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Baake, Michael; Gähler, Franz and Grimm, Uwe
(2012).
DOI: https://doi.org/10.1063/1.3688337
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.
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About
- Item ORO ID
- 33146
- Item Type
- Journal Item
- ISSN
- 1089-7658
- Project Funding Details
-
Funded Project Name Project ID Funding Body Not Set Not Set German Research Council (Deutsche Forschungsgemeinschaft (DFG)), within the CRC 701 Not Set Not Set Leverhulme Visiting Professorship grant - Academic Unit or School
-
Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM) - Copyright Holders
- © 2012 American Institute of Physics
- Related URLs
-
- http://arxiv.org/abs/1201.1423(Other)
- http://jmp.aip.org/(Other)
- Depositing User
- Uwe Grimm