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 [dim H (C)]+1 , where [dimH(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 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-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.

Spectral triples and characterization of aperiodic order

“…In [28] the abscissa of convergence was proven to be a fractal dimension of the space (namely, the upper box dimension). In [15], in the case of substitution tilings, it was identified with the exponent of complexity of the tiling, and further with the Hausdorff dimension of its discrete tiling space in [16]. Our result here relates the abscissa of convergence in general to the weak complexity exponent.…”

“…The following is relatively easy to prove, given that for a tiling #T A relation between the abscissa of convergence and the complexity exponent was first made in [15] where it is proved that s 0 = β if δ(r) = 1 r in the context of (primitive) substitution tilings of R d . In this context one can prove further that s 0 is the Hausdorff dimension of the discrete tiling space [16].…”

Wavelets and spectral triples for fractal representations of Cuntz algebras

“…It is known that the abscissa of convergence coincides with the upper Minkowski dimension of ∂B D ∼ = Λ ∞ D associated to the ultrametric d λ w [20, Theorem 2]. In the self-similar cases (when the weight is given as in Definition 4.5), the upper Minkowski dimension turns out to coincide with the Hausdorff dimension [16,Theorem 2.12]. In particular, when the scaling factor λ is just N, the Hausdorff dimensions of (Λ ∞ D , d w N D ) and S A coincide, where we equip S A with the metric induced by the Euclidean metric on [0, 1] 2 .…”

Fractal spectral triples on Kellendonk’sC∗-algebra of a substitution tiling

“…However, in the special case of tiling algebras, this spectral triple essentially measured the Euclidean distance between two tilings in the groupoid used to define the C * -algebra, and ignored the substitution system. Since Bellissard and Pearson's seminal result there have been a number of papers on spectral triples of tilings, see for example [13,14,17,19]. The survey article [11] explains these constructions and their relationship to one another.…”