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A metric characterisation of repulsive tilings | J. Savinien
; | Date: |
27 Oct 2014 | 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. | Source: | arXiv, 1410.7251 | Services: | Forum | Review | PDF | Favorites |
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