Abundance of C-robust homoclinic tangencies

A diffeomorphism f has a C-robust homoclinic tangency if there is a C-neighbourhood U of f such that every diffeomorphism in g ∈ U has a hyperbolic set Λg, depending continuously on g, such that the stable and unstable manifolds of Λg 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-robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of C-generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display C-robust homoclinic tangencies. keywords: chain recurrence set, dominated splitting, heterodimensional cycle, homoclinic class, homoclinic tangency, hyperbolic set. MSC 2000: 37C05, 37C20, 37C25, 37C29, 37C70.