Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T00:36:44.593Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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