Homoclinic tangencies leading to robust heterodimensional cycles

Abstract

We consider C r (r 1) diffeomorphisms f defined on manifolds of dimension 3 with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism f can be C r approximated by diffeomorphisms with C 1 robust heterodimensional cycles. As an application, we show that the classic Simon-Asaoka's examples of diffeomorphisms with C 1 robust homoclinic tangencies also display C 1 robust heterodimensional cycles. In a second application, we consider homoclinic tangencies associated to hyperbolic sets. When the entropy of these sets is large enough we obtain C 1 robust cycles after C 1 perturbations.

Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

“…Remark 1. The results of Theorem 1 and Corollary 1 below are also achieved in [4] with an additional assumption that the homoclinic tangencies are connected to pre-existing blenders.…”