Abstract
Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets in 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 of the thicknesses of two Cantor sets K K and L L is larger than 1, then K K and L 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.
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