Non-wandering sets with non-empty interiors

We study diffeomorphisms of a closed connected manifold whose non-wandering set has a non-empty interior and conjecture that C-1-generic diffeomorphisms whose non-wandering set has a non-empty interior are transitive. We prove this conjecture in three cases: hyperbolic diffeomorphisms, partially hyperbolic diffeomorphisms with two hyperbolic bundles, and tame diffeomorphisms (in the first case, the conjecture is folklore; in the second one, it follows by adapting the proof in Brin (1975 Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature Funct. Anal. Appl. 9 9-19)). We 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 interiors are weakly hyperbolic (the existence of a dominated or a volume hyperbolic splitting).