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

TL;DR

This paper extends the scale recurrence lemma to multidimensional conformal settings for Cantor sets in the complex plane and establishes a dimension formula for their images under certain functions.

Contribution

It introduces a multidimensional conformal version of the scale recurrence lemma and proves a dimension formula for images of product Cantor sets.

Findings

Proved a multidimensional conformal scale recurrence lemma.

Established a dimension formula for images of Cantor set products.

Applied the results to determine Hausdorff dimension of transformed Cantor sets.

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 \[HD(h(K_1\times K_2 \times ...\times K_n))=\min \{l,HD(K_1)+...+HD(K_n)\}.\]