Pearson and Bellissard recently built a spectral triple - the data of Riemanian noncommutative geometry - for ultrametric Cantor sets. They derived a family of Laplace-Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular self-similar Cantor sets. We use Bratteli diagrams to encode the self-similarity, and Cuntz-Krieger algebras to implement it. We show that the abscissa of convergence of the zeta-function of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace-Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels.Comment: 29 pages, 4 figure
We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent α in terms of the ranks of certain groups which appear in the construction. We give bounds for α. These computations apply to some well known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over Q. We show that such a link cannot be established for more general tilings, and we present a counter-example in dimension one.
In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph Λ, via the infinite path space Λ ∞ of Λ. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of Λ ∞ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary -Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph Λ. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are -regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure , and show that is a rescaled version of the measure on Λ ∞ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of 2 (Λ ∞ , ) which was constructed by Farsi et al. 2.5 below. The space of infinite paths of a stationary -Bratteli diagram is often a Cantor set, enabling us to study its associated Pearson-Bellissard spectral triple. Indeed, if the matrices 1 , … , are the adjacency matrices for a -graph Λ, then the space of infinite paths in Λ is homeomorphic to the Cantor set (also called ). In other words, the Pearson-Bellissard spectral triples for stationary -Bratteli diagrams can also be viewed as spectral triples for higher-rank graphs.We then proceed to study, in Section 3, the geometrical information encoded by these spectral triples. Theorem 3.14 establishes that the Pearson-Bellissard spectral triple associated to ( Λ , ) is finitely summable, with dimension ∈ (0, 1). Section 3.3 focuses on the Dixmier traces of the spectral triples, and establishes both an integral formula for the Dixmier trace (Theorems 3.23 and 3.28) and a concrete expression for the measure induced by the Dixmier trace (Theorem 3.26). These computations also reveal that the ultrametric Cantor sets ( Λ , ) are -regular in the sense of [59, Definition 11]. Other settings in the literature in which spectral triples on Cantor sets admit an integral formula for the Dixmier trace include [13,47,17,14].In full generality, Dixmier traces are defined on the Dixmier-Macaev (also called Lorentz) ideal 1,∞ ⊆ () inside the compact operators and are computed using a generalized limit (roughly speaking, a linear functional that lies between lim sup and lim inf). Although the theory of Dixmier traces can be quite intricate, many of the computations simplify substantially in our setting, and so our treatment of the general theory will be brief; we refer the interested reader to the extensive literature on Dixmier traces and other singular traces (cf. [19, 55, 54, 1...
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