Transverse Laplacians for Substitution Tilings

TL;DR

This paper extends the spectral triple framework to self-similar tiling spaces, providing explicit spectra, eigenvalue calculations, and heat kernel asymptotics for Laplace-Beltrami operators on these complex structures.

Contribution

It introduces a method to analyze Laplace-Beltrami operators on tiling spaces using Bratteli diagrams and Cuntz-Krieger algebras, with explicit spectral and asymptotic results.

Findings

Complete spectrum of Laplace-Beltrami operators determined

Explicit eigenvalue calculation method provided

Weyl asymptotics and heat kernel estimates obtained

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.