A Complex Gap Lemma

TL;DR

This paper extends Newhouse's Gap Lemma to complex dynamical Cantor sets in the plane, providing a criterion for their robust intersection based on a new notion of thickness.

Contribution

It introduces a complex version of the Gap Lemma and a new notion of thickness for planar holomorphic IFS Cantor sets, establishing intersection criteria.

Findings

If the product of thicknesses exceeds 1, the Cantor sets intersect.

The thickness varies continuously, enabling robust intersection criteria.

Provides a new tool for analyzing intersections in complex dynamical systems.

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.