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Mean equicontinuity and mean sensitivity on cellular automata

Part of: Topological dynamics

Published online by Cambridge University Press:  03 November 2020

LUGUIS DE LOS SANTOS BAÑOS  and
FELIPE GARCÍA-RAMOS
LUGUIS DE LOS SANTOS BAÑOS
Affiliation:
Instituto de Física, Universidad Autónoma de San Luis Potosí, Av. Dr. Manuel Nava 6, 78290San Luis, SLP, Mexico (e-mail: luguis.sb.25@gmail.com)
FELIPE GARCÍA-RAMOS*
Affiliation:
Instituto de Física, Universidad Autónoma de San Luis Potosí, Av. Dr. Manuel Nava 6, 78290San Luis, SLP, Mexico (e-mail: luguis.sb.25@gmail.com) CONACyT, Mexico City, Mexico
*
e-mail: fgramos@conacyt.mx
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Abstract

We show that a cellular automaton (or shift-endomorphism) on a transitive subshift is either almost equicontinuous or sensitive. On the other hand, we construct a cellular automaton on a full shift (hence a transitive subshift) that is neither almost mean equicontinuous nor mean sensitive.

Keywords

cellular automatamean equicontinuitymean sensitivityalmost equicontinuityalmost mean equicontinuity

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37B10: Symbolic dynamics 37B15: Cellular automata
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3704 - 3721
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Akin, E., Auslander, J. and Berg, K.. When is a transitive map chaotic? Convergence in Ergodic Theory and Probability. Walter de Gruyter & Co., Berlin, 1996, pp. 25–40.CrossRefGoogle Scholar
Auslander, J. and Yorke, J. A.. Interval maps, factors of maps, and chaos. Tohoku Math. J. 32(2) (1980), 177–188.CrossRefGoogle Scholar
Blanchard, F., Formenti, E. and Kurka, P.. Cellular automata in the Cantor, Besicovitch, and Weyl topological spaces. Complex Syst. 11(2) (1997), 107–124.Google Scholar
Blanchard, F. and Tisseur, P.. Some properties of cellular automata with equicontinuity points. Ann. Inst. Henri Poincaré Probab. Stat. 36(5) (2000), 569–582.CrossRefGoogle Scholar
Downarowicz, T. and Glasner, E.. Isomorphic extensions and applications. Topol. Methods Nonlinear Anal. 48(1) (2016), 321–338.Google Scholar
Fomin, S.. On dynamical systems with pure point spectrum. Dokl. Akad. Nauk SSSR 77(4) (1951), 29–32 (in Russian).Google Scholar
Fuhrmann, G., Gröger, M. and Lenz, D.. The structure of mean equicontinuous group actions. Preprint, 2018, arXiv:1812.10219.Google Scholar
García-Ramos, F.. Limit behaviour of $\mu$ -equicontinuous cellular automata. Theoret. Comput. Sci. 623 (2016), 2–14.CrossRefGoogle Scholar
García-Ramos, F.. Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy. Ergod. Th. & Dynam. Sys. 37(4) (2017), 1211–1237.CrossRefGoogle Scholar
García-Ramos, F., Jäger, T. and Ye, X.. Mean equicontinuity, almost automorphy and regularity. Israel J. Math., to appear.Google Scholar
García-Ramos, F., Li, J. and Zhang, R.. When is a dynamical system mean sensitive? Ergod. Th. & Dynam. Sys. 39(6) (2019), 1608–1636.CrossRefGoogle Scholar
Gilman, R. H.. Classes of linear automata. Ergod. Th. & Dynam. Sys. 7(1) (1987), 105–118.CrossRefGoogle Scholar
Guckenheimer, J.. Sensitive dependence to initial conditions for one dimensional maps. Comm. Math. Phys. 70(2) (1979), 133–160.CrossRefGoogle Scholar
Huang, W., Li, J., Thouvenot, J.-P., Xu, L. and Ye, X.. Bounded complexity, mean equicontinuity and discrete spectrum. Ergod. Th. & Dynam. Sys., to appear.Google Scholar
Kurka, P.. Languages, equicontinuity and attractors in cellular automata. Ergod. Th. & Dynam. Sys., 17(2) (1997), 417–433.CrossRefGoogle Scholar
Kurka, P.. Topological and Symbolic Dynamics (Cours Specials, 11). Société Mathématique de France, Paris, 2003.Google Scholar
Li, J., Tu, S. and Ye, X.. Mean equicontinuity and mean sensitivity. Ergod. Th. & Dynam. Sys. 35(8) (2015), 2587–2612.CrossRefGoogle Scholar
Li, J., Ye, X. and Yu, T.. Mean equicontinuity, bounded complexity and applications. Discrete Contin. Dyn. Syst. A, 25., in press.Google Scholar
Martin, B.. Damage spreading and $\mu$ -sensitivity on cellular automata. Ergod. Th. & Dynam. Sys. 27(2) (2007), 545–565.CrossRefGoogle Scholar
Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. 58(2) (1952), 116–136.CrossRefGoogle Scholar
Sablik, M.. Directional dynamics for cellular automata: a sensitivity to initial condition approach. Theoret. Comput. Sci. 400(1–3) (2008), 1–18. CrossRefGoogle Scholar