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Quasicrystals and Almost Periodicity

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Abstract

We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let denote the space of uniformly discrete subsets of the Euclidean space. Let denote the elements of that admit an autocorrelation measure. A Patterson set is an element of such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that S ∈ is a Patterson set if and only if S is almost periodic in (,), where denotes the Besicovitch topology. Let χ be an ergodic random element of . We prove that χ is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum. As an illustration, we study deformed model sets.

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  1. LaPCS, Bâtiment B, Domaine de Gerland, Université Claude Bernard Lyon 1, 50, avenue Tony-Garnier, 69366, Lyon Cedex 07, France

    Jean-Baptiste Gouéré

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  1. Jean-Baptiste Gouéré
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Communicated by J.L. Lebowitz

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Gouéré, JB. Quasicrystals and Almost Periodicity. Commun. Math. Phys. 255, 655–681 (2005). https://doi.org/10.1007/s00220-004-1271-8

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