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C1-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents

  • Jairo Bochi (a1)

Abstract

We prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Mañé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

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1.Abdenur, F. and Viana, M., Flavors of partial hyperbolicity, in preparation.
2.Alves, J., Bonatti, C. and Viana, M., SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351398.
3.Arbieto, A. and Matheus, C., A pasting lemma and some applications for conservative systems, Ergod. Theory Dynam. Syst. 27 (2007), 13991417.
4.Arnaud, M.-C., Bonatti, C. and Crovisier, S., Dynamiques symplectiques génériques, Ergod. Theory Dynam. Syst. 25 (2005), 14011436.
5.Avila, A., On the regularization of conservative maps, preprint (available at http://arxiv.org/abs/0810.1533).
6.Avila, A., Bochi, J. and Wilkinson, A., Nonuniform center bunching and the genericity of ergodicity among C 1 partially hyperbolic symplectomorphisms, preprint (available at http://arxiv.org/abs/0810.1533).
7.Bessa, M., The Lyapunov exponents of zero divergence three-dimensional vector fields, Ergod. Theory Dynam. Syst. 27 (2007), 14451472.
8.Bessa, M. and Dias, J. Lopes, Generic dynamics of 4-dimensional C 2 hamiltonian systems, Commun. Math. Phys. 281 (2008), 597619.
9.Bochi, J., Genericity of zero Lyapunov exponents, Ergod. Theory Dynam. Syst. 22 (2002), 16671696.
10.Bochi, J. and Fayad, B., Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2,ℝ) cocycles, Bull. Braz. Math. Soc. 37 (2006), 307349
11.Bochi, J. and Viana, M., Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps, Annales Inst. H. Poincaré 19 (2002), 113123.
12.Bochi, J. and Viana, M., Lyapunov exponents: how frequently are dynamical systems hyperbolic?, in Modern dynamical systems and applications (ed. Brin, M., Hasselblatt, B. and Pesin, Y.), pp. 271297 (Cambridge University Press 2004).
13.Bochi, J. and Viana, M., The Lyapunov exponents of generic volume preserving and symplectic maps, Annals Math. 161 (2005), 14231485.
14.Bonatti, C. and Díaz, L. J., Persistent nonhyperbolic transitive diffeomorphisms, Annals Math. 143(2) (1996), 357396.
15.Bonatti, C., Díaz, L. J. and Viana, M., Dynamics beyond uniform hyperbolicity (Springer, 2005).
16.Brin, M., Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funct. Analysis Applic. 9 (1975), 816.
17.Burns, K. and Wilkinson, A., On the ergodicity of partially hyperbolic systems, Annals Math., in press.
18.Burns, K., Dolgopyat, D. and Pesin, Y., Partial hyperbolicity, Lyapunov exponents, and stable ergodicity, J. Statist. Phys. 109 (2002), 927942.
19.Dolgopyat, D., On dynamics of mostly contracting diffeomorphisms, Commun. Math. Phys. 213 (2000), 181201.
20.Dolgopyat, D. and Pesin, Y., Every compact manifold carries a completely hyperbolic diffeomorphism, Ergod. Theory Dynam. Syst. 22 (2002), 409435.
21.Dolgopyat, D. and Wilkinson, A., Stable accessibility is C 1 dense: geometric methods in dynamics II, Astérisque 287 (2003), 3360.
22.Gourmelon, N., Adapted metrics for dominated splittings, Ergod. Theory Dynam. Syst. 27 (2007), 18391849.
23.Hasselblatt, B. and Pesin, Y., Partially hyperbolic dynamical systems, in Handbook of dynamical systems (ed. Hasselblatt, B. and Katok, A.), Volume 1B (Elsevier, 2006).
24.Horita, V. and Tahzibi, A., Partial hyperbolicity for symplectic diffeomorphisms, Annales Inst. H. Poincaré 23 (2006), 641661.
25.Mañé, R., Oseledec's theorem from the generic viewpoint, in Proc. of the International Congress of Mathematicians, Warszawa, 1983, Volume 2, pp. 12591276 (North-Holland, Amsterdam, 1983).
26.Mañé, R., The Lyapunov exponents of generic area preserving diffeomorphisms, in Proc. Int. Conf. on Dynamical Systems, Montevideo, 1995, Pitman Research Notes in Mathematics, Volume 362 pp. 110119 (Pitman, London, 1996).
27.Moreira, C. G. and Yoccoz, J.-C., Stable intersections of regular Cantor sets with large Hausdorff dimensions, Annals Math. 154 (2001), 4596.
28.Oseledets, V. I., A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc. 19 (1968), 197231.
29.Pugh, C. and Shub, M., Stable ergodicity and julienne quasiconformality, J. Eur. Math. Soc. 2 (2000), 125179.
30.Robinson, R. C., Generic properties of conservative systems, Am. J. Math. 92 (1970), 562603.
31.Hertz, F. Rodriguez, Hertz, M. A. Rodriguez, Tahzibi, A. and Ures, R., A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms, Electron. Res. Announc. Amer. Math. Soc. 14 (2007) 3588
32.Hertz, F. Rodriguez, Hertz, M. A. Rodriguez and Ures, R., A survey on partially hyperbolic dynamics, Fields Inst. Commun. 51 (2007), 3588.
33.Saghin, R. and Xia, Z., Partial hyperbolicity or dense elliptic periodic points for C 1-generic symplectic diffeomorphisms, Trans. Am. Math. Soc. 358 (2006), 51195138.
34.Tahzibi, A., Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math. 142 (2004), 315344.
35.Zehnder, E., Note on smoothing symplectic and volume preserving diffeomorphisms, in Geometry and topology, Volume III, Lecture Notes in Mathematics, No. 597, pp. 828854 (Springer, 1977).
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C1-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents

  • Jairo Bochi (a1)

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