Stable intersections of conformal regular Cantor sets with large Hausdorff dimensions

In this paper we prove that among pairs $K,K^{\prime }\subset C$ of conformal dynamically defined Cantor sets with sum of Hausdorff dimensions $HD\left(K\right)+HD\left(K^{\prime }\right)>2$, there is an open and dense subset of such pairs verifying $\text{int}(K^{\prime }-K)\ne \varnothing$. This is motivated by the work [11], 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.