On bi-Lyapunov stable homoclinic classes

  • A. Arbieto
  • B. Carvalho
  • W. Cordeiro
  • D. J. Obata
Article

Abstract

For a C1 generic diffeomorphism if a bi-Lyapunov stable homoclinic class is homogeneous then it does not have weak eigenvalues. Using this, we show that such homoclinic classes are hyperbolic if it has one of the following properties: shadowing, specification or limit shadowing.

Keywords

bi-Lyapunov stability hyperbolic generic properties homoclinic classes shadowing specification 

k]Mathematical subject classification

Primary: 37C20 Secondary: 37D20 

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References

  1. [1]
    F. Abdenur. Generic robustness of spectral decompositions. Ann.Sci.EcoleNorm. Sup. (4), 36(2) (2003), 213–224.MathSciNetMATHGoogle Scholar
  2. [2]
    F. Abdenur. Attractors of generic diffeomorphisms are persistent. Nonlinearity, 16(1) (2003), 301–311.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    F. Abdenur, C. Bonatti and L. Diaz. Non-wandering sets with non-emptyinteriors. Nonlinearity, 17(1) (2004), 175–191.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    F. Abdenur, C. Bonatti, S. Crovisier, L. Diaz and L. Wen. Periodic points and homoclinic classes. Ergodic Theory Dynam. Systems, 27(1) (2007), 1–22.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    F. Abdenur and L. Diaz. Pseudo-orbit shadowing in the C 1 topology. Discrete Contin. Dyn. Syst., 17(2) (2007), 223–245.MathSciNetMATHGoogle Scholar
  6. [6]
    C. Bonatti and S. Crovisier. Récurrence et généricité. Invent. Math., 158(1) (2004), 33–104.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    C. Bonatti, S. Crovisier, L. Diaz and N. Gourmelon. Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication. Preprint Arxiv.Google Scholar
  8. [8]
    C.M. Carballo, C.A. Morales and M.J. Pacifico. Homoclinic classes for generic C 1 vector fields. Ergodic Theory Dynam. Systems, 23(2) (2003), 403–415.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    J. Franks. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc., 158 (1971), 301–308.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    N. Gourmelon. AFrank’slemmathatpreservesinvariantmanifolds. Preprint Arxiv.Google Scholar
  11. [11]
    S. Gan and D. Yang. Expansive homoclinic classes. Nonlinearity, 22(4) (2009), 729–733.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    B. Hasselblatt and A. Katok. Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge (1995).Google Scholar
  13. [13]
    N. Haydn and D. Ruelle. Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification. Comm. Math. Phys., 148(1) (1992), 155–167.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    M. Hirsch, C. Pugh and M. Shub. Invariants manifolds. Lectures notes in Mathematics, 583 (1977), Springer-Verlag.Google Scholar
  15. [15]
    R. Mañé. An ergodic closing lemma. Annals of Math., 116 (1982), 503–540.MATHCrossRefGoogle Scholar
  16. [16]
    J. Palis. A global perspective for non-conservative dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire, 22(4) (2005), 485–507.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    J. Palis and W. de Melo. Geometric theory of dynamical systems. Anintroduction. Springer-Verlag, New York-Berlin, 1982. xii+198 pp.Google Scholar
  18. [18]
    S. Pilyugin. Shadowing in dynamical systems. Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. xviii+271 pp.MATHGoogle Scholar
  19. [19]
    R. Potrie. Generic bi-Lyapunov stable homoclinic classes. Nonlinearity, 23(7) (2010), 1631–1649.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    R. Potrie and M. Sambarino. Codimension one generic homoclinic classes with interior. Bull. Braz. Math. Soc. (N.S.), 41(1) (2010), 125–138.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    E. Pujals and M. Sambarino. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2), 151(3) (2000), 961–1023.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    K. Sakai, N. Sumi and K. Yamamoto. Diffeomorphisms satisfying the specification property. Proc. Amer. Math. Soc., 138(1) (2010), 315–321.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    M. Sambarino and J. Vieitez. On persistently expansive homoclinic classes. Discrete Contin. Dyn. Syst., 14 (2006), 465–481.MathSciNetMATHGoogle Scholar
  24. [24]
    K. Sigmund. Generic properties of invariant measures for Axiom A diffeomorphisms. Invent. Math., 11 (1970), 99–109.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 73 (1967), 747–817.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    X. Wen, S. Gan and L. Wen. C 1-stably shadowable chain components are hyperbolic. J. Differential Equations, 246(1) (2009), 340–357.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2013

Authors and Affiliations

  • A. Arbieto
    • 1
  • B. Carvalho
    • 1
  • W. Cordeiro
    • 1
  • D. J. Obata
    • 1
  1. 1.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de Janeiro, RJBrazil

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