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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

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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.

Information

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

References

[1]Arnoux, P., Mauduit, C., Shiokawa, I. and Tamura, J.. Complexity of sequences defined by billiards in the cube. Bull. Soc. Math. France 122(1) (1994), 1–12.Google Scholar
[2]Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated C *-algebra. Ergod. Th. & Dynam. Sys. 18(3) (1998), 509–537.CrossRefGoogle Scholar
[3]Baryshnikov, Y.. Complexity of trajectories in rectangular billiards. Comm. Math. Phys. 174(1) (1995), 43–56.Google Scholar
[4]Berthé, V. L. and Vuillon, L.. Tilings and rotations: a two-dimensional generalisation of Sturmian sequences. Discrete Math. 223 (2000), 27–53.Google Scholar
[5]Cassaigne, J.. Special factors of sequences with linear subword complexity. Developments in Language Theory, II (Magdeburg, 1995). World Scientific, Singapore, 1996, pp. 25–34.Google Scholar
[6]Forrest, A., Hunton, J. and Kellendonk, J.. Topological invariants for projection method patterns. Mem. Amer. Math. Soc. 159(758) (2002).Google Scholar
[7]Hansen, C. and Robinson, E. A. Jr. On the complexity of self-affine tilings. Preprint, 2002.Google Scholar
[8]Hubert, P. and Vuillon, L.. Complexity of cutting words on regular tilings. European J. Combin. 28(1) (2007), 429–438.Google Scholar
[9]Körner, T. W.. Fourier Analysis, 2nd edn. Cambridge University Press, Cambridge, 1989.Google Scholar
[10]Lagarias, J. C. and Pleasants, P. A. B.. Repetitive Delone sets and quasicrystals. Ergod. Th. & Dynam. Sys. 23(3) (2003), 831–867.CrossRefGoogle Scholar
[11]Oguey, C., Duneau, M. and Katz, A.. A geometrical approach of quasiperiodic tilings. Comm. Math. Phys. 118(1) (1988), 99–118.CrossRefGoogle Scholar
[12]Robinson, E. A. Jr. Symbolic dynamics and tilings of ℝd. Symbolic Dynamics and its Applications (Proceedings of Symposia in Applied Mathematics, 60). American Mathematical Society, Providence, RI, 2004, pp. 81–119.Google Scholar
[13]Sadun, L.. Tiling spaces are inverse limits. J. Math. Phys. 44(11) (2003), 5410–5414.Google Scholar
[14]Vuillon, L.. Local configurations in discrete planes. Bull. Belg. Math. Soc. 6 (1999), 625–636.Google Scholar