Abstract
For a C 1 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Abdenur. Generic robustness of spectral decompositions. Ann.Sci.EcoleNorm. Sup. (4), 36(2) (2003), 213–224.
F. Abdenur. Attractors of generic diffeomorphisms are persistent. Nonlinearity, 16(1) (2003), 301–311.
F. Abdenur, C. Bonatti and L. Diaz. Non-wandering sets with non-emptyinteriors. Nonlinearity, 17(1) (2004), 175–191.
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.
F. Abdenur and L. Diaz. Pseudo-orbit shadowing in the C 1 topology. Discrete Contin. Dyn. Syst., 17(2) (2007), 223–245.
C. Bonatti and S. Crovisier. Récurrence et généricité. Invent. Math., 158(1) (2004), 33–104.
C. Bonatti, S. Crovisier, L. Diaz and N. Gourmelon. Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication. Preprint Arxiv.
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.
J. Franks. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc., 158 (1971), 301–308.
N. Gourmelon. AFrank’slemmathatpreservesinvariantmanifolds. Preprint Arxiv.
S. Gan and D. Yang. Expansive homoclinic classes. Nonlinearity, 22(4) (2009), 729–733.
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).
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.
M. Hirsch, C. Pugh and M. Shub. Invariants manifolds. Lectures notes in Mathematics, 583 (1977), Springer-Verlag.
R. Mañé. An ergodic closing lemma. Annals of Math., 116 (1982), 503–540.
J. Palis. A global perspective for non-conservative dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire, 22(4) (2005), 485–507.
J. Palis and W. de Melo. Geometric theory of dynamical systems. Anintroduction. Springer-Verlag, New York-Berlin, 1982. xii+198 pp.
S. Pilyugin. Shadowing in dynamical systems. Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. xviii+271 pp.
R. Potrie. Generic bi-Lyapunov stable homoclinic classes. Nonlinearity, 23(7) (2010), 1631–1649.
R. Potrie and M. Sambarino. Codimension one generic homoclinic classes with interior. Bull. Braz. Math. Soc. (N.S.), 41(1) (2010), 125–138.
E. Pujals and M. Sambarino. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2), 151(3) (2000), 961–1023.
K. Sakai, N. Sumi and K. Yamamoto. Diffeomorphisms satisfying the specification property. Proc. Amer. Math. Soc., 138(1) (2010), 315–321.
M. Sambarino and J. Vieitez. On persistently expansive homoclinic classes. Discrete Contin. Dyn. Syst., 14 (2006), 465–481.
K. Sigmund. Generic properties of invariant measures for Axiom A diffeomorphisms. Invent. Math., 11 (1970), 99–109.
S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 73 (1967), 747–817.
X. Wen, S. Gan and L. Wen. C 1-stably shadowable chain components are hyperbolic. J. Differential Equations, 246(1) (2009), 340–357.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Arbieto, A., Carvalho, B., Cordeiro, W. et al. On bi-Lyapunov stable homoclinic classes. Bull Braz Math Soc, New Series 44, 105–127 (2013). https://doi.org/10.1007/s00574-013-0005-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-013-0005-y