Preprint A202/2003
Nonwandering sets with non-empty interior

Lorenzo Diaz | Abdenur, Flavio | Bonatti, Christian

**Keywords: **
Generic properties | homoclinic classes | nonwandering set

We study diffeomorphisms on a closed manifold whose nonwandering sets have non-empty interior and conjecture that $C^1$-generic diffeomorphisms whose nonwandering sets have non-empty interior are transitive. We first prove this conjecture in three cases: hyperbolic, partially hyperbolic with two hyperbolic bundles, and tame diffeomorphisms (in the first case, the conjecture is a folklore result, and in the second it follows by adapting the proof in [B]).
We also study this conjecture without global assumptions and prove that, generically, a homoclinic class with non-empty interior is either the whole manifold or else accumulated by infinitely many different homoclinic classes. Finally, we prove that generically homoclinic classes and non-wandering sets with non-empty interior are weakly hyperbolic (existence of a dominated splitting or volume hyperbolicity).