Transverse Laplacians for Substitution Tilings

Antoine Julien (1) , Jean Savinien (1)

1 ICJ - Institut Camille Jordan

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.

Domaines: Algèbres d'opérateurs [math.OA]

Antoine Julien, Jean Savinien. Transverse Laplacians for Substitution Tilings. 2010. ⟨hal-00529788⟩