We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2)154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z.303 (2023), 3], to prove that under the right hypothesis for the Cantor sets $K_1,\ldots ,K_n$ and the function $h:\mathbb {C}^{n}\to \mathbb {R}^{l}$ , the following formula holds: $$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

中文翻译：

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复平面康托集的尺度递推引理和维数公式

我们证明了 Moreira 和 Yoccoz 的尺度递推引理的多维共形版本[正则 Cantor 集与大 Hausdorff 维数的稳定交集。安.数学。 (2)154(1) (2001), 45–96] 对于复平面上的康托集。然后，我们使用这个新的递归引理，以及莫雷拉在[通过可微实映射的正则康托集的笛卡尔积的图像的几何性质。数学。 Z。303 (2023), 3]，证明在康托集的正确假设下 $K_1,\l点,K_n$ 和函数 $h:\mathbb {C}^{n}\到 \mathbb {R}^{l}$ ，以下公式成立： $$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{对齐*} $$