Nontransverse heterodimensional cycles: stabilisation and robust tangencies

Abstract

We consider three-dimensional diffeomorphisms having simultaneously heterodimensional cycles and heterodimensional tangencies associated to saddle-foci. These cycles lead to a completely nondominated bifurcation setting. For every r 2, we exhibit a class of such diffeomorphisms whose heterodimensional cycles can be C r stabilised and (simultaneously) approximated by diffeomorphisms with C r robust homoclinic tangencies. The complexity of our nondominated setting with plenty of homoclinic and heteroclinic intersections is used to overcome the difficulty of performing C r perturbations, r 2, which are remarkably more difficult than C 1 ones. Our proof is reminiscent of the Palis-Takens' approach to get surface diffeomorphisms with infinitely many sinks (Newhouse phenomenon) in the unfolding of homoclinic tangencies of surface diffeomorphisms. This proof involves a scheme of renormalisation along nontransverse heteroclinic orbits converging to a center-unstable Hénonlike family displaying blender-horseshoes. A crucial step is the analysis of the embeddings of these blender-horseshoes in a nondominated context.

Homoclinic tangencies leading to robust heterodimensional cycles

“…Similarly, in [5] it is proved that both types of robust cycles can be C 1 approximated by diffeomorphisms with heterodimensional cycles associated to saddles with nonreal multipliers. A nondominated C r setting, r 2, with simultaneous occurrence of robust homoclinic tangencies and robust heterodimensional cycles was explored in [25].…”