Complexity and cohomology for cut-and-projection tilings

By A. Julien. Ergodic Theory and Dynamical Systems, 2008

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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.

Subfields: Computational Theory and Mathematics, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25ae4a3cee0a9b3bf916d1610b31798722151549