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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
Posted:
October 7, 2009
MathSciNet review:
2557187
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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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Béguin, S.
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(2008b:37074), http://dx.doi.org/10.1007/s00574-006-0013-2
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G. D. Birkhoff :
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E. J. Brouwer, Beweis des ebenen Translationssatzes, Math.
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1511684, http://dx.doi.org/10.1007/BF01456888
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Brown and W.
D. Neumann, Proof of the Poincaré-Birkhoff fixed point
theorem, Michigan Math. J. 24 (1977), no. 1,
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0448339 (56 #6646)
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Patricia
H. Carter, An improvement of the
Poincaré-Birkhoff fixed point theorem, Trans. Amer. Math. Soc. 269 (1982), no. 1, 285–299. MR 637039
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Dalbono and C.
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Wei
Yue Ding, A generalization of the
Poincaré-Birkhoff theorem, Proc. Amer.
Math. Soc. 88 (1983), no. 2, 341–346. MR 695272
(84f:54053), http://dx.doi.org/10.1090/S0002-9939-1983-0695272-2
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John
Franks, Generalizations of the Poincaré-Birkhoff
theorem, Ann. of Math. (2) 128 (1988), no. 1,
139–151. MR
951509 (89m:54052), http://dx.doi.org/10.2307/1971464
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John
Franks, A variation on the Poincaré-Birkhoff theorem,
Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math.,
vol. 81, Amer. Math. Soc., Providence, RI, 1988,
pp. 111–117. MR 986260
(90e:58095), http://dx.doi.org/10.1090/conm/081/986260
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Guillou, Théorème de translation plane de Brouwer et
généralisations du théorème de
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no. 2, 331–351 (French). MR 1273787
(95h:55003), http://dx.doi.org/10.1016/0040-9383(94)90016-7
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Howard
Jacobowitz, Periodic solutions of
𝑥′′+𝑓(𝑥,𝑡)=0 via the
Poincaré-Birkhoff theorem, J. Differential Equations
20 (1976), no. 1, 37–52. MR 0393673
(52 #14482)
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Howard
Jacobowitz, Corrigendum: The existence of the second fixed point: a
correction to “Periodic solutions of
𝑥”+𝑓(𝑥,𝑡)=0 via the
Poincaré-Birkhoff theorem” (J. Differential Equations 20
(1976), no. 1, 37–52), J. Differential Equations
25 (1977), no. 1, 148–149. MR 0437857
(55 #10778)
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B. de Kerékjártó :
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Frédéric
Le Roux, Homéomorphismes de surfaces:
théorèmes de la fleur de Leau-Fatou et de la
variété stable, Astérisque 292
(2004), vi+210 (French, with English and French summaries). MR 2068866
(2005e:37091)
- [MU]
Rogério
Martins and Antonio
J. Ureña, The star-shaped condition on Ding’s version
of the Poincaré-Birkhoff theorem, Bull. Lond. Math. Soc.
39 (2007), no. 5, 803–810. MR 2365229
(2008k:54061), http://dx.doi.org/10.1112/blms/bdm064
- [P]
H. Poincaré :
Sur un théorème de géométrie,
Rend. Circ. Mat. Palermo, 33 (1912), 375-407.
- [BCL]
- F. Béguin, S. Crovisier, F. Le Roux :
Pseudo-rotations of the open annulus,
Bull. Braz. Math. Soc. (N.S.), 37 (2006), 275-306. MR 2266384 (2008b:37074)
- [Bi1]
- G. D. Birkhoff :
Proof of Poincaré's last geometric theorem,
Trans. Amer. Math. Soc., 14 (1913), 14-22.
- [Bi2]
- G. D. Birkhoff :
An extension of Poincaré's last geometric theorem,
Acta. Math., 47 (1925), 297-311.
- [Br]
- L. E. J. Brouwer :
Beweis des ebenen Translationssatzes,
Math. Ann., 72 (1912), 37-54. MR 1511684
- [BN]
- L. E. J. M. Brown, W. D. Newmann :
Proof of the Poincaré-Birkhoff fixed point theorem,
Michigan. Math. J., 24 (1977), 21-31. MR 0448339 (56:6646)
- [C]
- P. H. Carter :
An improvement of the Poincaré-Birkhoff fixed point theorem,
Trans. Amer. Math. Soc., 269 (1982), 285-299. MR 637039 (84h:54041)
- [DR]
- F. Dalbono, C. Rebelo :
Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems,
Rend. Sem. Math. Univ. Pol. Torino, 60 (2002), 233-263 (2003). MR 2010410 (2004j:37100)
- [D]
- W.-Y. Ding :
A generalization of the Poincaré-Birkhoff theorem,
Proc. Amer. Math. Soc., 88 (1983), 341-346. MR 695272 (84f:54053)
- [F1]
- J. Franks :
Generalizations of the Poincaré-Birkhoff theorem,
Annals of Math. (2), 128 (1988), 139-151. MR 951509 (89m:54052)
- [F2]
- J. Franks :
A variation on the Poincaré-Birkhoff Theorem,
Contemp. Math., 81, Amer. Math. Soc., Providence, RI, 1988, 111-117. MR 986260 (90e:58095)
- [G1]
- L. Guillou :
Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff,
Topology, 33 (1994), 331-351. MR 1273787 (95h:55003)
- [G2]
- L. Guillou :
A simple proof of P. Carter's theorem,
Proc. Amer. Math. Soc., 125 (1997), 1555-1559. MR 1372031 (97g:54055)
- [J1]
- H. Jacobowitz :
Periodic solutions of  via the Poincaré-Birkhoff theorem,
J. Diff. Equations, 20 (1976), 37-52. MR 0393673 (52:14482)
- [J2]
- H. Jacobowitz :
Corrigendum: The existence of the second fixed point: a correction to ``Periodic solutions of  via the Poincaré-Birkhoff theorem'',
J. Diff. Equations, 25 (1977), 148-149. MR 0437857 (55:10778)
- [K]
- B. de Kerékjártó :
The plane translation theorem of Brouwer and the last geometric theorem of Poincaré,
Acta Sci. Math. Szeged, 4 (1928-29), 86-102.
- [L]
- F. Le Roux :
Homéomorphismes de surfaces: théorèmes de la fleur de Leau-Fatou et de la variété stable,
Astérisque, 292 (2004). MR 2068866 (2005e:37091)
- [MU]
- R. Matins, A. J. Ureña :
The star-shaped condition on Ding's version of the Poincaré-Birkhoff theorem,
Bull. London Math. Soc., 39 (2007), no. 5, 803-810. MR 2365229 (2008k:54061)
- [P]
- H. Poincaré :
Sur un théorème de géométrie,
Rend. Circ. Mat. Palermo, 33 (1912), 375-407.
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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:
http://dx.doi.org/10.1090/S0002-9939-09-10105-3
PII:
S 0002-9939(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
Posted:
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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