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Communications in Mathematical Physics

, Volume 301, Issue 2, pp 285–318 | Cite as

Transverse Laplacians for Substitution Tilings

  • Antoine Julien
  • Jean Savinien
Article

Abstract

Pearson and Bellissard recently built a spectral triple – the data of Riemannian 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 ζ-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.

Keywords

Dirac Operator Dirichlet Form Beltrami Operator Spectral Triple Bratteli Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Université de Lyon, CNRS, Université Lyon 1, Institut Camille JordanVilleurbanne CedexFrance
  2. 2.SFB 701Universität BielefeldBielefeldGermany

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