Skip to main content
SpringerLink
Log in

Nonuniform hyperbolicity for C 1-generic diffeomorphisms

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the ergodic theory of non-conservative C 1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C 1-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C 1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.

In addition, confirming a claim made by R. Mañé in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin’s Stable Manifold Theorem, even if the diffeomorphism is only C 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. F. Abdenur, Generic robustness of spectral decompositions, Annales Scientifiques de l’École Normale Supérieure 36 (2003), 213–224.

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Abdenur, Ch. Bonatti, S. Crovisier, L. J. Dáz and L. Wen, Periodic points in homoclinic classes, Ergodic Theory and Dynamical Systems 27 (2007), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  3. M.-C. Arnaud, Ch. Bonatti and S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory and Dynamical Systems 25 (2005), 1401–1436.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Avila and J. Bochi, A generic C 1 map has no absolutely continuous invariant probability measure, Nonlinearity 19 (2006), 2717–2725.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems 22 (2002), 1667–1696.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bochi, C 1-generic symplectic diffeomorphisms: partial hyperbolicity and zero center Lyapunov exponents, Journal of the Institute of Mathematics of Jussieu 9 (2010), 49–93.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic diffeomorphisms, Annals of Mathematics 161 (2005), 1423–1485.

    Article  MathSciNet  MATH  Google Scholar 

  8. Ch. Bonatti and S. Crovisier, Récurrence et généricité, Inventiones Mathematicae 158 (2004), 33–104.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ch. Bonatti and L. J. Dáz, Connexions hétéroclines et généricité d’une infinité de puits ou de sources, Annales Scientifiques de l’École Normale Supérieure 32 (1999), 135–150.

    Article  MATH  Google Scholar 

  10. Ch. Bonatti, L. J. Diaz and E. Pujals, A C 1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Annals of Mathematics 158 (2003), 355–418.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ch. Bonatti, L. J. Dáz, E. R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles, Astérisque 286 (2003), 187–222.

    Google Scholar 

  12. Ch. Bonatti, L. J. Dáz and M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences (Mathematical Physics), Vol. 102, Springer-Verlag, Berlin, 2004.

    Google Scholar 

  13. C. Bonatti, N. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergodic Theory and Dynamical Systems 26 (2006), 1307–1338.

    Article  MathSciNet  MATH  Google Scholar 

  14. Ch. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel Journal of Mathematics 115 (2000), 157–193.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Transactions of the American Mathematical Society 154 (1971), 377–397.

    MathSciNet  MATH  Google Scholar 

  16. C. Conley, Isolated invariant sets and Morse index, CBMS Regional Conference Series in Mathematics, Vol. 38, American Mathematical Society, Providence, RI, 1978.

    MATH  Google Scholar 

  17. L. J. Dáz and A. Gorodetski, Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes, Ergodic Theory and Dynamical Systems 29 (2009), 1479–1513.

    Article  MathSciNet  Google Scholar 

  18. D. Dolgopyat and A. Wilkinson, Stable accessibility is C 1-dense, Astérisque 287 (2003), 33–60.

    MathSciNet  Google Scholar 

  19. S. Gan, Horseshoe and entropy for C 1 surface diffeomorphisms, Nonlinearity 15 (2002), 841–848.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Hayashi, Connecting invariant manifolds and the solution of the C 1 stability and Ω-stability conjectures for flows, Annals of Mathematics 145 (1997), 81–137; 150 (1999), 353–356.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin, 1977.

    MATH  Google Scholar 

  22. V. Horita and A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 23 (2006), 641–661.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Kuratowski, Topology II, Academic Press - PWN - Polish Sci. Publishers, Warszawa, 1968.

    Google Scholar 

  24. F. Ledrappier, Quelques propriétés des exposants caractéristiques, in École d’été de probabilités de Saint-Flour, 1982, Lecture Notes in Mathematics, Vol. 1097, Springer-Verlag, Berlin, 1984, pp. 305–396.

    Chapter  Google Scholar 

  25. R. Mañé, An ergodic closing lemma, Annals of Mathematics 116 (1982), 503–540.

    Article  MathSciNet  Google Scholar 

  26. R. Mañé Oseledec’s theorem from the generic viewpoint, in Proceedings of the International Congress of Mathematicians (Warsaw, 1983), Vols. 1–2, 1984, pp. 1269–1276.

  27. R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  28. C. Morales and M. J. Pacífico, Lyapunov stability of ω-limit sets, Discrete and Continuous Dynamical Systems 8 (2002), 671–674.

    Article  MathSciNet  MATH  Google Scholar 

  29. Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Mathematical Surveys 324 (1977), 55–114.

    Article  MathSciNet  Google Scholar 

  30. V. Pliss, On a conjecture of Smale, Differencialnye Uravnenija 8 (1972), 268–282.

    MathSciNet  MATH  Google Scholar 

  31. C. Pugh, An improved closing lemma and a general density theorem, American Journal of Mathematics 89 (1967), 1010–1021.

    Article  MathSciNet  MATH  Google Scholar 

  32. C. Pugh, The C 1+α hypothesis in Pesin theory, Institut des Hautes Études Scientifiques. Publications Mathématiques 59 (1984), 143–161.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Ruelle, A measure associated with Axiom A attractors, American Journal of Mathematics 98 (1976), 619–654.

    Article  MathSciNet  MATH  Google Scholar 

  34. K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Inventiones Mathematicae 11 (1970), 99–109.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Yang, Ergodic measures far away from tangencies, preprint IMPA (2009).

  36. J. Yang, Lyapunov stable chain recurrence classes, preprint IMPA (2008).

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, PUC-Rio de Janeiro, 22460-010, Rio de Janeiro RJ, Brazil

    Flavio Abdenur

  2. UMR 5584, CNRS — Institut de Mathématiques de Bourgogne, BP 47 870, 21078, Dijon Cedex, France

    Christian Bonatti

  3. UMR 7539, CNRS — LAGA, Université Paris 13 99, Av. J.-B. Clément, 93430, Villetaneuse, France

    Sylvain Crovisier

Authors
  1. Flavio Abdenur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Bonatti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sylvain Crovisier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Flavio Abdenur.

Additional information

Partially supported by a CNPq/Brazil research grant.

Partially supported by the ANR project “DynNonHyp” BLAN08-2 313375. The last two authors, however, don’t support the fact that the short term projects funded by the ANR replace the long term research funded by the CNRS.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abdenur, F., Bonatti, C. & Crovisier, S. Nonuniform hyperbolicity for C 1-generic diffeomorphisms. Isr. J. Math. 183, 1–60 (2011). https://doi.org/10.1007/s11856-011-0041-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-011-0041-5

Keywords

Search

Navigation