Finite entropy for multidimensional cellular automata

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13 Scopus citations

Abstract

Let X = S where is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T:X → X (continuous, commuting with the action of ). Shereshevsky [Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag. Math. (N.S.) 4(2) (1993), 203-210] proved that for =ℤd with d > 1 no CA can be forward expansive, raising the following conjecture: for G = ℤd, d > 1, the topological entropy of any CA is either zero or infinite. Morris and Ward [Entropy bounds for endomorphisms commuting with K actions. Israel J. Math. 106 (1998), 1-11] proved this for linear CAs, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exists a d-dimensional CA with finite, non-zero topological entropy. We also discuss a measure-theoretic counterpart of this question for measure-preserving CAs.

Original languageEnglish
Pages (from-to)1243-1260
Number of pages18
JournalErgodic Theory and Dynamical Systems
Volume28
Issue number4
DOIs
StatePublished - 1 Aug 2008
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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