On maximal transitive sets of generic diffeomorphisms

Abstract. – We construct locally generic C¹-diffeomorphisms of 3-manifolds with maximal transitive Cantor sets without periodic points. The locally generic diffeomorphisms constructed also exhibit strongly pathological features generalizing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). Two of these features are: coexistence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of infinitely many nontrivial (nonhyperbolic) homoclinic classes.¶We prove that these phenomena are associated to the existence of a homoclinic class H(P,f) with two specific properties:¶– in a C¹-robust way, the homoclinic class H(P,f) does not admit any dominated splitting,¶– there is a periodic point P^′ homoclinically related to P such that the Jacobians of P^′ and P are greater than and less than one, respectively.