A Metric Characterisation of Repulsive Tilings

A tiling of R 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 R, with finite local complexity. From a spectral triple built on the discrete hull Ξ of T , and its Connes distance, we derive two metrics dsup and dinf on Ξ. We show that T is repulsive if and only if dsup and dinf are Lipschitz equivalent. This generalises previous works for subshifts by J. Kellendonk, D. Lenz, and the author.