Stable intersections of Cantor sets and positive density of persistent tangencies for homoclinic bifurcations of automorphisms of $\mathbb{C}^2$

Abstract

Let $\{f_\mu\}_{\mu \in \mathbb{D}}$ be a family of automorphisms of $\mathbb{C}^2$ unfolding a generic homoclinic tangency associated to a fixed point $p$ belonging to a horseshoe. We prove that if the linearized versions of the Cantor sets representing the local intersections of the stable and unstable manifolds of $p$ with the horseshoe have stable intersections, then the set of parameters $\mu$ corresponding to automorphisms with persistent tangencies has positive density at $\mu = 0$.