Une version feuilletée équivariante du théorème de translation de Brouwer

Le Calvez, Patrice

Publications Mathématiques de l'IHÉS, Tome 102 (2005), p. 1-98 / Harvested from Numdam

Résumé

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 $\mathbf {R}$, 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 $g\ge 1$ has infinitely many contractible periodic points.

Détail

Publié le : 2005-01-01

MR 2217051 | Zbl pre05017589 | Zbl 1152.37015 | 2 citations dans Numdam

DOI : https://doi.org/10.1007/s10240-005-0034-1

Bibliographie

1. G. D. Birkhoff, An extension of Poincaré's last geometric theorem, Acta Math., 47 (1925), 297-311. | JFM 52.0573.02

2. L. E. J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann., 72 (1912), 37-54. | JFM 43.0569.02 | MR 1511684

3. M. Brown, A new proof of Brouwer's lemma on translation arcs, Houston J. Math., 10 (1984), 35-41. | Zbl 0551.57005

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. | MR 953953 | Zbl 0668.54024

5. P. H. Carter, An improvment of the Poincaré-Birkhoff theorem, Trans. Am. Math. Soc., 269 (1982), 285-299. | MR 637039 | Zbl 0507.55002

6. C. Conley, Isolated invariants sets and Morse index, CBMS Reg. Conf. Series in Math., Am. Math. Soc., 38 (1978). | MR 511133 | Zbl 0397.34056

7. A. Fathi, An orbit closing proof of Brouwer's lemma on translation arcs, Enseign. Math., 33 (1987), 315-322. | Zbl 0649.54022

8. A. Floer, Proof of the Arnold conjectures for surfaces and generalizations to certain Kähler manifolds, Duke Math. J., 51 (1986), 1-32. | MR 835793 | Zbl 0607.58016

9. M. Flucher, Fixed points of measure preserving torus homeomorphism, Manuscr. Math., 68 (1990), 271-293. | MR 1065931 | Zbl 0722.58027

10. J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139-151. | MR 951509 | Zbl 0676.58037

11. J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dyn. Syst., 8* (1988), 99-107. | MR 967632 | Zbl 0634.58023

12. J. Franks, A new proof of the Brouwer plane translation theorem, Ergodic Theory Dyn. Syst., 12 (1992), 217-226. | MR 1176619 | Zbl 0767.58025

13. J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418. | MR 1161099 | Zbl 0766.53037

14. J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math., 2 (1996), 1-19. | MR 1371312 | Zbl 0891.58033

15. J. Franks, The Conley index and non-existence of minimal homeomorphisms, Ill. J. Maths, 43 (1999), 457-464. | MR 1700601 | Zbl 0948.37005

16. J. Franks and M. Handel, Periodic points of Hamiltonian surface diffeomorphims, Geom. Topol., 7 (2003), 713-756. | MR 2026545 | Zbl 1034.37028

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. L. Guillou, Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, Topology, 33 (1994), 331-351. | MR 1273787 | Zbl 0924.55001

19. M.-E. Hamstrom, The space of homeomorphisms on a torus, Ill. J. Math., 9 (1965), 59-65. | MR 170334 | Zbl 0127.13505

20. M.-E. Hamstrom, Homotopy groups of the space of homeomorphisms on a 2-manifold, Ill. J. Math., 10 (1966), 563-573. | MR 202140 | Zbl 0151.33002

21. M. Handel, There are no minimal homeomorphisms of the multipunctured plane, Ergodic Theory Dyn. Syst., 12 (1992), 75-83. | MR 1162400 | Zbl 0769.58037

22. M. Handel, A fixed point theorem for planar homeomorphisms, Topology, 38 (1999), 235-264. | MR 1660349 | Zbl 0928.55001

23. M. Handel and W. P. Thurston, New proofs of some results of Nielsen, Adv. Math., 56 (1985), 173-191. | MR 788938 | Zbl 0584.57007

24. H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser, Basel-Boston-Berlin (1994). | MR 1306732 | Zbl 0805.58003

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. | JFM 54.0612.02

26. B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math., 14 (1983). | MR 685755 | Zbl 0512.55003

27. P. Le Calvez, Propriétés dynamiques de l'anneau et du tore, Astérisque, 204 (1991). | Zbl 0784.58033

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. | MR 1440768 | Zbl 0877.58011

29. P. Le Calvez, Une propriété dynamique des homéomorphismes du plan au voisinage d'un point fixe d'indice > 1, Topology, 38 (1999), 25-35. | Zbl 0976.54046

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. | Numdam | Zbl 1017.37017

31. P. Le Calvez, Une version feuilletée du théorème de translation de Brouwer, Comment. Math. Helv., 21 (2004), 229-259. | MR 2059431 | Zbl 1073.37045

32. P. Le Calvez et A. Sauzet, Une démonstration dynamique du théorème de translation de Brouwer, Expo. Math., 14 (1996), 277-287. | MR 1409005 | Zbl 0859.54029

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. F. Le Roux, Homéomorphismes de surfaces, théorème de la fleur de Leau-Fatou et de la variété stable, Astérisque, 292 (2004). | MR 2068866 | Zbl 1073.37046

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. | Zbl 0974.37040

36. N. A. Nikishin, Fixed points of diffeomorphisms of two-dimensional spheres preserving an oriented plane, Funkts. Anal. Prilozh., 8 (1974), 84-85. | MR 346846 | Zbl 0298.57019

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. | MR 912377 | Zbl 0632.58014

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. | MR 1181727 | Zbl 0766.58023

39. A. Sauzet, Application des décompositions libres à l'étude des homéomorphismes de surface, Thèse de l'Université Paris 13 (2001).

40. M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pac. J. Math., 193 (2000), 419-461. | MR 1755825 | Zbl 1023.57020

41. S. Schwartzman, Asymptotic cycles, Ann. Math., 68 (1957), 270-284. | MR 88720 | Zbl 0207.22603

42. J.-C. Sikorav, Points fixes d'une application symplectique homologue à l'identité, J. Differ. Geom., 22 (1985), 49-79. | Zbl 0555.58013

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. | MR 353372 | Zbl 0331.55006