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 locally periodic patterns. We consider a non-periodic, repetitive tiling T of \({\mathbb {R}}^d\), with finite local complexity. From a spectral triple built on the discrete hull \(\varXi \) of T and its Connes distance, we derive two metrics \(d_{\mathrm{sup}}\) and \(d_{\mathrm{inf}}\) on \(\varXi \). We show that T is repulsive if and only if \(d_{\mathrm{sup}}\) and \(d_{\mathrm{inf}}\) are Lipschitz equivalent. This generalises the previous works on subshifts by J. Kellendonk, D. Lenz, and the author.