Une version feuilletée équivariante du théorème de translation de Brouwer

Patrice Le Calvez

Publications Mathématiques de l'IHÉS (2005)

  • Volume: 102, page 1-98
  • ISSN: 0073-8301

Abstract

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The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a Brouwer line: a proper topological embedding C of R, disjoint from its image and separating f(C) and f–1(C). Suppose that f commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines. We apply this result to give simple proofs of previous results about area-preserving homeomorphisms of surfaces and to prove the following theorem: any hamiltonian homeomorphism of a closed surface of genus g ≥ 1 has infinitely many contractible periodic points.

How to cite

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Le Calvez, Patrice. "Une version feuilletée équivariante du théorème de translation de Brouwer." Publications Mathématiques de l'IHÉS 102 (2005): 1-98. <http://eudml.org/doc/104214>.

@article{LeCalvez2005,
author = {Le Calvez, Patrice},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Brouwer's plan translation theorem; Brouwer line; Brouwer homeomorphism; foliation; area-preserving homeomorphism; orientation preserving homeomorphism},
language = {fre},
pages = {1-98},
publisher = {Springer},
title = {Une version feuilletée équivariante du théorème de translation de Brouwer},
url = {http://eudml.org/doc/104214},
volume = {102},
year = {2005},
}

TY - JOUR
AU - Le Calvez, Patrice
TI - Une version feuilletée équivariante du théorème de translation de Brouwer
JO - Publications Mathématiques de l'IHÉS
PY - 2005
PB - Springer
VL - 102
SP - 1
EP - 98
LA - fre
KW - Brouwer's plan translation theorem; Brouwer line; Brouwer homeomorphism; foliation; area-preserving homeomorphism; orientation preserving homeomorphism
UR - http://eudml.org/doc/104214
ER -

