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Slopes of Multidimensional Subshifts

  • Emmanuel Jeandel
  • Etienne MoutotEmail author
  • Pascal Vanier
Article
  • 14 Downloads
Part of the following topical collections:
  1. Computer Science Symposium in Russia (2018)

Abstract

In this paper we study the directions of periodicity of multidimensional subshifts of finite type (SFTs) and of multidimensional effectively closed and sofic subshifts. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope representing that direction. In this paper, we prove that \({{\Sigma }^{0}_{1}}\) sets of non-commensurable \(\mathbb {Z}^{2}\) vectors are exactly the sets of slopes of 2D SFTs and that \({{\Sigma }^{0}_{2}}\) sets of non-commensurable vectors are exactly the sets of slopes of 3D SFTs, and exactly the sets of slopes of 2D and 3D sofic and effectively closed subshifts.

Keywords

Subshifts SFTs Computability Slopes Periodicity 

Notes

Acknowledgements

The authors would like to particularly thank Ville Salo for some very useful remarks on a previous version of this paper that led to a better exposition and to some corrections.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Campus Scientifique - BP 239Vandoeuvre-les-NancyFrance
  2. 2.Université de Lyon, ENS de Lyon, UCBL, CNRSLIPLyon CedexFrance
  3. 3.Faculté des Sciences et TechnologieCréteil CedexFrance

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