Heteroclinic Cycles and Homoclinic Closures for Generic Diffeomorphisms

We prove some C ¹ generic results about orbit-connecting, in particular about heteroclinic cycles and homoclinic closures. As a consequence we obtain a three-ways C ¹ density theorem: Diffeomorphisms with either infinitely many weakly transitive components or a heterodimensional cycle are C ¹ dense in the complement of the C ¹ closure of Axiom A and no-cycle diffeomorphisms.