Embeddings of self-similar ultrametric Cantor sets

作者：Julien Antoine; Savinien Jean^*

来源：Topology and Its Applications, 2011, 158(16): 2148-2157.

DOI：10.1016/j.topol.2011.07.009

• Self-similar Cantor sets
• Tiling spaces
• Ultrametric spaces
• Embeddings in Euclidean spaces

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 RIdimh(C)+1 where [dim(H)(C)] 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 sequence of refining 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 Rd+1.
We also show that C is bi-Holder 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.

• 出版日期2011-10-1