Embeddings of self-similar ultrametric Cantor sets

Abstract

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 i m H ( C ) ] + 1 , where [ d i m 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 R d + 1 . We also show that C is bi-Hölder embeddable in the real line. The image of C in R turns out to be the ω-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson–Bellissard via noncommutative geometry.