A metric characterisation of repulsive tilings

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.