Nous donnons un exemple d’une variété symplectique contenant une hypersurface stable telle que les hypersurfaces voisines sont instables.
We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.
Classification : 53D40, 53D25
Mots clés : stabilité, structure Hamiltonienne, feuilletage caractéristique
@article{AIF_2010__60_7_2449_0, author = {Cieliebak, Kai and Frauenfelder, Urs and Paternain, Gabriel P.}, title = {Stability is not open}, journal = {Annales de l'Institut Fourier}, pages = {2449--2459}, publisher = {Association des Annales de l'institut Fourier}, volume = {60}, number = {7}, year = {2010}, doi = {10.5802/aif.2614}, mrnumber = {2849269}, zbl = {1235.53089}, language = {en}, url = {archive.numdam.org/item/AIF_2010__60_7_2449_0/} }
Cieliebak, Kai; Frauenfelder, Urs; Paternain, Gabriel P. Stability is not open. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2449-2459. doi : 10.5802/aif.2614. http://archive.numdam.org/item/AIF_2010__60_7_2449_0/
[1] Certain smooth ergodic systems, Uspehi Mat. Nauk, Volume 22 (1967) no. 5 (137), pp. 107-172 | MR 224771 | Zbl 0177.42002
[2] Compactness results in symplectic field theory, Geom. Topol., Volume 7 (2003), pp. 799-888 | Article | MR 2026549 | Zbl 1131.53312
[3] A Floer homology for exact contact embeddings, Pacific J. Math., Volume 239 (2009) no. 2, pp. 251-316 | Article | MR 2461235 | Zbl pre05541980
[4] Symplectic topology of Mañé’s critical values, Geometry and Topology, Volume 14 (2010), pp. 1765-1870 | Article | MR 2679582 | Zbl pre05769287
[5] Compactness for punctured holomorphic curves, J. Symplectic Geom., Volume 3 (2005) no. 4, pp. 589-654 (Conference on Symplectic Topology) | MR 2235856 | Zbl 1113.53053
[6] First steps in stable Hamiltonian topology (2010) (arXiv:1003.5084)
[7] Introduction to symplectic field theory, Geom. Funct. Anal. (2000) no. Special Volume, Part II, pp. 560-673 (GAFA 2000 (Tel Aviv, 1999)) | MR 1826267 | Zbl 0989.81114
[8] Geodesic flows on manifolds of negative curvature with smooth horospheric foliations, Ergodic Theory Dynam. Systems, Volume 11 (1991) no. 4, pp. 653-686 | Article | MR 1145615 | Zbl 0727.58035
[9] Horospheric foliations and relative pinching, J. Differential Geom., Volume 39 (1994) no. 1, pp. 57-63 | MR 1258914 | Zbl 0795.53026
[10] Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems, Volume 14 (1994) no. 4, pp. 645-666 | Article | MR 1304137 | Zbl 0821.58032
[11] Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin, 1977 | MR 501173 | Zbl 0355.58009
[12] Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994 | MR 1306732 | Zbl 0805.58003
[13] Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math. (N.S.), Volume 19 (1993) no. 1, pp. 1-30 | MR 1231509 | Zbl 0798.58055
[14] Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, Volume 54, Cambridge University Press, Cambridge, 1995 (With a supplementary chapter by Katok and Leonardo Mendoza) | MR 1326374 | Zbl 0878.58019
[15] Synchronisation of canonical measures for hyperbolic attractors, Comm. Math. Phys., Volume 106 (1986) no. 2, pp. 267-275 | Article | MR 855312 | Zbl 0618.58026
[16] Geodesic flows, Progress in Mathematics, Volume 180, Birkhäuser Boston Inc., Boston, MA, 1999 | MR 1712465 | Zbl 0930.53001
[17] Anosov flows, Amer. J. Math., Volume 94 (1972), pp. 729-754 | Article | MR 377930 | Zbl 0257.58007
[18] On uniformly quasiconformal Anosov systems, Math. Res. Lett., Volume 12 (2005) no. 2-3, pp. 425-441 | MR 2150895 | Zbl 1081.37015