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

TL;DR

This paper demonstrates that for a family of complex automorphisms unfolding a homoclinic tangency, the parameters with persistent tangencies form a set of positive density, under certain stable intersection conditions of associated Cantor sets.

Contribution

It establishes the positive density of parameters with persistent tangencies in complex automorphisms, extending understanding of homoclinic bifurcations in $ ext{C}^2$.

Findings

Parameters with persistent tangencies have positive density at the bifurcation point.

Stable intersections of associated Cantor sets imply persistent tangencies.

Results apply to automorphisms unfolding generic homoclinic tangencies.

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 $\mu = 0$.