Finite type approximations of Gibbs measures on sofic subshifts - Archive ouverte HAL
Article Dans Une Revue Nonlinearity Année : 2005

Finite type approximations of Gibbs measures on sofic subshifts

L. Ramirez
  • Fonction : Auteur
E. Ugalde
  • Fonction : Auteur

Résumé

Consider a Hölder continuous potential $\phi$ 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 $\mu_\phi$ on $X$ associated to $\phi$. 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 $(\mu_{\phi}^m)$. In this paper, we prove that these measures weakly converge at exponential speed to $\mu_\phi$ (in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of $\mu_\phi$. 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. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.

Dates et versions

hal-00142140 , version 1 (17-04-2007)

Identifiants

Citer

J. -R. Chazottes, L. Ramirez, E. Ugalde. Finite type approximations of Gibbs measures on sofic subshifts. Nonlinearity, 2005, 18, pp.445-463. ⟨hal-00142140⟩
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