ROBUST HETERODIMENSIONAL CYCLES AND $C^1$-GENERIC DYNAMICS

Abstract

A diffeomorphism f has a heterodimensional cycle if there are (transitive) hyperbolic sets � andhaving different indices (dimension of the unstable bundle) such that the unstable manifold ofmeets the stable one ofand vice-versa. This cycle has co-index one if index (�) = index (�) ± 1. This cycle is robust if, for every g close to f, the continuations ofandfor g have a heterodimensional cycle. We prove that any co-index one heterodimensional cycle associated to 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 di- mension). 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.