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Monatshefte für Mathematik

, Volume 170, Issue 2, pp 205–225 | Cite as

Metrics on tiling spaces, local isomorphism and an application of Brown’s lemma

  • Rui Pacheco
  • Helder Vilarinho
Article

Abstract

We give an application of a topological dynamics version of multidimensional Brown’s lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points \(F\), one can find a patch such that, for each scale factor \(\lambda \), there is a vector \(\vec {t}_\lambda \) so that copies of this patch appear in the tilling “nearly” centered on \(\lambda F+\vec {t}_\lambda \) once we allow “bounded perturbations” in the structure of the homothetic copies of \(F\). Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group “actions” on patches and we discuss the local isomorphism property of tilings within this setting.

Keywords

Tiling spaces Multiple topological recurrence Local isomorphism Brown’s lemma 

Mathematics Subject Classification (1991)

37B20 37B50 05B45 05D10 

References

  1. 1.
    Brown, T.C.: On locally finite semigroups. Ukraine Math. J. 20, 732–738 (1968)Google Scholar
  2. 2.
    Brown, T.C.: An interesting combinatorial method in the theory of locally finite semigroups. Pac. J. Math. 36, 285–289 (1971)zbMATHCrossRefGoogle Scholar
  3. 3.
    Brown, T. C.: A partition of the non-negative integers, with applications. Integers, 5, no. 2 (2005)Google Scholar
  4. 4.
    De La Llave, R., Windsor, A.: An application of topological multiple recurrence to tiling. Discrete Contin. Dyn. Syst. Ser. S 2(2), 315–324 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Frank, N.P., Sadun, L.: Topology of (some) tiling spaces without finite local complexity. Discrete Contin. Dyn. Syst. 23(3), 847–865 (2009)Google Scholar
  6. 6.
    Frank, N.P., Sadun, L.: Fusion tilings with infinite local complexity. arXiv:1201.3911v2 [math.DS] (2012)Google Scholar
  7. 7.
    Fürstenberg, H.: Recurrence in Ergodic theory and combinatorial number theory. Princeton University Press, Princeton (1981)Google Scholar
  8. 8.
    Gottschalk, W.H.: Orbit-closure decomposition and almost periodic properties. Bull. Am. Math. Soc. 50, 915–919 (1944)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Grünbaum, B., Shephard, G.C.: Tilings and Patterns. Freeman, New York (1986)Google Scholar
  10. 10.
    Hindman, N., Strauss, D.: Algebra in the Stone–Čech Compactification. Walter de Gruyter, Berlin (1998)zbMATHCrossRefGoogle Scholar
  11. 11.
    Montgomery, D., Zippin, L.: Topological Transformation Groups. In: Tracts in Pure and Applied Mathematics, vol. 1, 3rd edn. Interscience Publishers, New York (1965)Google Scholar
  12. 12.
    Penrose, R.: Pentaplexy. Bull. Inst. Math. Appl. 10, 266–271 (1974)Google Scholar
  13. 13.
    Radin, C., Wolff, M.: Space tilings and local isomorphism. Geometriae Dedicata 42, 355–360 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Robinson, E.A. Jr.: Symbolic dynamics and tilings of \(\mathbb{R}^d\). Symbolic dynamics and its applications. In: Proceedings of Symposia in Applied Mathematics, vol. 60, pp. 81–119. American Mathematical Society, Providence (2004)Google Scholar
  15. 15.
    Sadun, L.: Topology of tiling spaces. In: University Lecture Series, vol. 46. American Mathematical Society, Providence (2008)Google Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Universidade da Beira InteriorCovilhãPortugal

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