Fractal spectral triples on Kellendonk's $C^*$-algebra of a substitution tiling

TL;DR

This paper introduces a new class of noncommutative spectral triples based on fractal trees on substitution tilings, providing a self-similar geodesic distance framework that respects the hierarchical structure of the tilings.

Contribution

It constructs fractal spectral triples on Kellendonk's $C^*$-algebra using fractal trees, defining a self-similar geodesic distance via Perron-Frobenius theory, and demonstrates their properties with the Penrose tiling.

Findings

Spectral triples are $ heta$-summable.

Distance is self-similar and respects substitution hierarchy.

Explicit construction on Penrose tiling.

Abstract

We introduce a new class of noncommutative spectral triples on Kellendonk's $C^*$-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is $\theta$-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.