Spectral triples from stationary Bratteli diagrams

TL;DR

This paper develops spectral triples for stationary Bratteli diagrams' path spaces, analyzing their zeta functions, heat kernels, and Dirichlet forms, with applications to Pisot substitution tiling spaces and their dynamical systems.

Contribution

It introduces a method to construct spectral triples on stationary Bratteli diagrams and explores Dirichlet forms in the context of tiling spaces, linking them to elliptic operators.

Findings

Constructed spectral triples for stationary Bratteli diagrams.

Identified two types of Dirichlet forms in tiling spaces.

Connected Dirichlet forms to elliptic differential operators.

Abstract

We construct spectral triples for path spaces of stationary Bratteli diagrams and study their associated mathematical objects, in particular their zeta function, their heat kernel expansion and their Dirichlet forms. One of the main difficulties to properly define a Dirichlet form concerns its domain. We address this question in particular in the context of Pisot substitution tiling spaces for which we find two types of Dirichlet forms: one of transversal type, and one of longitudinal type. Here the eigenfunctions under the translation action can serve as a good core for a non-trivial Dirichlet form. We find that the infinitesimal generators can be interpreted as elliptic differential operators on the maximal equicontinuous factor of the tiling dynamical system.