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Transverse Laplacians for Substitution Tilings

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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.

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Authors and Affiliations

  1. Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622, Villeurbanne Cedex, France

    Antoine Julien & Jean Savinien

  2. SFB 701, Universität Bielefeld, Bielefeld, Germany

    Jean Savinien

Authors
  1. Antoine Julien
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  2. Jean Savinien
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Correspondence to Jean Savinien.

Additional information

Communicated by A. Connes

Work supported by the NSF grants no. DMS-0300398 and no. DMS-0600956.

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Julien, A., Savinien, J. Transverse Laplacians for Substitution Tilings. Commun. Math. Phys. 301, 285–318 (2011). https://doi.org/10.1007/s00220-010-1150-4

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