Robust heterodimensional cycles and C1-generic dynamics

A diffeomorphism f has a heterodimensional cycle if there are (transitive) hyperbolic sets Lambda and Sigma having different indices (dimension of the unstable bundle) such that the unstable manifold of Lambda meets the stable one of Sigma and vice versa. This cycle has co-index 1 if index(Lambda) = index(Sigma) +/- 1. This cycle is robust if, for every g close to f, the continuations of Lambda and Sigma for g have a heterodimensional cycle. We prove that any co-index 1 heterodimensional cycle associated with a pair of hyperbolic saddles generates C-1-robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles. We also derive some consequences from this result for C-1-generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle.