Stable intersections of regular conformal Cantor sets with large Hausdorff dimensions

TL;DR

This paper extends the understanding of intersections of conformal Cantor sets in the complex plane, showing that for large Hausdorff dimensions, most pairs have non-empty interior of their difference set.

Contribution

It adapts and generalizes previous real-line results to conformal Cantor sets in the complex plane, introducing new concepts and detailed analysis.

Findings

Open and dense subset of pairs with non-empty difference interior

Extension of real-line results to complex conformal sets

Methodology applicable to complex dynamical systems

Abstract

In this paper we prove that among pairs $K,\,K' \subset \mathbb{C}$ of conformal dynamically defined Cantor sets with sum of Hausdorff dimensions $HD(K)+HD(K')>2$, there is an open and dense subset of such pairs verifying $\text{int}(K-K')\neq \emptyset$. This is motivated by the work \cite{MY}, where Moreira and Yoccoz proved a similar statement for dynamically defined Cantor sets in the real line. Here we adapt their argument to the context of conformal Cantor sets in the complex plane, this requires the introduction of several new concepts and a more detailed analysis in some parts of the argument.