Equilibrium States And Growth Of Quasi-lattice Ordered Monoids

Each multiplicative real-valued homomorphism on a quasi-lattice ordered monoid gives rise to a quasi-periodic dynamics on the associated Toeplitz $ C^*$-algebra; here we study the KMS equilibrium states of the resulting $ C^*$-dynamical system. We show that under a nondegeneracy assumption on the homomorphism there is a critical inverse temperature $ \beta _c$ such that at each inverse temperature $ \beta \geq \beta _c$ there exists a unique KMS state. Strictly above $ \beta _c$, the KMS states are generalised Gibbs states with density operators determined by analytic extension to the upper half-plane of the unitaries implementing the dynamics. These are faithful Type I states. The critical value $ \beta _c$ is the largest real pole of the partition function of the system and is related to the clique polynomial and skew-growth function of the monoid, relative to the degree map given by the logarithm of the multiplicative homomorphism. Motivated by the study of equilibrium states, we give a proof of the inversion formula for the growth series of a quasi-lattice ordered monoid in terms of the clique polynomial as in recent work of Albenque-Nadeau and McMullen for the finitely generated case and in terms of the skew-growth series as in recent work of Saito. Specifically, we show that $ e^{-\beta _c}$ is the smallest pole of the growth series and thus is the smallest positive real root of the clique polynomial. We use this to show that equilibrium states in the subcritical range can only occur at inverse temperatures that correspond to roots of the clique polynomial in the interval $ (e^{-\beta _c},1)$, but we are not aware of any examples in which such roots exist.