Stable intersections of conformal Cantor sets

Abstract

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 $\mathbb {C}^2$ . Then we study limit geometries, that is, 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 $\mathrm{Aut}(\mathbb {C}^2)$ can be seen as a case of stable intersection of Cantor sets in our sense and give some (not optimal) estimate on how ‘thick’ those sets have to be.