Generic bi-Lyapunov stable homoclinic classes

Details Generic bi-Lyapunov stable homoclinic classes Generic bi-Lyapunov stable homoclinic classes

2009-03-24

arxiv.org/abs/0903.4090v4

Mathematics

Dynamical Systems

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

Scientific paper

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.

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