Meyer sets, topological eigenvalues, and Cantor fiber bundles

Abstract

We introduce two new characterizations of Meyer sets. A repetitive Delone set in ${\mathbb{R}}^d$ 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.

minor errors corrected, references added. To appear in the Journal of the LMS

References in corpus (3)

• On the Dynamics of G-Solenoids. Applications to Delone Sets
• On the Bragg Diffraction Spectra of a Meyer Set
• Fusion: a general framework for hierarchical tilings of $R^d$

Cited by in corpus (1)

• Fusion: a general framework for hierarchical tilings of $R^d$