Finite type approximations of Gibbs measures on sofic subshifts

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