On the existence of non-trivial homoclinic classes

We show that, for Cl-generic diffeornorphisms, every chain recurrent class C that has a partially hyperbolic splitting E(s) circle plus E(c) circle plus E(u) with dim E(c) = 1 either is an isolated hyperbolic periodic orbit, or is accumulated by non-trivial homoclinic classes. We also prove that, for C(1) -generic diffeomorphisms, any chain recurrent class that has a dorninated splitting E circle plus F with dim(E) = 1 either is a homoclinic class, or the bundle E is uniformly contracting. As a corollary we prove in dimension three a conjecture of Palis, which announces that any C(1) -generic diffeomorphism is either Morse-Smale, or has a non-trivial homoclinic class.Mathematics, AppliedMathematicsSCI(E)9ARTICLE1473-15082