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Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type

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Abstract

In this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction …). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d—that is a subshift whose set of forbidden patterns can be generated by a Turing machine—can be obtained by applying dynamical operations on a subshift of finite type of dimension d+1—a subshift that can be defined by a finite set of forbidden patterns. This result improves Hochman’s (Invent. Math. 176(1):131–167, 2009).

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References

  1. Aubrun, N., Sablik, M.: An order on sets of tilings corresponding to an order on languages. In: 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009), vol. 3, pp. 99–110 (2009)

    Google Scholar 

  2. Beal, M.P.: Codage Symbolique. Masson, Paris (1993)

    Google Scholar 

  3. Berger, R.: The Undecidability of the Domino Problem. Am. Math. Soc., Providence (1966)

    Google Scholar 

  4. Boyle, M.: Open problems in symbolic dynamics. Contemp. Math. 468, 69–118 (2008)

    Article  MathSciNet  Google Scholar 

  5. Dale, M.: Nonrecursive tilings of the plane. II. J. Symb. Log. 39(2), 286–294 (1974)

    Article  MATH  Google Scholar 

  6. Durand, B., Levin, L., Shen, A.: Complex tilings. In: STOC’01: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 732–739. ACM, New York (2001)

    Chapter  Google Scholar 

  7. Durand, B., Romashchenko, A.E., Shen, A.: Fixed point and aperiodic tilings. In: Developments in Language Theory, pp. 276–288 (2008)

    Chapter  Google Scholar 

  8. Durand, B., Romashchenko, A.E., Shen, A.: Fixed-point tile sets and their applications (2010). arXiv:0910.2415

  9. Hanf, W.: Nonrecursive tilings of the plane. I. J. Symb. Log. 39(2), 283–285 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Theory Comput. Syst. 3(4), 320–375 (1969)

    MathSciNet  MATH  Google Scholar 

  11. Hochman, M.: On the dynamics and recursive properties of multidimensional symbolic systems. Invent. Math. 176(1), 131–167 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kitchens, B.: Symbolic Dynamics. Springer, New York (1998)

    Book  MATH  Google Scholar 

  13. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  14. Mozes, S.: Tilings, substitution systems and dynamical systems generated by them. J. Anal. Math. (Jerus.) 53, 139–186 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pavlov, R., Schraudner, M.: Classification of sofic projective subdynamics of multidimensional shifts of finite type (2010, submitted)

  16. Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12(3), 177–209 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rogers, H. Jr.: Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge (1987)

    Google Scholar 

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Acknowledgements

The authors are grateful to Michael Schraudner for useful discussions and important remarks about the redaction. We also want to thank the anonymous referee for his rigorous and detailed review which helped us to clarify the paper and Mike Boyle for some comments about sub-action concepts. Moreover, this research is partially supported by projects ANR EMC and ANR SubTile.

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Authors and Affiliations

  1. LIP–UMR 5668, ENS Lyon–CNRS–UCBL–INRIA, 46 Allée d’Italie Lyon, 69364, Lyon Cedex, France

    Nathalie Aubrun

  2. LATP, Aix-Marseille Université, 39, rue F. Joliot Curie, 13453, Marseille Cedex 13, France

    Mathieu Sablik

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  1. Nathalie Aubrun
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  2. Mathieu Sablik
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Correspondence to Nathalie Aubrun.

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Aubrun, N., Sablik, M. Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type. Acta Appl Math 126, 35–63 (2013). https://doi.org/10.1007/s10440-013-9808-5

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