Equilibrium and monoidal growth

Abstract:

Each multiplicative real-valued homomorphism on a quasi-lattice ordered monoid gives rise to a quasi-periodic dynamics on the associated Toeplitz C*-algebra. We study the KMS equilibrium states of the resulting C*-dynamical system and, as a byproduct, we obtain 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.