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Slopes of 3-Dimensional Subshifts of Finite Type

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10846)

Abstract

In this paper we study the directions of periodicity of three-dimensional subshifts of finite type (SFTs) and in particular their slopes. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope being the angles of the periodicity vector. In this paper, we prove that any \(\varSigma ^0_2\) set may be realized as a a set of slopes of an SFT.

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References

  1. Aubrun, N., Sablik, M.: Simulation of effective subshifts by two-dimensional subshifts of finite type. Acta Applicandae Math. 126(1), 35–63 (2013)

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Berger, R.: The Undecidability of the Domino Problem. Ph.D. thesis, Harvard University (1964)

    Google Scholar 

  3. Berger, R.: The Undecidability of the Domino Problem. No. 66 in Memoirs of the American Mathematical Society, The American Mathematical Society (1966)

    MathSciNet  CrossRef  Google Scholar 

  4. Culik II, K., Kari, J.: An aperiodic set of Wang cubes. J. Univers. Comput. Sci. 1(10), 675–686 (1995)

    MathSciNet  MATH  Google Scholar 

  5. Durand, B., Romashchenko, A., Shen, A.: Fixed-point tile sets and their applications. J. Comput. Syst. Sci. 78(3), 731–764 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Gurevich, Y., Koryakov, I.: Remarks on Berger’s paper on the domino problem. Siberian Math. J. 13(2), 319–320 (1972)

    CrossRef  MATH  Google Scholar 

  7. Hochman, M., Meyerovitch, T.: A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. 171(3), 2011–2038 (2010)

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Jeandel, E., Rao, M.: An aperiodic set of 11 Wang tiles. CoRR abs/1506.06492 (2015). http://arxiv.org/abs/1506.06492

  9. Jeandel, E., Vanier, P.: Slopes of tilings. In: Kari, J. (ed.) JAC, pp. 145–155. Turku Center for Computer Science (2010)

    Google Scholar 

  10. Jeandel, E., Vanier, P.: Characterizations of periods of multi-dimensional shifts. Ergod. Theor. Dyn. Syst. 35(2), 431–460 (2015). http://journals.cambridge.org/article_S0143385713000606

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Kari, J.: The nilpotency problem of one-dimensional cellular automata. SIAM J.Comput. 21(3), 571–586 (1992)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Kari, J.: A small aperiodic set of Wang tiles. Discrete Math. 160(1–3), 259–264 (1996)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  14. Meyerovitch, T.: Growth-type invariants for \(\mathbb{Z}^d\) subshifts of finite type and arithmetical classes of real numbers. Inventiones Math. 184(3), 567–589 (2010)

    MathSciNet  CrossRef  Google Scholar 

  15. Myers, D.: Non recursive tilings of the plane II. J. Symbolic Log. 39(2), 286–294 (1974)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. Ollinger, N.: Two-by-two substitution systems and the undecidability of the domino problem. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 476–485. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69407-6_51

    CrossRef  MATH  Google Scholar 

  17. Poupet, V.: Yet another aperiodic tile set. In: Journées Automates Cellulaires (JAC), pp. 191–202. TUCS (2010)

    Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MATH  Google Scholar 

  20. Wang, H.: Proving theorems by pattern recognition I. Commun. ACM 3(4), 220–234 (1960)

    CrossRef  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous reviewers who pointed out a mistake in a previous version of the paper.

This work was supported by grant TARMAC ANR 12 BS02 007 01.

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Correspondence to Pascal Vanier .

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Moutot, E., Vanier, P. (2018). Slopes of 3-Dimensional Subshifts of Finite Type. In: Fomin, F., Podolskii, V. (eds) Computer Science – Theory and Applications. CSR 2018. Lecture Notes in Computer Science(), vol 10846. Springer, Cham. https://doi.org/10.1007/978-3-319-90530-3_22

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  • DOI: https://doi.org/10.1007/978-3-319-90530-3_22

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