KMS and ground states for the normalized dynamics of the tame $C^{\ast }$-algebras ${\mathcal{O}}_{m,n}$

We study KMS and ground states of the tame $C^{\ast }$-algebras ${\mathcal{O}}_{m,n}$ of Ara, Exel and Katsura under the dynamics ${\sigma }_t(s_i)=m^{it}{s}_i$, ${\sigma }_t(t_j)=n^{it}{t}_j$. Normalized restriction to the full corner $A=q{\mathcal{O}}_{m,n}q$, generated by the partial isometries $v_{ij}=s_i^{\ast }{t}_j$ of common energy $\log (n\mathrm{/}m)$, is at each inverse temperature $\beta$ an affine weak-$\ast$ homeomorphism of the two systems. Let $m,n\ge 2$. When $m=n$ the corner dynamics is trivial, so the KMS simplices are identified with trace simplices. When $m\mathrm{\ne }n$, KMS states occur only at $\beta =1$, and there they are not unique. Conformal branching models surject the KMS${}_1$ simplex onto a transportation polytope of dimension $(m-1)(n-1)$, whose vertices lift to extremal, hence factorial, KMS${}_1$ states, the fibre over each vertex being itself the KMS${}_1$ simplex of a quotient of $A$.