A metric characterisation of repulsive tilings

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 $Ξ$ of $T$, and its Connes distance, we derive two metrics $d_{sup}$ and $d_{inf}$ on $Ξ$. 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.

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