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Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures

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Abstract

We give explicit C 1-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly.

The conditions of the criterion are met on a C 1-dense and open subset of the set of diffeomorphisms having a robust cycle. As a corollary, there exists a C 1-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy.

The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs.

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References

  1. Abdenur F., Bonatti C., Crovisier S., Díaz L.J., Wen L.: Periodic points and homoclinic classes. Ergodic Theory Dynam. Syst. 27(1), 1–22 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avila, A., Crovisier, S., Wilkinson, A.: Diffeomorphisms with positive metric entropy. ar**v:1408.4252

  3. Baladi V., Bonatti C., Schmitt B.: Abnormal escape rates from nonuniformly hyperbolic sets. Ergodic Theory Dynam. Syst. 19(5), 1111–1125 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baraviera A., Bonatti C.: Removing zero Lyapunov exponents. Ergodic Theory Dynam. Syst. 23(6), 1655–1670 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bochi J., Bonatti C., Díaz L.J.: Robust vanishing of all Lyapunov exponents for iterated function systems. Math. Z. 276(1-2), 469–503 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonatti C.: Towards a global view of dynamical systems, for the C 1-topology. Ergodic Theory Dynam. Syst. 31(4), 959–993 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonatti C., Crovisier S.: Récurrence et généricité. Invent. Math. 158(1), 33–104 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bonatti C., Crovisier S., Díaz L.J., Gourmelon N.: Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication. Ergodic Theory Dynam. Syst. 33(3), 739–776 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bonatti C., Díaz L.J.: Persistent nonhyperbolic transitive diffeomorphisms. Ann. Math. 143(2), 357–396 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bonatti C., Díaz L.J.: Robust heterodimensional cycles and C 1-generic dynamics. J. Inst. Math. Jussieu 7(3), 469–525 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bonatti C., Díaz L.J.: Abundance of C 1-robust homoclinic tangencies. Trans. Am. Math. Soc. 364(10), 5111–5148 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bonatti C., Díaz L.J., Gorodetski A.: Non-hyperbolic ergodic measures with large support. Nonlinearity 23(3), 687–705 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Bonatti C., Díaz L.J., Kiriki S.: Stabilization of heterodimensional cycles. Nonlinearity 25(4), 931–960 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Bonatti, C., Díaz, L.J., Viana, M.: Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Encyclopaedia of Mathematical Sciences. Mathematical Physics, III, vol. 102. Springer, Berlin (2005)

  15. Cao Y., Luzzatto S., Rios I.: Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: horseshoes with internal tangencies. Discrete Contin. Dyn. Syst. 15(1), 61–71 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crovisier S., Pujals E.R.: Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Invent. Math. 201(2), 385–517 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Díaz L.J., Gorodetski A.: Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes. Ergodic Theory Dynam. Syst. 29(5), 1479–1513 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gorodetski, A., Ilyashenko, Yu.S., Kleptsyn, V.A., Nalsky, M.B.: Nonremovability of zero Lyapunov exponents.(Russian) Funktsional. Anal. i Prilozhen. 39(1), 27–38 [translation in Funct. Anal. Appl. 39(1), 21–30 (2005)]

  19. Guckenheimer J., Williams R.: Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50, 59–72 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hayashi S.: Connecting invariant manifolds and the solution of the C 1 stability and \({\Omega}\)-stability conjectures for flows. Ann. Math. 145(1), 81–137 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kleptsyn, V.A., Nalsky, M. B.: Stability of the existence of nonhyperbolic measures for C 1-diffeomorphisms. Funktsional. Anal. i Prilozhen. 41(4), 30–45 (2007) [translation in Funct. Anal. Appl. 41(4), 271–283 (2007)]

  22. Kupka I.: Contribution à la théorie des champs génériques. Contrib. Differ. Equations 2, 457–484 (1963)

    MathSciNet  MATH  Google Scholar 

  23. Leplaideur R., Oliveira K., Rios I.: Equilibrium states for partially hyperbolic horseshoes. Ergodic Theory Dynam. Syst. 31(1), 179–195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mañé R.: Contributions to stability conjecture. Topology 17(4), 383–396 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mañé R.: An ergodic closing lemma. Ann. Math. 116(3), 503–540 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Moreira C.G.: There are no C 1-stable intersections of regular Cantor sets. Acta Math. 206(2), 311–323 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Moreira, C.G., Silva, W.: On the geometry of horseshoes in higher dimensions. ar**v:1210.2623 (preprint).

  28. Moreira C.G., Yoccoz J.-C.: Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. Math. 154(1), 45–96 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Newhouse S.: Diffeomorphisms with infinitely many sinks. Topology 13, 9–18 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Palis J.: A global view of dynamics and a conjecture on the denseness and finitude of attractors. Astérisque 261, 335–347 (2000)

    MathSciNet  MATH  Google Scholar 

  31. Pesin Y.: Families of invariant manifolds for diffeomorphisms with non-zero characteristic Lyapunov exponents. Izvestiya Akademii Nauk SSSR 40, 1332–1379 (1976)

    MathSciNet  Google Scholar 

  32. Pujals E.R., Sambarino M.: Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. Math. 151(3), 961–1023 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shub M.: Topological transitive diffeomorphisms on T 4. Lecture Notes Math. 206, 39–40 (1971)

    Article  Google Scholar 

  34. Shub M., Wilkinson A.: Pathological foliations and removable zero exponents. Invent. Math. 139(3), 495–508 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Smale S.: Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Norm. Sup. Pisa 17, 97–116 (1963)

    MathSciNet  MATH  Google Scholar 

  36. Walters P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer, New York (1981)

    Google Scholar 

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Jairo Bochi

  2. Institut de Mathématiques de Bourgogne, Dijon, France

    Christian Bonatti

  3. Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil

    Lorenzo J. Díaz

Authors
  1. Jairo Bochi
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  2. Christian Bonatti
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  3. Lorenzo J. Díaz
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Correspondence to Jairo Bochi.

Additional information

Communicated by K. Khanin

The authors received support from CNPq, FAPERJ, PRONEX, and Ciência Sem Fronteiras CAPES (Brazil); Balzan–Palis Project, ANR (France); Fondecyt and PIA-Conicyt (Anillo ACT 1103 DySyRF) (Chile), and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS). The authors acknowledge the kind hospitality of Institut de Mathématiques de Bourgogne and Departamento de Matemática PUC-Rio while preparing this paper.

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Bochi, J., Bonatti, C. & Díaz, L.J. Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures. Commun. Math. Phys. 344, 751–795 (2016). https://doi.org/10.1007/s00220-016-2644-5

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  • DOI: https://doi.org/10.1007/s00220-016-2644-5

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