Stable Intersections of Conformal Cantor Sets

TL;DR

This paper studies stable intersections of conformal Cantor sets, linking their geometric properties to dynamical systems and automorphisms of complex two-space, providing criteria for stability and applications to Newhouse regions.

Contribution

It introduces a new class of conformal Cantor sets, relates them to complex dynamical structures, and establishes criteria for their stable intersections with applications to automorphisms of $\\C^2$.

Findings

Established a criterion for stable intersection of conformal Cantor sets.

Connected stable intersections to the existence of Newhouse regions in $Aut(\C^2)$.

Provided estimates on the 'thickness' needed for stable intersections.

Abstract

We investigate stable intersections of conformal Cantor sets and their consequences to dynamical systems. First we define this type of Cantor set and relate it to horseshoes appearing in automorphisms of $\C^2$. Then we study limit geometries, objects related to the asymptotic shape of the Cantor sets, to obtain a criterion that guarantees stable intersection between some configurations. Finally we show that the Buzzard construction of a Newhouse region on $Aut(\C^2)$ can be seem as a case of stable intersection of Cantor sets in our sense and give some (not optimal) estimative on how \say{thick} those sets have to be.