Finite entropy for multidimensional cellular automata

Abstract

AbstractLet $X=S^{\mathbb {G}}$ where $\mathbb {G}$ 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 $\mathbb {G}$). 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 $\mathbb {G}=\mathbb {Z}^d$ with d>1 no CA can be forward expansive, raising the following conjecture: for $G=\mathbb {Z}^d$, 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.