References

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  1. 1. G. D. Birkhoff, An extension of Poincaré’s last geometric theorem, Acta Math., 47 (1925), 297–311. Zbl52.0573.02JFM52.0573.02
  2. 2. L. E. J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann., 72 (1912), 37–54. Zbl43.0569.02MR1511684JFM43.0569.02
  3. 3. M. Brown, A new proof of Brouwer’s lemma on translation arcs, Houston J. Math., 10 (1984), 35–41. Zbl0551.57005
  4. 4. M. Brown, E. Slaminka, and W. Transue, An orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line, Topol. Appl., 29 (1988), 213–217. Zbl0668.54024MR953953
  5. 5. P. H. Carter, An improvment of the Poincaré-Birkhoff theorem, Trans. Am. Math. Soc., 269 (1982), 285–299. Zbl0507.55002MR637039
  6. 6. C. Conley, Isolated invariants sets and Morse index, CBMS Reg. Conf. Series in Math., Am. Math. Soc., 38 (1978). Zbl0397.34056MR511133
  7. 7. A. Fathi, An orbit closing proof of Brouwer’s lemma on translation arcs, Enseign. Math., 33 (1987), 315–322. Zbl0649.54022
  8. 8. A. Floer, Proof of the Arnold conjectures for surfaces and generalizations to certain Kähler manifolds, Duke Math. J., 51 (1986), 1–32. Zbl0607.58016MR835793
  9. 9. M. Flucher, Fixed points of measure preserving torus homeomorphism, Manuscr. Math., 68 (1990), 271–293. Zbl0722.58027MR1065931
  10. 10. J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139–151. Zbl0676.58037MR951509
  11. 11. J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dyn. Syst., 8* (1988), 99–107. Zbl0634.58023MR967632
  12. 12. J. Franks, A new proof of the Brouwer plane translation theorem, Ergodic Theory Dyn. Syst., 12 (1992), 217–226. Zbl0767.58025MR1176619
  13. 13. J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403–418. Zbl0766.53037MR1161099
  14. 14. J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math., 2 (1996), 1–19. Zbl0891.58033MR1371312
  15. 15. J. Franks, The Conley index and non-existence of minimal homeomorphisms, Ill. J. Maths, 43 (1999), 457–464. Zbl0948.37005MR1700601
  16. 16. J. Franks and M. Handel, Periodic points of Hamiltonian surface diffeomorphims, Geom. Topol., 7 (2003), 713–756. Zbl1034.37028MR2026545
  17. 17. J.-M. Gambaudo et P. Le Calvez, Infinité d’orbites périodiques pour les difféomorphismes conservatifs de l’anneau, Mém. Soc. Math Fr., 79 (1999), 125–146. 
  18. 18. L. Guillou, Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, Topology, 33 (1994), 331–351. Zbl0924.55001MR1273787
  19. 19. M.-E. Hamstrom, The space of homeomorphisms on a torus, Ill. J. Math., 9 (1965), 59–65. Zbl0127.13505MR170334
  20. 20. M.-E. Hamstrom, Homotopy groups of the space of homeomorphisms on a 2-manifold, Ill. J. Math., 10 (1966), 563–573. Zbl0151.33002MR202140
  21. 21. M. Handel, There are no minimal homeomorphisms of the multipunctured plane, Ergodic Theory Dyn. Syst., 12 (1992), 75–83. Zbl0769.58037MR1162400
  22. 22. M. Handel, A fixed point theorem for planar homeomorphisms, Topology, 38 (1999), 235–264. Zbl0928.55001MR1660349
  23. 23. M. Handel and W. P. Thurston, New proofs of some results of Nielsen, Adv. Math., 56 (1985), 173–191. Zbl0584.57007MR788938
  24. 24. H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser, Basel–Boston–Berlin (1994). Zbl0805.58003MR1306732
  25. 25. 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. JFM54.0612.02
  26. 26. B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math., 14 (1983). Zbl0512.55003MR685755
  27. 27. P. Le Calvez, Propriétés dynamiques de l’anneau et du tore, Astérisque, 204 (1991). Zbl0784.58033
  28. 28. P. Le Calvez, Une généralisation du théorème de Conley-Zehnder aux homéomorphismes du tore de dimension deux, Ergodic Theory Dyn. Syst., 17 (1997), 71–86. Zbl0877.58011MR1440768
  29. 29. P. Le Calvez, Une propriété dynamique des homéomorphismes du plan au voisinage d’un point fixe d’indice &gt; 1, Topology, 38 (1999), 25–35. Zbl0976.54046
  30. 30. P. Le Calvez, Dynamique des homéomorphismes du plan au voisinage d’un point fixe, Ann. Sci. Éc. Norm. Supér., IV Sér., 36 (2003), 139–171. Zbl1017.37017
  31. 31. P. Le Calvez, Une version feuilletée du théorème de translation de Brouwer, Comment. Math. Helv., 21 (2004), 229–259. Zbl1073.37045MR2059431
  32. 32. P. Le Calvez et A. Sauzet, Une démonstration dynamique du théorème de translation de Brouwer, Expo. Math., 14 (1996), 277–287. Zbl0859.54029MR1409005
  33. 33. P. Le Calvez et J.-C. Yoccoz, Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant. Manuscrit. 
  34. 34. F. Le Roux, Homéomorphismes de surfaces, théorème de la fleur de Leau-Fatou et de la variété stable, Astérisque, 292 (2004). Zbl1073.37046MR2068866
  35. 35. S. Matsumoto, Arnold conjecture for surface homeomorphisms, Proceedings of the French–Japanese Conference “Hyperspace Topologies and Applications” (La Bussière, 1997), Topol. Appl., 104 (2000), 191–214. Zbl0974.37040
  36. 36. N. A. Nikishin, Fixed points of diffeomorphisms of two-dimensional spheres preserving an oriented plane, Funkts. Anal. Prilozh., 8 (1974), 84–85. Zbl0298.57019MR346846
  37. 37. S. Pelikan and E. Slaminka, A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds, Ergodic Theory Dyn. Syst., 7 (1987), 463–479. Zbl0632.58014MR912377
  38. 38. D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Commun. Pure Appl. Math., 45 (1992), 1303–1360. Zbl0766.58023MR1181727
  39. 39. A. Sauzet, Application des décompositions libres à l’étude des homéomorphismes de surface, Thèse de l’Université Paris 13 (2001). 
  40. 40. M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pac. J. Math., 193 (2000), 419–461. Zbl1023.57020MR1755825
  41. 41. S. Schwartzman, Asymptotic cycles, Ann. Math., 68 (1957), 270–284. Zbl0207.22603MR88720
  42. 42. J.-C. Sikorav, Points fixes d’une application symplectique homologue à l’identité, J. Differ. Geom., 22 (1985), 49–79. Zbl0555.58013
  43. 43. C. P. Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics, Invent. Math., 26 (1974), 187–200. Zbl0331.55006MR353372

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