Une version feuilletée équivariante du théorème de translation de Brouwer
Publications Mathématiques de l'IHÉS, Tome 102 (2005), pp. 1-98

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.

@article{PMIHES_2005__102__1_0,
     author = {Le Calvez, Patrice},
     title = {Une version feuillet\'ee \'equivariante du th\'eor\`eme de translation de {Brouwer}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--98},
     year = {2005},
     publisher = {Springer},
     volume = {102},
     doi = {10.1007/s10240-005-0034-1},
     mrnumber = {2217051},
     zbl = {1152.37015},
     language = {fr},
     url = {https://www.numdam.org/articles/10.1007/s10240-005-0034-1/}
}
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
SP  - 1
EP  - 98
VL  - 102
PB  - Springer
UR  - https://www.numdam.org/articles/10.1007/s10240-005-0034-1/
DO  - 10.1007/s10240-005-0034-1
LA  - fr
ID  - PMIHES_2005__102__1_0
ER  - 
%0 Journal Article
%A Le Calvez, Patrice
%T Une version feuilletée équivariante du théorème de translation de Brouwer
%J Publications Mathématiques de l'IHÉS
%D 2005
%P 1-98
%V 102
%I Springer
%U https://www.numdam.org/articles/10.1007/s10240-005-0034-1/
%R 10.1007/s10240-005-0034-1
%G fr
%F PMIHES_2005__102__1_0
Le Calvez, Patrice. Une version feuilletée équivariante du théorème de translation de Brouwer. Publications Mathématiques de l'IHÉS, Tome 102 (2005), pp. 1-98. doi: 10.1007/s10240-005-0034-1

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

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

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

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. | Zbl | MR

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

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

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

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

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

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

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

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

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

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

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

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

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. | Zbl | MR

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

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

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

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

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

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

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

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

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

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. | Zbl | MR

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

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. | Zbl | Numdam

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

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

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). | Zbl | MR | Numdam

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

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

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. | Zbl | MR

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. | Zbl | MR

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. | Zbl | MR

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

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

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. | Zbl | MR

Cité par Sources :