Ergodic properties and KMS conditions on $C^*$-symbolic dynamical systems

Abstract

A C^* -symbolic dynamical system ({\mathcal A}, \rho, \Sigma) consists of a unital C^* -algebra {\mathcal A} and a finite family { \rho_\alpha }_{\alpha \in \Sigma} of endomorphisms \rho_\alpha of {\mathcal A} indexed by symbols \alpha of \Sigma satisfying some conditions. The endomorphisms \rho_\alpha, \alpha \in \Sigma yield both a subshift \Lambda_\rho and a C^* -algebra {\mathcal O}_\rho . We will study ergodic properties of the positive operator lambda_\rho = \sum_{\alpha \in \Sigma}\rho_\alpha on {\mathcal A} . We will next introduce KMS conditions for continuous linear functionals on {\mathcal O}_\rho under gauge action at inverse temperature taking its value in complex numbers. We will study relationships among the eigenvectors of lambda_\rho in {\mathcal A}^* , the continuous linear functionals on {\mathcal O}_\rho satisfying KMS conditions and the invariant measures on the associated one-sided shifts. We will finally present several examples of continuous linear functionals satisfying KMS conditions.