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Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms

Authors: Artur Avila and Jairo Bochi
Journal: Trans. Amer. Math. Soc. 364 (2012), 2883-2907
MSC (2010): Primary 37D25, 37D30, 37C20
Published electronically: February 14, 2012
MathSciNet review: 2888232
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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.

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  • [ABC] F. Abdenur, C. Bonatti, S. Crovisier. Nonuniform hyperbolicity for $ C^1$-generic diffeomorphisms. Israel J. Math. 183 (2011), 1-60. MR 2811152
  • [AM] A. Arbieto, C. Matheus. A pasting lemma and some applications for conservative systems. Ergodic Theory Dynam. Systems 27 (2007), 1399-1417. MR 2358971
  • [A1] M.-C. Arnaud. Création de connexions en topologie $ C^1$. Ergodic Theory Dynam. Systems 21 (2001), 339-381. MR 1827109
  • [A2] -. Le ``closing lemma'' en topologie $ C^1$. Mém. Soc. Math. Fr. (N.S.) No. 74 (1998). MR 1662930
  • [Av] A. Avila. On the regularization of conservative maps. Acta Math. 205 (2010), no. 1, 5-18. MR 2736152
  • [ABW] A. Avila, J. Bochi, A. Wilkinson. Nonuniform center bunching and the genericity of ergodicity among $ C^1$ partially hyperbolic symplectomorphisms. Ann. Sci. Ec. Norm. Sup. 42 (2009), 931-979. MR 2567746
  • [BP] L. Barreira, Y. Pesin. Nonuniform hyperbolicity: Dynamics of systems with nonzero Lyapunov exponents. Cambridge, 2007. MR 2348606
  • [B1] J. Bochi. Genericity of zero Lyapunov exponents. Ergodic Theory Dynam. Systems 22 (2002), 1667-1696. MR 1944399
  • [B2] -. $ C^1$-generic symplectic diffeomorphisms: Partial hyperbolicity and zero centre Lyapunov exponents. J. Inst. Math. Jussieu, 9 (2010), 49-93. MR 2576798
  • [BFP] J. Bochi, B. Fayad, E. Pujols. A remark on conservative diffeomorphisms. C. R. Math. Acad. Sci. Paris 342 (2006), no. 10, 763-766. MR 2227756
  • [BV] J. Bochi, M. Viana. The Lyapunov exponents of generic volume preserving and symplectic maps. Annals of Math. 161 (2005), 1423-1485. MR 2180404
  • [BC] C. Bonatti, S. Crovisier. Récurrence et généricité. Invent. Math. 158 (2004), 33-104. MR 2090361
  • [BDV] C. Bonatti, L. Díaz, M. Viana. Dynamics beyond uniform hyperbolicity. Springer, 2005. MR 2105774
  • [CV] C. Castaing, M. Valadier. Convex analysis and measurable multifunctions. Lecture Notes in Math. vol. 580. Springer, 1977. MR 0467310
  • [C] S. Crovisier. Perturbation de la dynamique de difféomorphismes en topologie $ C^1$. arXiv:0912.2896
  • [Ga] S. Gan. A generalized shadowing lemma. Discrete Contin. Dyn. Syst. 8 (2002), no. 3, 627-632. MR 1897871
  • [G] N. Gourmelon. Adapted metrics for dominated splittings. Ergod. Th. Dynam. Sys. 27 (2007), 1839-1849. MR 2371598
  • [HPS] M. W. Hirsch, C. Pugh, M. Shub. Invariant Manifolds. Lecture Notes in Math. vol. 583. Springer, 1977. MR 0501173
  • [J] G. W. Johnson. An unsymmetric Fubini theorem. Amer. Math. Monthly 91 (1984), 131-133. MR 0729555
  • [M1] R. Mañé. An ergodic closing lemma. Annals of Math. 116 (1982), 503-540. MR 0678479
  • [M2] -. Oseledec's theorem from the generic viewpoint. Proceedings of the ICM, Warsaw (1983), vol. 2, 1259-1276. MR 0804776
  • [M3] -. Ergodic theory and differentiable dynamics. Springer, 1987. MR 0889254
  • [P] R. R. Phelps. Lectures on Choquet's theorem. 2nd ed. Lecture Notes in Math. vol. 1597. Springer, 2001. MR 1835574
  • [PS] C. Pugh, M. Shub. Stable ergodicity and julienne quasi-conformality. J. Eur. Math. Soc. 2 (2000), no. 1, 1-52. MR 1750453
  • [R] R. C. Robinson. Generic properties of conservative systems. Am. J. Math. 92 (1970), 562-603. MR 0273640
  • [RRTU] 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.
  • [T] A. Tahzibi. Stably ergodic diffeomorphisms which are not partially hyperbolic. Israel J. Math. 142 (2004), 315-344. MR 2085722

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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

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

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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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