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.
@InProceedings{grandjean_et_al:LIPIcs.ICALP.2018.128, author = {Grandjean, Anael and Hellouin de Menibus, Benjamin and Vanier, Pascal}, 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 = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.128}, 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} }
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