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Relating diffraction and spectral data of aperiodic tilings: towards a Bloch theorem. (English) Zbl 1467.52029

Summary: The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by Čech cohomology of the tiling space) and the spectral properties (of Hamiltonians defined on such tilings) defined by \(K\)-theory, and to show their equivalence in dimensions \(\leq 3\). A theorem makes precise the conditions for this relationship to hold. It can be viewed as an extension of the “Bloch Theorem” to a large class of aperiodic tilings. The idea underlying this result is based on the relationship between cohomology and \(K\)-theory traces and their equivalence in low dimensions.

MSC:

52C23 Quasicrystals and aperiodic tilings in discrete geometry
19K14 \(K_0\) as an ordered group, traces
37B52 Tiling dynamics
58J42 Noncommutative global analysis, noncommutative residues
19K56 Index theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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