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≥slant 2, we exhibit a class of such diffeomorphisms whose heterodimensional cycles can be Cr stabilised and (simultaneously) approximated by diffeomorphisms with Cr robust homoclinic tangencies. The complexity of our nondominated setting with plenty of homoclinic and heteroclinic intersections is used to overcome the difficulty of performing Cr perturbations, r≥slant 2, which are remarkably more difficult than C1 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 H\'enon-like family displaying blender-horseshoes. A crucial step is the analysis of the embeddings of these blender-horseshoes in a nondominated context.