License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.128
URN: urn:nbn:de:0030-drops-91323
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9132/
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Grandjean, Anael ; Hellouin de Menibus, Benjamin ; Vanier, Pascal

Aperiodic Points in Z²-subshifts

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Abstract

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.

BibTeX - Entry

@InProceedings{grandjean_et_al:LIPIcs:2018:9132,
  author =	{Anael Grandjean and Benjamin Hellouin de Menibus and Pascal Vanier},
  title =	{{Aperiodic Points in Z²-subshifts}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{128:1--128:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9132},
  URN =		{urn:nbn:de:0030-drops-91323},
  doi =		{10.4230/LIPIcs.ICALP.2018.128},
  annote =	{Keywords: Subshifts of finite type, Wang tiles, periodicity, aperiodicity, computability, tilings}
}

Keywords: Subshifts of finite type, Wang tiles, periodicity, aperiodicity, computability, tilings
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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