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Homological Pisot substitutions and exact regularity

Barge, M, Bruin, H, Jones, L and Sadun, L (2011) Homological Pisot substitutions and exact regularity Israel Journal of Mathematics. 1 - 20. ISSN 0021-2172

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Abstract

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and the first rational Čech cohomology is d-dimensional. We construct examples of such “homological Pisot” substitutions whose tiling flows do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.

Item Type: Article
Additional Information: The original publication is available at http://www.springerlink.com
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Symplectic Elements
Date Deposited: 09 Dec 2011 11:00
Last Modified: 23 Sep 2013 18:55
URI: http://epubs.surrey.ac.uk/id/eprint/37233

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