Homoclinic tangencies leading to robust heterodimensional cycles

TL;DR

This paper demonstrates that certain diffeomorphisms with homoclinic tangencies can be approximated by those with robust heterodimensional cycles, revealing new connections between tangencies and cycle robustness in higher-dimensional dynamics.

Contribution

It establishes conditions under which homoclinic tangencies lead to robust heterodimensional cycles in $C^r$ diffeomorphisms, extending previous results to generic settings and specific examples.

Findings

Homoclinic tangencies related to saddles can be approximated by diffeomorphisms with robust heterodimensional cycles.

Classic examples with $C^1$ robust tangencies also exhibit $C^1$ robust heterodimensional cycles.

High entropy hyperbolic sets with homoclinic tangencies can produce $C^1$ robust cycles after perturbations.

Abstract

We consider $C^r$ ($r\geqslant 1$) diffeomorphisms $f$ defined on manifolds of dimension $\geqslant 3$ with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism $f$ can be {$C^r$} approximated by diffeomorphisms with {$C^1$} robust heterodimensional cycles. As an application, we show that the classic Simon-Asaoka's examples of diffeomorphisms with $C^1$ robust homoclinic tangencies also display {$C^1$} robust heterodimensional cycles. In a second application, we consider homoclinic tangencies associated to hyperbolic sets. When the entropy of these sets is large enough we obtain $C^1$ robust cycles after $C^1$ perturbations.