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Hulls of aperiodic solids and gap labeling theorems. (English) Zbl 0972.52014
Baake, Michael (ed.) et al., Directions in mathematical quasicrystals. Providence, RI: AMS, American Mathematical Society. CRM Monogr. Ser. 13, 207-258 (2000).
Summary: We review the basic constructions liable to replace Bloch theory for aperiodic solids. Point sets describing atomic positions lead to the notion of the hull, a topological dynamical system with an action of the translation group. We establish that quantities like the hull, the diffraction measure or the electronic density of states are uniquely determined by the Gibbs state describing thermal equilibrium of the solid. We recall the construction of the corresponding noncommutative Brillouin zone for electrons or phonons. We describe its topology through its algebraic $$K$$-theory. The gap labeling theory is reviewed and completed by a general conjecture for the case of a transversally totally discontinuous hull and by results obtained for two-dimensional media.
For the entire collection see [Zbl 0955.00025].

##### MSC:
 52C23 Quasicrystals and aperiodic tilings in discrete geometry 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 19K14 $$K_0$$ as an ordered group, traces 82D25 Statistical mechanical studies of crystals 47C15 Linear operators in $$C^*$$- or von Neumann algebras 37A55 Dynamical systems and the theory of $$C^*$$-algebras