Nontransverse heterodimensional cycles: stabilisation and robust tangencies

TL;DR

This paper demonstrates the stabilization of heterodimensional cycles and the approximation by diffeomorphisms with robust tangencies in three-dimensional systems, using complex nondominated bifurcation techniques and renormalization methods.

Contribution

It introduces a new framework for $C^r$ stabilization of heterodimensional cycles and robust tangencies in nondominated settings, extending previous results to higher smoothness classes.

Findings

Existence of $C^r$ stabilizations for heterodimensional cycles

Approximation by diffeomorphisms with robust homoclinic tangencies

Use of renormalization and blender-horseshoes in nondominated contexts

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 $r\geqslant 2$, 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, $r\geqslant 2$, 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 H\'enon-like family displaying blender-horseshoes. A crucial step is the analysis of the embeddings of these blender-horseshoes in a nondominated context.