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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