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Complexity and cohomology for cut-and-projection tilings

Published online by Cambridge University Press:  23 June 2009

ANTOINE JULIEN
ANTOINE JULIEN*
Affiliation:
Université de Lyon, Université Lyon 1, INSA de Lyon, F-69621, Ecole Centrale de Lyon, CNRS UMR5208, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France (email: julien@math.univ-lyon1.fr)
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Abstract

We consider a subclass of tilings: the tilings obtained by cut-and-projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent α in terms of the ranks of certain groups which appear in the construction. We give bounds for α. These computations apply to some well-known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over ℚ. We show that such a link cannot be established for more general tilings, and we present a counterexample in dimension one.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 489 - 523
Copyright
Copyright © Cambridge University Press 2009

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