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Article Dans Une Revue Communications in Mathematical Physics Année : 2005

Quasicrystals and almost periodicity

Résumé

We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let U denote the space of uniformly discrete subsets of the Euclidean space. Let A denote the elements of U that admit an autocorrelation measure. A Patterson set is an element of A such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that S in A is a Patterson set if and only if S is almost periodic in (U,T), where T denotes the Besicovitch topology. Let chi be an ergodic random element of U. We prove that chi 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.

Dates et versions

hal-00078642 , version 1 (06-06-2006)

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Jean-Baptiste Gouéré. Quasicrystals and almost periodicity. Communications in Mathematical Physics, 2005, 255(3), pp.655 - 681. ⟨10.1007/s00220-004-1271-8⟩. ⟨hal-00078642⟩
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