Skip to main content

Codimension one generic homoclinic classes with interior

Abstract

We study C 1-generic diffeomorphisms with a homoclinic class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.

This is a preview of subscription content, access via your institution.

References

  1. F. Abdenur, C. Bonatti, S. Crovisier and L. Diaz. Generic diffeomorphisms on compact surfaces. Fund. Math., 187 (2005), 127–159.

    MATH  Article  MathSciNet  Google Scholar 

  2. F. Abdenur, C. Bonatti, S. Crovisier, L. Diaz and L. Wen. Periodic points and homoclinic classes. Ergodic Theory Dynam. Systems, 27 (2007), 1–22.

    MATH  Article  MathSciNet  Google Scholar 

  3. F. Abdenur, C. Bonatti and L. Diaz. Nonwandering sets with non empy interior. Nonlinearity, 17 (2004), 175–191.

    MATH  Article  MathSciNet  Google Scholar 

  4. C. Bonatti and S. Crovisier. Recurrence et Genericite. Inventiones Math., 158 (2004), 33–104.

    MATH  Article  MathSciNet  Google Scholar 

  5. C. Bonatti, L.J. Díaz and M. Viana. Dynamics Beyond Uniform Hyperbolicity. Springer-Verlag (2005).

  6. C.M. Carballo, C.A. Morales and M.J. Pacifico. Homoclinic classes for generic C 1 vector fields. Ergodic Theory Dynam. Systems, 23 (2003), 403–415.

    MATH  Article  MathSciNet  Google Scholar 

  7. S. Crovisier. Partial hyperbolicity far from homoclinic bifurcations. Preprint Arxiv (2008).

  8. J. Franks. Nessesary conditions for stability of diffeomorphisms. Transactions of the A.M.S., 158 (1971), 301–308.

    MATH  Article  MathSciNet  Google Scholar 

  9. N. Gourmelon. Addapted metrics for dominated splitting. Ergod. Th. and Dyn. Sys., 27 (2007), 1839–1849.

    MATH  MathSciNet  Google Scholar 

  10. N. Gourmelon. Generation of homoclinic tangencies by C 1 perturbations. Disc. and Cont. Dynamical Systems A, 26 (2010), 1–42.

    MATH  Article  MathSciNet  Google Scholar 

  11. M. Hirsch, C. Pugh and M. Shub. Invariant Manifolds. Springer Lecture Notes in Math., 583 (1977).

  12. P. Lessa and M. Sambarino. Invariant Manifolds for codimension one dominated splitting. Preprint.

  13. S. Liao. On the stability conjecture. Chinese Annals of Math., 1 (1980), 9–30.

    MATH  Google Scholar 

  14. R. Mañe. A proof of the C 1 stability conjecture. Publications del IHES (1987).

  15. E. Pujals. Some simple questions related to the Cr stability conjecture. Nonlinearity, 21 (2008).

  16. E.R. Pujals and M. Sambarino. Integrability on codimension one dominated splitting. Bull. Braz. Math. Soc., N.S., 38 (2007), 1–19.

    MATH  Article  MathSciNet  Google Scholar 

  17. M. Shub. Global Stability of Dynamical Systems. Springer-Verlag (1987).

  18. L. Wen. Homoclinic tangencies and dominated splitting. Nonlinearity (2002).

  19. L. Wen. The selecting lemma of Liao. Disc. and Cont. Dyn. Sys., 20 (2008), 159–175.

    MATH  Article  Google Scholar 

  20. J. Yang. Lyapunov stable homoclinic classes. Preprint Arxiv (2008).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rafael Potrie.

About this article

Cite this article

Potrie, R., Sambarino, M. Codimension one generic homoclinic classes with interior. Bull Braz Math Soc, New Series 41, 125–138 (2010). https://doi.org/10.1007/s00574-010-0006-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-010-0006-z

Keywords

  • Dominated splitting
  • C1 generic dynamics
  • homoclinic classes

Mathematical subject classification

  • 37D30
  • 37C20
  • 37C70