Partial hyperbolicity and homoclinic tangencies
We show that any diffeomorphism of a compact manifold can be approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.
Periodic orbits and chain-transitive sets of C1-diffeomorphisms
We prove that the chain-transitive sets of C1-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology...
Properties of dynamical systems with the asymptotic average shadowing property
This article investigates under what conditions nontransitivity can coexist with the asymptotic average shadowing property. We show that there is a large class of maps satisfying both conditions simultaneously and that it is possible to find such examples even among maps on a compact interval. We also study the limit shadowing property and its relation to the asymptotic average shadowing property.