Meyer sets, topological eigenvalues, and Cantor fiber bundles | Zendy

Kellendonk Johannes | Zendy; Sadun Lorenzo | Zendy

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Meyer sets, topological eigenvalues, and Cantor fiber bundles

Author(s) -

Kellendonk Johannes,

Sadun Lorenzo

Publication year - 2014

Publication title -

journal of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.441

H-Index - 62

eISSN - 1469-7750

pISSN - 0024-6107

DOI - 10.1112/jlms/jdt062

Subject(s) - mathematics , topological conjugacy , fiber bundle , conjugate , torus , cantor set , pure mathematics , eigenvalues and eigenvectors , open set , expansive , set (abstract data type) , canonical bundle , compact space , totally disconnected space , bundle , topology (electrical circuits) , combinatorics , discrete mathematics , mathematical analysis , physics , locally compact space , computer science , geometry , materials science , compressive strength , quantum mechanics , composite material , thermodynamics , programming language

We introduce two new characterizations of Meyer sets. A repetitive Delone set in ℝ d of 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.

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