Abundance of $C^1$-robust homoclinic tangencies

TL;DR

This paper demonstrates the abundance of $C^1$-robust homoclinic tangencies in higher-dimensional manifolds using blender-horseshoes, revealing their generic presence in certain homoclinic classes.

Contribution

It introduces blender-horseshoes as a local mechanism to generate $C^1$-robust homoclinic tangencies in manifolds of dimension three or higher.

Findings

Blender-horseshoes generate robust tangencies.

Homoclinic classes with saddles of different indices often exhibit robust tangencies.

Such tangencies are generic in certain non-dominated homoclinic classes.

Abstract

A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a $C^1$-neighbourhood $\cU$ of $f$ such that every diffeomorphism in $g\in \cU$ has a hyperbolic set $\La_g$, depending continuously on $g$, such that the stable and unstable manifolds of $\La_g$ have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with $C^1$-robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of $C^1$-generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display $C^1$-robust homoclinic tangencies.