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 r{\geqslant } 2 , we exhibit a class of such diffeomorphisms whose heterodimensional cycles can be C r C^r stabilised and (simultaneously) approximated by diffeomorphisms with C r 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 C^r perturbations, r ⩾ 2 r\geqslant 2 , which are remarkably more difficult than C 1 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 Hénon-like family displaying blender-horseshoes. A crucial step is the analysis of the embeddings of these blender-horseshoes in a nondominated context.