Abundance of $C^{1}$-robust homoclinic tangencies

Abstract

A diffeomorphism f has a C 1 -robust homoclinic tangency if there is a C 1 -neighbourhood U of f such that every diffeomorphism in g ∈ U has a hyperbolic setg, depending contin- uously on g, such that the stable and unstable manifolds ofg have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a lo- cal 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.