Consider a H\"older continuous potential $\phi$ defined on the full shift$A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specifiedsofic subshift. It is well-known that there is a unique Gibbs measure$\mu_\phi$ on $X$ associated to $\phi$. Besides, there is a natural nestedsequence 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 weakly convergeat exponential speed to $\mu_\phi$ (in the classical distance metrizing weaktopology). We also establish a strong mixing property (ensuring weakBernoullicity) of $\mu_\phi$. Finally, we prove that the measure-theoreticentropy 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.We stress that we use basic algebraic tools (contractive properties of iteratedmatrices) and symbolic dynamics.
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机译：考虑在全移位$ A ^ \ nn $上定义的H \“较旧的连续势$ \ phi $，其中$ A $是有限字母。令$ X \ subset A ^ \ nn $是指定的子移位。众所周知，在与$ \ phi $关联的$ X $上有一个唯一的吉布斯度量$ \ mu_ \ phi $，此外，自然存在一个有限类型为$（X_m）$的子移位的自然嵌套序列，收敛到sofic子移位$ X $。对于这个序列，我们可以将一系列Gibbs度量$（\ mu _ {\ phi} ^ m）$关联起来。在本文中，我们证明了这些度量将指数速度微弱地收敛到$ \ mu_ \ phi $（在我们还建立了$ \ mu_ \ phi $的强混合特性（确保弱Bernoullicity），最后证明了$ \ mu_ \ phi ^ m $的量度理论熵收敛到$ \ mu_ \ phi $呈指数级增长。我们指出如何将结果扩展到更一般的子移位和势。我们强调，我们使用基本的代数工具（迭代矩阵的收缩性质）和符号动力学。
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