Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms
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- by Artur Avila and Jairo Bochi PDF
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Abstract:
We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure, then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independent interest that hold $C^r$-generically for any $r \ge 1$; namely, the continuity of the ergodic decomposition, the persistence of invariant sets, and the $L^1$-continuity of Lyapunov exponents.References
- Flavio Abdenur, Christian Bonatti, and Sylvain Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math. 183 (2011), 1–60. MR 2811152, DOI 10.1007/s11856-011-0041-5
- Alexander Arbieto and Carlos Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems 27 (2007), no. 5, 1399–1417. With an appendix by David Diica and Yakov Simpson-Weller. MR 2358971, DOI 10.1017/S014338570700017X
- Marie-Claude Arnaud, Création de connexions en topologie $C^1$, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 339–381 (French, with French summary). MR 1827109, DOI 10.1017/S0143385701001183
- Marie-Claude Arnaud, Le “closing lemma” en topologie $C^1$, Mém. Soc. Math. Fr. (N.S.) 74 (1998), vi+120 (French, with English and French summaries). MR 1662930, DOI 10.24033/msmf.387
- Artur Avila, On the regularization of conservative maps, Acta Math. 205 (2010), no. 1, 5–18. MR 2736152, DOI 10.1007/s11511-010-0050-y
- Artur Avila, Jairo Bochi, and Amie Wilkinson, Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 931–979 (English, with English and French summaries). MR 2567746, DOI 10.24033/asens.2113
- Luis Barreira and Yakov Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, vol. 115, Cambridge University Press, Cambridge, 2007. Dynamics of systems with nonzero Lyapunov exponents. MR 2348606, DOI 10.1017/CBO9781107326026
- Jairo Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems 22 (2002), no. 6, 1667–1696. MR 1944399, DOI 10.1017/S0143385702001165
- Jairo Bochi, $C^1$-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents, J. Inst. Math. Jussieu 9 (2010), no. 1, 49–93. MR 2576798, DOI 10.1017/S1474748009000061
- Jairo Bochi, Bassam R. Fayad, and Enrique Pujals, A remark on conservative diffeomorphisms, C. R. Math. Acad. Sci. Paris 342 (2006), no. 10, 763–766 (English, with English and French summaries). MR 2227756, DOI 10.1016/j.crma.2006.03.028
- Jairo Bochi and Marcelo Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math. (2) 161 (2005), no. 3, 1423–1485. MR 2180404, DOI 10.4007/annals.2005.161.1423
- Christian Bonatti and Sylvain Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), no. 1, 33–104 (French, with English and French summaries). MR 2090361, DOI 10.1007/s00222-004-0368-1
- Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR 2105774
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
- S. Crovisier. Perturbation de la dynamique de difféomorphismes en topologie $C^1$. arXiv:0912.2896
- Shaobo Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst. 8 (2002), no. 3, 627–632. MR 1897871, DOI 10.3934/dcds.2002.8.627
- Nikolaz Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dynam. Systems 27 (2007), no. 6, 1839–1849. MR 2371598, DOI 10.1017/S0143385707000272
- M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
- G. W. Johnson, An unsymmetric Fubini theorem, Amer. Math. Monthly 91 (1984), no. 2, 131–133. MR 729555, DOI 10.2307/2322111
- Ricardo Mañé, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503–540. MR 678479, DOI 10.2307/2007021
- Ricardo Mañé, Oseledec’s theorem from the generic viewpoint, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 1269–1276. MR 804776
- Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
- Robert R. Phelps, Lectures on Choquet’s theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. MR 1835574, DOI 10.1007/b76887
- Charles Pugh and Michael Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 1, 1–52. MR 1750453, DOI 10.1007/s100970050013
- R. Clark Robinson, Generic properties of conservative systems, Amer. J. Math. 92 (1970), 562–603. MR 273640, DOI 10.2307/2373361
- F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, R. Ures. New criteria for ergodicity and non-uniform hyperbolicity. Duke Math. J. 160, no. 3, (2011), 599–629.
- Ali Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math. 142 (2004), 315–344. MR 2085722, DOI 10.1007/BF02771539
Additional Information
- Artur Avila
- Affiliation: Laboratoire de Probabilités et Modèles aléatoires, CNRS UMR 7599, Université de Paris VI, Paris Cedex 05, France
- Address at time of publication: IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil
- Email: artur@math.sunysb.edu
- Jairo Bochi
- Affiliation: Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Mq. S. Vicente, 225, 22453-900, Rio de Janeiro, RJ, Brazil
- Email: jairo@mat.puc-rio.br
- Received by editor(s): January 18, 2010
- Received by editor(s) in revised form: May 3, 2010
- Published electronically: February 14, 2012
- Additional Notes: Both authors were partially supported by CNPq–Brazil. This research was partially conducted during the period that the first author served as a Clay Research Fellow.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2883-2907
- MSC (2010): Primary 37D25, 37D30, 37C20
- DOI: https://doi.org/10.1090/S0002-9947-2012-05423-7
- MathSciNet review: 2888232