Scale recurrence lemma and dimension formula for Cantor sets in the complex plane

Pub Date : 2024-03-25 DOI:10.1017/etds.2024.15

CARLOS GUSTAVO T. DE A. MOREIRA, ALEX MAURICIO ZAMUDIO ESPINOSA

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Abstract

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:

$$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

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复平面中的康托集合的尺度递推稃和维度公式

我们证明了莫雷拉和约科兹[大豪斯多夫维度规则康托集合的稳定交集。Ann. of Math. (2)154(1) (2001)，45-96]，用于复平面中的康托集合。然后，我们利用这一新的递推公设，结合莫雷拉在 [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps.Math.Z.303(2023), 3]中的观点，证明在对康托集 $K_1,\ldots ,K_n$ 和函数 $h：\mathbb {C}^{n}\to \mathbb {R}^{l}$, 下面的公式成立：$$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*}$$

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