Cantor sets in higher dimension I: Criterion for stable intersections

TL;DR

This paper extends the criterion for stable intersections of Cantor sets from one dimension to higher dimensions, providing a method to construct explicit examples in various settings.

Contribution

It introduces a new criterion for stable intersections of higher-dimensional Cantor sets under mild conditions, generalizing previous one-dimensional results.

Findings

Established a higher-dimensional stable intersection criterion.

Developed a method for constructing explicit stably intersecting Cantor sets.

Applied the criterion to both real and complex cases.

Abstract

We study the geometry of dynamically defined Cantor sets in arbitrary dimensions, introducing a criterion for $\mathcal{C}^{1+\alpha}$ stable intersections of such Cantor sets, under a mild bunching condition. This condition is naturally satisfied for perturbations of conformal Cantor sets and, in particular, always holds in dimension one. Our work extends the celebrated recurrent compact set criterion of Moreira and Yoccoz for stable intersection of Cantor sets in the real line to higher-dimensional spaces. Based on this criterion, we develop a method for constructing explicit examples of stably intersecting Cantor sets in any dimension. This construction operates in the most fragile and critical regimes, where the Hausdorff dimension of one of the Cantor sets is arbitrarily small and both Cantor sets are nearly homothetical. All results and examples are provided in both real and complex settings.