𝒞 0 -rigidity of characteristics in symplectic geometry
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 5, p. 857-864

The paper concerns a 𝒞 0 -rigidity result for the characteristic foliations in symplectic geometry. A symplectic homeomorphism (in the sense of Eliashberg-Gromov) which preserves a smooth hypersurface also preserves its characteristic foliation.

Cet article porte sur un résultat de rigidité 𝒞 0 du feuilletage caractéristique en géométrie symplectique. Un homéomorphisme symplectique (au sens d’Eliashberg-Gromov) qui préserve une hypersurface lisse préserve également son feuilletage caractéristique.

DOI : https://doi.org/10.24033/asens.2111
Classification:  53D05,  57R17
Keywords: symplectic geometry
@article{ASENS_2009_4_42_5_857_0,
     author = {Opshtein, Emmanuel},
     title = {$\mathcal {C}^0$-rigidity of characteristics in symplectic geometry},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {5},
     year = {2009},
     pages = {857-864},
     doi = {10.24033/asens.2111},
     zbl = {1186.53054},
     mrnumber = {2571960},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_5_857_0}
}
Opshtein, Emmanuel. $\mathcal {C}^0$-rigidity of characteristics in symplectic geometry. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 5, pp. 857-864. doi : 10.24033/asens.2111. http://www.numdam.org/item/ASENS_2009_4_42_5_857_0/

[1] D. Burns & R. Hind, Symplectic rigidity for Anosov hypersurfaces, Ergodic Theory Dynam. Systems 26 (2006), 1399-1416. | Zbl 1109.37025

[2] K. Cieliebak, Symplectic boundaries: creating and destroying closed characteristics, Geom. Funct. Anal. 7 (1997), 269-321. | Zbl 0876.53017

[3] Y. M. Eliashberg, A theorem on the structure of wave fronts and its application in symplectic topology, Funktsional. Anal. i Prilozhen. 21 (1987), 65-72. | Zbl 0655.58015

[4] Y. M. Eliashberg & M. Gromov, Convex symplectic manifolds, in Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math. 52, Amer. Math. Soc., 1991, 135-162. | Zbl 0742.53010

[5] Y. M. Eliashberg & H. Hofer, Towards the definition of symplectic boundary, Geom. Funct. Anal. 2 (1992), 211-220. | MR 1159830 | Zbl 0756.53016

[6] Y. M. Eliashberg & H. Hofer, Unseen symplectic boundaries, in Manifolds and geometry (Pisa, 1993), Sympos. Math., XXXVI, Cambridge Univ. Press, 1996, 178-189. | MR 1410072 | Zbl 0870.53020

[7] D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. 95 (1972), 66-82. | MR 288785 | Zbl 0231.58009

[8] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. | MR 809718 | Zbl 0592.53025

[9] H. Hofer & E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, 1994. | MR 1306732 | Zbl 0805.58003

[10] F. Lalonde & D. Mcduff, Local non-squeezing theorems and stability, Geom. Funct. Anal. 5 (1995), 364-386. | MR 1334871 | Zbl 0837.58014

[11] F. Laudenbach & J.-C. Sikorav, Hamiltonian disjunction and limits of Lagrangian submanifolds, Int. Math. Res. Not. 1994 (1994). | MR 1266111 | Zbl 0812.53031

[12] D. Mcduff & D. Salamon, Introduction to symplectic topology, second éd., Oxford Mathematical Monographs, Oxford Univ. Press, 1998. | MR 1698616 | Zbl 0844.58029

[13] E. Opshtein, Maximal symplectic packings in 2 , Compos. Math. 143 (2007), 1558-1575. | MR 2371382 | Zbl 1133.53057

[14] G. Paternain, L. Polterovich & K. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Mosc. Math. J. 3 (2003), 593-619, 745. | MR 2025275 | Zbl 1048.53058

[15] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225-255. | MR 433464 | Zbl 0335.57015