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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 , pure mathematics , conjugate , torus , eigenvalues and eigenvectors , open set , compact space , canonical bundle , set (abstract data type) , expansive , cantor set , totally disconnected space , bundle , topology (electrical circuits) , combinatorics , mathematical analysis , physics , computer science , locally compact space , geometry , materials science , quantum mechanics , composite material , compressive strength , 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.