Embedding of self-similar ultrametric Cantor sets

TL;DR

This paper investigates the geometric and spectral properties of self-similar ultrametric Cantor sets derived from stationary Bratteli diagrams, establishing bi-Lipschitz embeddings and computing their Hausdorff dimensions.

Contribution

It provides explicit calculations of Hausdorff dimensions, demonstrates bi-Lipschitz embeddability in Euclidean spaces, and links the spectral properties of these sets to noncommutative geometry.

Findings

Cantor sets are bi-Lipschitz embeddable in R^(d+1)

Hausdorff dimension equals the abscissa of convergence of a related zeta-function

The spectrum of a Laplacian on the set is characterized as the omega-spectrum

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+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.