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 $\mu =0$.

24 pages

References in corpus (1)

• Complex Horseshoes and the Dynamics of Mappings of Two Complex Variables