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

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 $K_1,\mathrm{.}\mathrm{.}\mathrm{.},K_n$ and the function $h:{\mathbb{C}}^n\to {\mathbb{R}}^l$, the following formula holds $$HD(h(K_1\times K_2\times \mathrm{.}\mathrm{.}\mathrm{.}\times K_n))=\min \{l,HD(K_1)+\mathrm{.}\mathrm{.}\mathrm{.}+HD(K_n)\}\mathrm{.}$$