2020/11/17 by Lorenzo J. Díaz, Díaz, Lorenzo J., Sebastián A. Pérez +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2011.08926
openalex publication_date 2020/11/17 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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 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\geqslant 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 Hénon-like family displaying blender-horseshoes. A crucial step is the analysis of the embeddings of these blender-horseshoes in a nondominated context.