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.