Finite Entropy for Multidimensional Cellular Automata

Let X = S where 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 G). Shereshevsky [14] proved that for G = Z with d > 1 no CA can be forward expansive, raising the following conjecture: For G = Z, d > 1 the topological entropy of any CA is either zero or infinite. Morris and Ward [11], proved this for linear CA’s, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exist a d-dimensional CA with finite, nonzero topological entropy. We also discuss a measure-theoretic counterpart of this question for measure-preserving CA’s.