Homoclinic tangencies leading to robust heterodimensional cycles

Abstract

We consider \(C^r\) (\(r\geqslant 1\)) diffeomorphisms f defined on manifolds of dimension \(\geqslant 3\) 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.