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29 March 2024
 
  » arxiv » 1410.7251

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A metric characterisation of repulsive tilings
J. Savinien ;
Date 27 Oct 2014
AbstractA 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.
Source arXiv, 1410.7251
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