自相似镶嵌的管公式和复维数
Tube formulas and complex dimensions of self-similar tilings
摘要：
利用第二作者在“Canonical self-affine tilings by iterated function systems”中构造的自相似tilings，定义了欧几里得空间中自相似集几何的生成函数.这个管状zeta函数编码与分形集的补相关的缩放和曲率特性，以及相关的映射系统。这使得人们可以获得自相似镶嵌的复杂尺寸作为管状zeta函数的极点，从而开发出自相似镶嵌的管公式，其中\$ \mathbb{R}^d $. The resulting power series in $ ε $ is a fractal extension of Steiner's classical tube formula for convex bodies $ K \ci \bRd $. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer $ i= 0，1，.，D-1美元，就像斯坦纳的一样。然而，我们的公式还包含每个复杂维度的项。这为“复杂层面”一词提供了进一步的理由。它还将分形弦理论的几个方面扩展到更高维度，并为第一作者和Machiel van Frankenhuijsen在“分形几何与复维度”中获得的分形弦管公式提供了新的见解。
英文摘要：
We use the self-similar tilings constructed by the second author in "Canonical self-affine tilings by iterated function systems" to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in \$\mathbb{R}^d$. The resulting power series in $ε$ is a fractal extension of Steiner's classical tube formula for convex bodies $K \ci \bRd$. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer $i=0,1,...,d-1$, just as Steiner's does. However, our formula also contains terms for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in "Fractal Geometry and Complex Dimensions" by the first author and Machiel van Frankenhuijsen.
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