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Spectral and topological properties of a family of generalised Thue-Morse sequences

Baake, Michael; Gähler, Franz and Grimm, Uwe (2012). Spectral and topological properties of a family of generalised Thue-Morse sequences. Journal of Mathematical Physics, 53(3) 032701.

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DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1063/1.3688337
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Abstract

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is known as the Devil’s staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures.
For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.

Item Type: Journal Article
Copyright Holders: 2012 American Institute of Physics
ISSN: 1089-7658
Project Funding Details:
Funded Project NameProject IDFunding Body
Not SetNot SetGerman Research Council (Deutsche Forschungsgemeinschaft (DFG)), within the CRC 701
Not SetNot SetLeverhulme Visiting Professorship grant
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Related URLs:
Item ID: 33146
Depositing User: Uwe Grimm
Date Deposited: 06 Mar 2012 13:54
Last Modified: 02 Dec 2012 12:01
URI: http://oro.open.ac.uk/id/eprint/33146
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