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Computational Characterization of Surface Entropies for ℤ² Subshifts of Finite Type

Authors Antonin Callard, Pascal Vanier

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Author Details

Antonin Callard
  • Université Paris-Saclay, ENS Paris-Saclay, Département Informatique, 91190 Gif-sur-Yvette, France
Pascal Vanier
  • Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC, 14000 Caen, France


The authors would like to thank Ronnie Pavlov for answering their many questions about surface entropy when they started this work, and Benjamin Hellouin de Menibus for the relecturing. Finally, we warmly thank the anonymous reviewers for their many helpful remarks and improvements.

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Antonin Callard and Pascal Vanier. Computational Characterization of Surface Entropies for ℤ² Subshifts of Finite Type. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 122:1-122:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Subshifts of finite type (SFTs) are sets of colorings of the plane that avoid a finite family of forbidden patterns. In this article, we are interested in the behavior of the growth of the number of valid patterns in SFTs. While entropy h corresponds to growths that are squared exponential 2^{hn²}, surface entropy (introduced in Pace’s thesis in 2018) corresponds to the eventual linear term in exponential growths. We give here a characterization of the possible surface entropies of SFTs as the Π₃ real numbers of [0,+∞].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Models of computation
  • surface entropy
  • arithmetical hierarchy of real numbers
  • 2D subshifts
  • symbolic dynamics


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  1. Nathalie Aubrun and Mathieu Sablik. Simulation of effective subshifts by two-dimensional subshifts of finite type. Acta Applicandae Mathematicae, 126(1):35-63, 2013. URL:
  2. Robert Berger. The Undecidability of the Domino Problem. Number 66 in Memoirs of the American Mathematical Society. The American Mathematical Society, 1966. Google Scholar
  3. Bruno Durand, Andrei Romashchenko, and Alexander Shen. Effective Closed Subshifts in 1D Can Be Implemented in 2D. In Fields of Logic and Computation, number 6300 in Lecture Notes in Computer Science, pages 208-226. Springer, 2010. URL:
  4. Bruno Durand, Andrei Romashchenko, and Alexander Shen. Fixed-point tile sets and their applications. Journal of Computer and System Sciences, 78(3):731-764, May 2012. URL:
  5. Yuri Gurevich and I Koryakov. Remarks on Berger’s paper on the domino problem. Siberian Math. Journal, pages 319-320, 1972. Google Scholar
  6. David Harel. Recurring Dominoes: Making the Highly Undecidable Highly Understandable. Annals of Discrete Mathematics, 24:51-72, 1985. Google Scholar
  7. Michael Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. Inventiones Mathematicae, 176(1):2009, April 2009. Google Scholar
  8. Michael Hochman and Tom Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. Annals of Mathematics, 171(3):2011-2038, May 2010. URL:
  9. Konrad Jacobs and Michael Keane. 0-1-sequences of Toeplitz type. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 13(2):123-131, 1969. URL:
  10. Emmanuel Jeandel and Pascal Vanier. Characterizations of periods of multidimensional shifts. Ergodic Theory and Dynamical Systems, 35(2):431-460, 2015. URL:
  11. S.C. Kleene. Two Papers on the Predicate Calculus, chapter Finite Axiomatizability of Theories in the Predicate Calculus Using Additional Predicate Symbols, pages 31-71. Number 10 in Memoirs of the American Mathematical Society. American Mathematical Society, 1952. Google Scholar
  12. Douglas A. Lind and Brian Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, New York, NY, USA, 1995. Google Scholar
  13. Tom Meyerovitch. Growth-type invariants for ℤ^d subshifts of finite type and arithmetical classes of real numbers. Inventiones Mathematicae, 184(3), 2010. URL:
  14. Dennis Pace. Surface Entropy of Shifts of Finite Type. PhD thesis, University of Denver, 2018. Google Scholar
  15. Xizhong Zheng and Klaus Weihrauch. Arithmetical hierarchy of real numbers. In Mathematical Foundations of Computer Science (MFCS), pages 23-33, 1999. URL:
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