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