A Complex Gap Lemma

Abstract

Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets of the plane associated to a holomorphic IFS. Our main result is a complex version of Newhouse's Gap Lemma : we show that under some assumptions, if the product t(K)t(L) of the thicknesses of two Cantor sets K and L is larger than 1, then K and L have non empty intersection. Since in addition this thickness varies continuously, this gives a criterion to get a robust intersection between two Cantor sets in the plane.