Finite type approximations of Gibbs measures on sofic subshifts

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

VIVO