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Glider automata on all transitive sofic shifts

Part of: Topological dynamics

Published online by Cambridge University Press:  30 September 2021

JOHAN KOPRA Open the ORCID record for JOHAN KOPRA [Opens in a new window]
JOHAN KOPRA*
Affiliation:
Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland
*
(e-mail: jtjkop@utu.fi)
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Abstract

For any infinite transitive sofic shift X we construct a reversible cellular automaton (that is, an automorphism of the shift X) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible cellular automaton which is sensitive with respect to all directions. As another application we prove a finitary version of Ryan’s theorem: the automorphism group $\operatorname {\mathrm {Aut}}(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of S-gap shifts these results do not extend beyond the sofic case.

Keywords

sofic shiftssynchronized shiftscellular automataautomorphisms

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37B15: Cellular automata
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 12 , December 2022 , pp. 3716 - 3744
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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