超度量Cantor集在欧几里得空间中的双Lipshitz嵌入
Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces
摘要：
一个超度量康托尔集可以被看作是一个有根的加权树的边界，称为Michon树。Assouad维度的概念被重新解释为Michon树。超度量康托尔集的Assouad维数是有限的当且仅当该空间是双Lipschitz可嵌入有限维欧几里得空间的。这个结果，由于Assouad和完善Luukkainen-Movahedi-Lankarani是重新证明的Michon树形式主义。它适用于回答嵌入问题的一些空间，可以被看作是自然的边界树：线性重复子移位，Sturmian子移位，和边界的Galton-Watson树与随机权重。其中一些给出了具有有限豪斯多夫维数的不可嵌入空间的例子。
英文摘要：
An ultrametric Cantor set can be seen as the boundary of a rooted weighted tree called the Michon tree. The notion of Assouad dimension is re-interpreted as seen on the Michon tree. The Assouad dimension of an ultrametric Cantor set is finite if and only if the space is bi-Lipschitz embeddable in a finite dimensional Euclidean space. This result, due to Assouad and refined by Luukkainen--Movahedi-Lankarani is re-proved in the Michon tree formalism. It is applied to answer the embedding question for some spaces which can be seen naturally as boundary of trees: linearly repetitive subshifts, Sturmian subshifts, and the boundary of Galton--Watson trees with random weights. Some of these give examples of nonembeddable spaces with finite Hausdorff dimension.
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