Generic bi-Lyapunov stable homoclinic classes

Abstract

We study, for $C^1$ generic diffeomorphisms, homoclinic classes which are Lyapunov stable both for backward and forward iterations. We prove they must admit a dominated splitting and show that under some hypothesis they must be the whole manifold. As a consequence of our results we also prove that in dimension 2 the class must be the whole manifold and in dimension 3, these classes must have nonempty interior. Many results on Lyapunov stable homoclinic classes for $C^1$-generic diffeomorphisms are also deduced.

26 pages. This version includes an appendix that will not appear in the published version

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• A Franks' lemma that preserves invariant manifolds
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Cited by in corpus (9)

• Partial hyperbolicity and attracting regions in 3-dimensional manifolds
• A Franks' lemma that preserves invariant manifolds
• A C1 generic condition for existence of symbolic extensions of volume preserving diffeomorphisms
• Hyperbolicity versus weak periodic orbits inside homoclinic classes
• Dirac physical measures on saddle-type fixed points
• Local perturbations of conservative $C^1$-diffeomorphisms
• An Isotopic Perturbation Lemma Along Periodic Orbits
• Partially Hyperbolic Sets with a Dynamically Minimal Invariant Lamination
• On the dominated splitting of Lyapunov stable aperiodic classes