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Each multiplicative real-valued homomorphism on a quasi-lattice orderedmonoid gives rise to a quasi-periodic dynamics on the associated ToeplitzC*-algebra; here we study the KMS equilibrium states of the resultingC*-dynamical system. We show that, under a nondegeneracy assumption on thehomomorphism, there is a critical inverse temperature $\beta_c$ such that ateach inverse temperature $\beta \geq \beta_c$ there exists a unique KMS state.Strictly above $\beta_c$, the KMS states are generalised Gibbs states withdensity operators determined by analytic extension to the upper half-plane ofthe unitaries implementing the dynamics. These are faithful Type~I states. Thecritical value $\beta_c$ is the largest real pole of the partition function ofthe system and is related to the clique polynomial and skew-growth function ofthe monoid, relative to the degree map given by the logarithm of themultiplicative homomorphism. Motivated by the study of equilibrium states, wegive a proof of the inversion formula for the growth series of a quasi-latticeordered monoid in terms of the clique polynomial as in recent work ofAlbenque--Nadeau and McMullen for the finitely generated case, and in terms ofthe 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 thesmallest positive real root of the clique polynomial. We use this to show thatequilibrium states in the subcritical range can only occur at inversetemperatures 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 rootsexist.

Equilibrium states and growth of quasi-lattice ordered monoids