Scale Recurrence Lemma and Dimension Formula for Cantor Sets in the Complex Plane

Abstract

We will prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz \cite{MY} for Cantor sets in the complex plane. We then use this new recurrence lemma, together with the ideas in \cite{M}, to prove that under the right hypothesis for the Cantor sets $K_1,...,K_n$ and the function $h:{\mathbb{C}}^n\to {\mathbb{R}}^l$, the following formula holds \[HD(h(K_1\times K_2 \times ...\times K_n))=\min \{l,HD(K_1)+...+HD(K_n)\}.\]

References in corpus (2)

• Local entropy averages and projections of fractal measures
• Moreira's Theorem on the arithmetic sum of dynamically defined Cantor sets

Cited by in corpus (1)

• Stable intersections of regular conformal Cantor sets with large Hausdorff dimensions