Finite type approximations of Gibbs measures on sofic subshifts

Consider a Holder continuous potential phi defined on the full shift A(N), where A is a finite alphabet. Let X subset of A(N) be a specified sofic subshift. It is well known that there is a unique Gibbs measure mu(phi) on X associated with phi. In addition, there is a natural nested sequence of subshifts of finite type (X-m) converging to the sofic subshift X. To this sequence we can associate a sequence of Gibbs measures (mu(phi)(m)). In this paper, we prove that these measures converge weakly at exponential speed to mu(phi) (in the classical distance metrizing weak topology). We also establish a mixing property that implies that mu(phi) is Bernoulli. Finally, we prove that the measure-theoretic entropy of mu(phi)(m) converges to the one of mu(phi) exponentially fast. We indicate how to extend our results to more general subshifts and potentials.