From PCA's to equilibrium systems and back

Stationary measures for probabilistic cellular automata (PCA's) ind dimensions give rise to space-time histories whose statistics may naturally be described by Gibbs states ind+1 dimensions for an interaction energy ? obtained from the PCA. In this note we study the converse question: Do all Gibbs states for this ? correspond to statistical space-time histories for the PCA? Our main result states that the answer is yes, at least for translation invariant or periodic Gibbs states. Thus ergodicity questions for PCA's can, at least partially, be formulated as questions of uniqueness of Gibbs states.