Inventiones mathematicae

, Volume 184, Issue 3, pp 567–589 | Cite as

Growth-type invariants for ℤd subshifts of finite type and arithmetical classes of real numbers

Article

Abstract

We discuss some numerical invariants of multidimensional shifts of finite type (SFTs) which are associated with the growth rates of the number of admissible finite configurations. Extending an unpublished example of Tsirelson (A strange two-dimensional symbolic system, 1992), we show that growth complexities of the form exp (nα+o(1)) are possible for non-integer α’s. In terminology of de Carvalho (Port. Math. 54(1):19–40, 1997), such subshifts have entropy dimension α. The class of possible α’s are identified in terms of arithmetical classes of real numbers of Weihrauch and Zheng (Math. Log. Q. 47(1):51–65, 2001).

Mathematics Subject Classification (2000)

37B10 37B40 37B50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahn, Y.-H., Dou, D., Park, K.K.: Entropy dimensions and variational principle. Trends Math. 10(2), 163–165 (2008) Google Scholar
  2. 2.
    Blume, F.: Possible rates of entropy convergence. Ergod. Theory Dyn. Syst. 17(1), 45–70 (1997) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    de Carvalho, M.: Entropy dimension of dynamical systems. Port. Math. 54(1), 19–40 (1997) MATHGoogle Scholar
  4. 4.
    Einsiedler, M., Lind, D., Miles, R., Ward, T.: Expansive subdynamics for algebraic \({\Bbb{Z}}\sp d\)-actions. Ergod. Theory Dyn. Syst. 21(6), 1695–1729 (2001) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ferenczi, S., Park, K.K.: Entropy dimensions and a class of constructive examples. Discrete Contin. Dyn. Syst. 17(1), 133–141 (2007) MathSciNetMATHGoogle Scholar
  6. 6.
    Hochman, M.: On the dynamics and recursive properties of multidimensional symbolic systems. Invent. Math. 176(1), 131–167 (2009) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hochman, M., Meyerovitch, T.: A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. (2) 171(3), 2010–2038 (2010) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Katok, A., Thouvenot, J.-P.: Slow entropy type invariants and smooth realization of commuting measure-preserving transformations. Ann. Inst. Henri Poincaré Probab. Stat. 33(3), 323–338 (1997) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kim, K.H., Ormes, N.S., Roush, F.W.: The spectra of nonnegative integer matrices via formal power series. J. Am. Math. Soc. 13(4), 773–806 (2000) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995) MATHCrossRefGoogle Scholar
  11. 11.
    Lind, D.A.: The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Theory Dyn. Syst. 4(2), 283–300 (1984) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Meyerovitch, T.: Finite entropy for multidimensional cellular automata. Ergod. Theory Dyn. Syst. 28(1), 61–83 (2008) MathSciNetGoogle Scholar
  13. 13.
    Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12, 177–209 (1971) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Simpson, S.G.: Medvedev degrees of 2-dimensional subshifts of finite type (2007) Google Scholar
  15. 15.
    Tsirelson, B.: A strange two-dimensional symbolic system. Unpublished notes from Tel Aviv University Math Colloquim (1992) Google Scholar
  16. 16.
    Zheng, X., Weihrauch, K.: The arithmetical hierarchy of real numbers. Math. Log. Q. 47(1), 51–65 (2001) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Pacific Institute for the Mathematical ScienceVancouverCanada

Personalised recommendations