Spectral triples and aperiodic order

TL;DR

This paper constructs spectral triples for compact metric spaces, introduces a new metric, and explores their properties and implications for aperiodic order, including subshifts and tilings, with connections to complexity and Laplace operators.

Contribution

It develops a novel approach to spectral triples on compact metric spaces and characterizes high order properties of aperiodic structures through Lipschitz equivalence of metrics.

Findings

Lipschitz equivalence of metrics characterizes high order in subshifts and tilings.

For episturmian subshifts, equivalence holds iff the subshift is repulsive.

The zeta-function's abscissa relates to the complexity exponent of the structure.

Abstract

We construct spectral triples for compact metric spaces (X, d). This provides us with a new metric d_s on X. We study its relation with the original metric d. When X is a subshift space, or a discrete tiling space, and d satisfies certain bounds we advocate that the property of d_s and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove that d_s and d are Lipschitz equivalent if and only if the subshift is repulsive (or power free). For Sturmian subshifts this is equivalent to linear recurrence. For repetitive tilings we show that if their patches have equi-distributed frequencies then the two metrics are Lipschitz equivalent. Moreover, we study the zeta-function of the spectral triple and relate its abscissa of convergence to the complexity exponent of the subshift or the tiling. Finally, we derive Laplace operators from the spectral triples and compare our construction with that of Pearson and Bellissard.