Transverse Laplacians for Substitution Tilings

Abstract

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.

29 pages, 4 figures

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Cited by in corpus (10)

• Dynamical Systems on Spectral Metric Spaces
• Spectral triples for the Sierpinski Gasket
• Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces
• Fractal spectral triples on Kellendonk's $C^{\ast }$-algebra of a substitution tiling
• Spectral triples and aperiodic order
• Embedding of self-similar ultrametric Cantor sets
• A noncommutative Sierpinski Gasket
• Spectral triples from stationary Bratteli diagrams
• Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs
• Spaces of Random Plane Triangulations and the Density of States