Hostname: page-component-7dd5485656-k8lnt Total loading time: 0 Render date: 2025-10-30T23:09:20.662Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  26 March 2009

Jairo Bochi
Jairo Bochi
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente, 225, Rio de Janeiro, CEP 22453-900, Brazil (jairo@mat.puc-rio.br)
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

symplectic diffeomorphismspartial hyperbolicityLyapunov exponentsgeneric properties

MSC classification

Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 1 , January 2010 , pp. 49 - 93
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Abdenur, F. and Viana, M., Flavors of partial hyperbolicity, in preparation.Google Scholar
2.Alves, J., Bonatti, C. and Viana, M., SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351–398.CrossRefGoogle Scholar
3.Arbieto, A. and Matheus, C., A pasting lemma and some applications for conservative systems, Ergod. Theory Dynam. Syst. 27 (2007), 1399–1417.CrossRefGoogle Scholar
4.Arnaud, M.-C., Bonatti, C. and Crovisier, S., Dynamiques symplectiques génériques, Ergod. Theory Dynam. Syst. 25 (2005), 1401–1436.CrossRefGoogle Scholar
5.Avila, A., On the regularization of conservative maps, preprint (available at http://arxiv.org/abs/0810.1533).Google Scholar
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).Google Scholar
7.Bessa, M., The Lyapunov exponents of zero divergence three-dimensional vector fields, Ergod. Theory Dynam. Syst. 27 (2007), 1445–1472.CrossRefGoogle Scholar
8.Bessa, M. and Dias, J. Lopes, Generic dynamics of 4-dimensional C 2 hamiltonian systems, Commun. Math. Phys. 281 (2008), 597–619.CrossRefGoogle Scholar
9.Bochi, J., Genericity of zero Lyapunov exponents, Ergod. Theory Dynam. Syst. 22 (2002), 1667–1696.CrossRefGoogle Scholar
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), 307–349CrossRefGoogle Scholar
11.Bochi, J. and Viana, M., Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps, Annales Inst. H. Poincaré 19 (2002), 113–123.CrossRefGoogle Scholar
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. 271–297 (Cambridge University Press 2004).Google Scholar
13.Bochi, J. and Viana, M., The Lyapunov exponents of generic volume preserving and symplectic maps, Annals Math. 161 (2005), 1423–1485.CrossRefGoogle Scholar
14.Bonatti, C. and Díaz, L. J., Persistent nonhyperbolic transitive diffeomorphisms, Annals Math. 143(2) (1996), 357–396.CrossRefGoogle Scholar
15.Bonatti, C., Díaz, L. J. and Viana, M., Dynamics beyond uniform hyperbolicity (Springer, 2005).Google Scholar
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), 8–16.CrossRefGoogle Scholar
17.Burns, K. and Wilkinson, A., On the ergodicity of partially hyperbolic systems, Annals Math., in press.Google Scholar
18.Burns, K., Dolgopyat, D. and Pesin, Y., Partial hyperbolicity, Lyapunov exponents, and stable ergodicity, J. Statist. Phys. 109 (2002), 927–942.CrossRefGoogle Scholar
19.Dolgopyat, D., On dynamics of mostly contracting diffeomorphisms, Commun. Math. Phys. 213 (2000), 181–201.CrossRefGoogle Scholar
20.Dolgopyat, D. and Pesin, Y., Every compact manifold carries a completely hyperbolic diffeomorphism, Ergod. Theory Dynam. Syst. 22 (2002), 409–435.CrossRefGoogle Scholar
21.Dolgopyat, D. and Wilkinson, A., Stable accessibility is C 1 dense: geometric methods in dynamics II, Astérisque 287 (2003), 33–60.Google Scholar
22.Gourmelon, N., Adapted metrics for dominated splittings, Ergod. Theory Dynam. Syst. 27 (2007), 1839–1849.CrossRefGoogle Scholar
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).Google Scholar
24.Horita, V. and Tahzibi, A., Partial hyperbolicity for symplectic diffeomorphisms, Annales Inst. H. Poincaré 23 (2006), 641–661.Google Scholar
25.Mañé, R., Oseledec's theorem from the generic viewpoint, in Proc. of the International Congress of Mathematicians, Warszawa, 1983, Volume 2, pp. 1259–1276 (North-Holland, Amsterdam, 1983).Google Scholar
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. 110–119 (Pitman, London, 1996).Google Scholar
27.Moreira, C. G. and Yoccoz, J.-C., Stable intersections of regular Cantor sets with large Hausdorff dimensions, Annals Math. 154 (2001), 45–96.CrossRefGoogle Scholar
28.Oseledets, V. I., A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc. 19 (1968), 197–231.Google Scholar
29.Pugh, C. and Shub, M., Stable ergodicity and julienne quasiconformality, J. Eur. Math. Soc. 2 (2000), 125–179.CrossRefGoogle Scholar
30.Robinson, R. C., Generic properties of conservative systems, Am. J. Math. 92 (1970), 562–603.CrossRefGoogle Scholar
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) 35–88Google Scholar
32.Hertz, F. Rodriguez, Hertz, M. A. Rodriguez and Ures, R., A survey on partially hyperbolic dynamics, Fields Inst. Commun. 51 (2007), 35–88.Google Scholar
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), 5119–5138.CrossRefGoogle Scholar
34.Tahzibi, A., Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math. 142 (2004), 315–344.CrossRefGoogle Scholar
35.Zehnder, E., Note on smoothing symplectic and volume preserving diffeomorphisms, in Geometry and topology, Volume III, Lecture Notes in Mathematics, No. 597, pp. 828–854 (Springer, 1977).CrossRefGoogle Scholar