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.