2):339-364
[2]	Abate M, Raissy J, Saracco A. Toeplitz operators and Carleson measures in strongly pseudoconvex domains. J Funct Anal, 2012, 263(11):3449-3491
[3]	Abate M, Saracco A. Carleson measures and uniformly discrete sequences in strongly pseudocomvex domains. J London Math Soc, 2011, 83(2):587-605
[4]	Arcozzi N, Rochberg R, Sawyer E. Carleson measures for analytic Besov spaces. Rev Mat Iberoamericana, 2002, 18:443-510
[5]	Arcozzi N, Rochberg R, Sawyer E. Carleson measures and interpolating sequences. Mem Amer Math Soc, 2006, 182(859)
[6]	Arcozzi N, Rochberg R, Sawyer E. Carleson measures for the Drury-Arveson Hardy space and other BesovSobolev spaces on complex balls. Adv Math, 2008, 218:1107-1180
[7]	Carleson L. An interpolation problem for bounded analytic functions. Amer J Math, 1958, 80:921-930
[8]	Carleson L. Interpolations by bounded analytic functions and the corona problem. Ann of Math, 1962, 76:547-559
[9]	Cima J A, Wogen W R. A Carleson measure theorem for the Bergman space on the ball. J Operator Theory, 1982, 7(1):157-165
[10]	Duren P L, Weir R. The pseudohyperbolic metric and Bergman spaces in the ball. Trans Amer Math Soc, 2007, 359:63-76
[11]	Hartmann A, Massaneda X, Nicolau A, Ortega-Cerda J. Reverse Carleson measures in Hardy spaces. Collectanea Math, 2014, 65:357-365
[12]	Hastings W W. A Carleson measure theorem for Bergman spaces. Proc Amer Math Soc, 1975, 52:237-241
[13]	Lefèvre P, Li D, Queffelec H, Rodriguez-Piazza L. Nevanlinna counting function and Carleson function of analytic maps. Math Ann, 2011, 351:305-326
[14]	Lefèvre P, Li D, Queffelec H, Rodriguez-Piazza L. Some revisited results about composition operators on Hardy spaces. Rev Mat Iberoamericana, 2012, 28:57-76
[15]	Luecking D. A technique for characterizing Carleson measure on Bergman spaces. Proc Amer Math Soc, 1983, 87:656-660
[16]	Nazarov F, Treil S, Volberg A. The Tb-theorem on non-homogeneous spaces. Acta Math, 2003, 190:151- 239
[17]	Ouyang C, Xu Q. BMO functions and Carleson measures with values in uniformly convex spaces. Canad J Math, 2010, 62:827-844
[18]	Ouyang C, Yang W, Zhao R. Characterizations of Bergman spaces and Bloch space in the unit ball of Cn. Trans Amer Math Soc, 1995, 347:4301-4313
[19]	Pau J, Zhao R. Carleson measures and Toeplitz operators for weighted Bergman spaces on the unit ball. Michigan Math J, 2015, 64:759-796
[20]	Peng R, Ouyang C. Carleson measures for Besov-Sobolev spaces with applications in the unit ball of Cn. Acta Math Sci, 2013, 33B:1219-1230
[21]	Volberg A, Wick B D. Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball. Amer J Math, 2012, 134:949-992
[22]	Zhu K. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, Vol 226. New York:Springer-Verlag, 2005

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

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

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 22

相关文章 2

Metrics

本文评价

推荐阅读 10

[1]	陈中川. RECURRENCE FOR WEIGHTED TRANSLATIONS ON GROUPS[J]. 数学物理学报(英文版), 2016, 36(2): 443-452.
[2]	田学廷, 孙文祥. DIFFEOMORPHISMS WITH VARIOUS C¹ STABLE PROPERTIES[J]. 数学物理学报(英文版), 2012, 32(2): 552-558.