On the equivalence of certain ergodic properties for Gibbs states

We extend our previous work by proving that for translation invariant Gibbs states on Z(d) with a translation invariant interaction potential Psi = (P si(A)) satisfying Sigma(A There Exists 0) \A\(-1)[diam(A)](d)\\Psi(A)\\ < i nfinity the following hold: (1) the Kolmogorov-property implies a trivial f ull tail and (2) the Bernoulli-property implies Folner independence. The ex istence of bilaterally deterministic Bernoulli Shifts tells us that neither (1) nor (2) is, in general, true for random fields without some further as sumption (even when d = 1).