The $\mathrm{GL}_n$-Connes-Marcolli Systems

Abstract: In this paper, we generalize the results of Laca, Larsen, and Neshveyev on the $\mathrm{GL}_2$-Connes-Marcolli system to the $\mathrm{GL}_n$ systems. We introduce the $\mathrm{GL}_n$-Connes-Marcolli systems and discuss the question of the existence and the classification of KMS equilibrium states at different inverse temperatures $\beta$. In particular, using an ergodicity argument, we prove that in the range $n-1 <\beta\leq n$, there is only one KMS state. We show that there are no KMS states for $\beta<n-1$ and not an integer, while we construct KMS states for integer values of $\beta$ in the range $1\leq\beta\leq n-1$, and we classify extremal KMS states for $\beta>n$.