A metric characterisation of repulsive tilings

TL;DR

This paper characterizes repulsive tilings, a key feature of aperiodic order, using spectral triples and Connes distances, establishing a Lipschitz equivalence criterion for repulsiveness.

Contribution

It introduces a novel metric-based characterization of repulsive tilings using spectral triples and extends previous subshift results to higher-dimensional tilings.

Findings

Repulsive tilings are characterized by Lipschitz equivalence of two derived metrics.

The approach generalizes previous subshift characterizations to $ ext{R}^d$ tilings.

Provides a new link between spectral geometry and aperiodic order.

Abstract

A tiling of $\mathbb{R}^d$ is repulsive if no $r$-patch can repeat arbitrarily close to itself, relative to $r$. This is a characteristic property of aperiodic order, for a non repulsive tiling has arbitrarily large local periodic patterns. We consider an aperiodic, repetitive tiling $T$ of $\mathbb{R}^d$, with finite local complexity. From a spectral triple built on the discrete hull $\Xi$ of $T$, and its Connes distance, we derive two metrics $d_{sup}$ and $d_{inf}$ on $\Xi$. We show that $T$ is repulsive if and only if $d_{sup}$ and $d_{inf}$ are Lipschitz equivalent. This generalises previous works for subshifts by J. Kellendonk, D. Lenz, and the author.