Finite type approximations of Gibbs measures on sofic subshifts

Abstract

Consider a Hölder continuous potential $ϕ$ defined on the full shift $A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $μ_ϕ$ on $X$ associated to $ϕ$. Besides, 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 weakly converge at exponential speed to $μ_ϕ$ (in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of $μ_ϕ$. 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. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.