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Quantified block gluing for multidimensional subshifts of finite type: aperiodicity and entropy

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Abstract

It is possible to extend the notion of block gluing for subshifts studied in [PS15] adding a gap function which gives the distance which allows to concatenate two rectangular blocks of the language. In this article, we study the interplay between this intensity and computational properties. In particular, we prove that there exists block gluing SFTs with linear gap which are aperiodic and that all the non-negative right-recursively enumerable (Π1-computable) numbers can be realized as entropy of such subshifts of finite type. As block gluing with linear gap implies transitivity, this last point provides a solution to Problem 9.1 in [HM10] about the characterization of the entropies of transitive subshift of finite type.

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Acknowledgements

We thank the anonymous referee for their careful reading and many remarks.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, AGH University of Science and Technology, D-13, ul. Grammar 8a, Krakow, Poland

    Silvère Gangloff

  2. Institutde Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, F-31062, Toulouse Cedex 9, France

    Mathieu Sablik

Authors
  1. Silvère Gangloff
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  2. Mathieu Sablik
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Correspondence to Mathieu Sablik.

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Gangloff, S., Sablik, M. Quantified block gluing for multidimensional subshifts of finite type: aperiodicity and entropy. JAMA 144, 21–118 (2021). https://doi.org/10.1007/s11854-021-0172-5

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  • DOI: https://doi.org/10.1007/s11854-021-0172-5

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