Aperiodic Points in Z²-subshifts

Authors Anael Grandjean, Benjamin Hellouin de Menibus , Pascal Vanier

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Anael Grandjean
  • Laboratoire d'Algorithmique, Complexité et Logique, Université Paris-Est Créteil, France
Benjamin Hellouin de Menibus
  • Laboratoire de Recherche en Informatique, Université Paris-Sud, CNRS, CentraleSupélec, Université Paris-Saclay, France
Pascal Vanier
  • Laboratoire d'Algorithmique, Complexité et Logique, Université Paris-Est Créteil, France

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Anael Grandjean, Benjamin Hellouin de Menibus, and Pascal Vanier. Aperiodic Points in Z²-subshifts. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 128:1-128:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We consider the structure of aperiodic points in Z^2-subshifts, and in particular the positions at which they fail to be periodic. We prove that if a Z^2-subshift contains points whose smallest period is arbitrarily large, then it contains an aperiodic point. This lets us characterise the computational difficulty of deciding if an Z^2-subshift of finite type contains an aperiodic point. Another consequence is that Z^2-subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some Z-subshift. Finally, we use this result to characterize sets of possible slopes of periodicity for Z^3-subshifts of finite type.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Subshifts of finite type
  • Wang tiles
  • periodicity
  • aperiodicity
  • computability
  • tilings


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