Meyer sets, topological eigenvalues, and Cantor fiber bundles

We introduce two new characterizations of Meyer sets. A repetitive Delone set in $\R^d$ with finite local complexity is topologically conjugate to a Meyer set if and only if it has $d$ linearly independent topological eigenvalues, which is if and only if it is topologically conjugate to a bundle over a $d$-torus with totally disconnected compact fiber and expansive canonical action. "Conjugate to" is a non-trivial condition, as we show that there exist sets that are topologically conjugate to Meyer sets but are not themselves Meyer. We also exhibit a diffractive set that is not Meyer, answering in the negative a question posed by Lagarias, and exhibit a Meyer set for which the measurable and topological eigenvalues are different.Comment: minor errors corrected, references added. To appear in the Journal of the LM

Equicontinuous factors, proximality and Ellis semigroup for Delone sets

We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. We consider in particular, the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.Comment: 65 page

Bragg spectrum and Gap Labelling of aperiodic solids

The diffraction spectrum of an aperiodic solid is related to the group of eigenvalues of the dynamical system associated with the solid. Those eigenvalues with continuous eigenfunctions constitute the topological Bragg spectrum. We relate the topological Bragg spectrum to the gap-labelling group, which is the group of possible gap labels for the spectrum of a Schr\"odinger operator describing the electronic motion in the solid