Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers

We show that the group $\mathbb{Q}⋊{\mathbb{Q}}_{+}^{\ast }$ of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup ${\mathbb N \rtimes \mathbb N^\times}$\mathrm{.}TheassociatedToeplitz$C^$-algebra${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$ is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of this Toeplitz algebra in terms of generators and relations, and use this to show that the $C^{\ast }$-algebra ${\mathcal{Q}}_{\mathbb{N}}$ recently introduced by Cuntz is the boundary quotient of $({\mathbb Q \rtimes \mathbb Q^+}, {\mathbb N \rtimes \mathbb N^\times})$inthesenseofCrispandLaca\mathrm{.}TheToeplitzalgebra${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$carriesanaturaldynamics$\sigma$,whichinducestheoneconsideredbyCuntzonthequotient${\mathcal Q\mathbb N}$,andourmainresultisthecomputationoftheKMS$\beta$(equilibrium)statesofthedynamicalsystem$({\mathcal T}({\mathbb N \rtimes \mathbb N^\times}), {\mathbb R},\sigma)$ for all values of the inverse temperature $\beta$. For $\beta \in [1,2]$ there is a unique KMS${}_{\beta }$ state, and the KMS${}_1$ state factors through the quotient map onto ${\mathcal Q\mathbb N}$,givingtheuniqueKMSstatediscoveredbyCuntz\mathrm{.}At$\beta =2$thereisaphasetransition,andfor$\beta>2$theKMS$_\beta$ states are indexed by probability measures on the circle. There is a further phase transition at $\beta =\mathrm{\infty }$, where the KMS${}_{\mathrm{\infty }}$ states are indexed by the probability measures on the circle, but the ground states are indexed by the states on the classical Toeplitz algebra $\mathcal{T}(\mathbb{N})$.