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A topological classification of D-dimensional cellular automata

Kůrka proposed a topological classification for cellular automata (CAs) on the one-dimensional full shift space [D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. New York: Cambridge University Press, 1995]. and we extended part of this classification to higher dimensional subshift spaces in [R.H. Gilman. Periodic behaviour of linear automata, in Dynamical Systems: Proceedings of the Special Year Held at the University of Maryland, College Park, 1986–1987, J.C. Alexander, ed., Lecture Notes in Mathematics, 1342, Berlin: Springer, pp. 216–219, 1988]. Here, we focus on expansiveness, showing that there exist subshift spaces in all dimensions on which there expansive CAs. This counters a result of Shereshevsky that there are no expansive CAs on full shift spaces in dimension greater than one [J. Milnor. On the entropy geometry of cellular automata. Complex Syst. 2 : 357–386, 1988]. We also supplement portions of [R.H. Gilman. Periodic behaviour of linear automata, in Dynamical systems: Proceedings of the Special Year Held at the University of Maryland, College Park, 1986–1987 J.C. Alexander, ed., Lecture Notes in Mathematics, 1342, Berlin: Springer, pp. 216–219, 1988] in order to give a topological classification of CAs on D-dimensional subshift spaces.