2009/08/07 by Antoine Julien, Jean Savinien
Materials Science · Mathematics · #Advanced Operator Algebra Research #Laplace transform #Mathematical analysis #Mathematics #Metric space #Noncommutative geometry #Pure mathematics #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #Spectral geometry #Substitution (logic) #Ultrametric space #math.OA #msc:37B50 #msc:47C15
paper · pdf · doi:10.1007/s00220-010-1150-4
29 pages, 4 figures
arxiv created 2009/08/07 · openalex publication_date 2010/10/21 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.