自相似超度量Cantor集的嵌入
Embedding of self-similar ultrametric Cantor sets
摘要：
研究了由定常Bratteli图产生的自相似超度量Cantor集。证明了这样的康托尔集C是双Lipschitz可嵌入R^（d+1）的，其中d表示其豪斯多夫维数的整数部分.我们计算这个豪斯多夫维数明确和表明，它是一个zeta函数的收敛与自然神经覆盖的C（给出的Bratteli图）的横坐标。作为推论，我们证明了R^d的（本原）置换镶嵌的断面在R^（d+1）中是双Lipschitz可嵌入的.我们还证明了C是bi-Hoelder可嵌入实数线的。C在R中的像是C上拉普拉斯算子的Ω谱（特征值集的极限点），该拉普拉斯算子是由皮尔逊-贝利斯萨德通过非交换几何引入的。
英文摘要：
We study self-similar ultrametric Cantor sets arising from stationary Bratteli diagrams. We prove that such a Cantor set C is bi-Lipschitz embeddable in R^(d+1), where d denotes the integer part of its Hausdorff dimension. We compute this Hausdorff dimension explicitly and show that it is the abscissa of convergence of a zeta-function associated with a natural nerve of coverings of C (given by the Bratteli diagram). As a corollary we prove that the transversal of a (primitive) substitution tiling of R^d is bi-Lipschitz embeddable in R^(d+1) . We also show that C is bi-Hoelder embeddable in the real line. The image of C in R turns out to be the omega-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson-Bellissard via noncommutative geometry.
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