Phase transition on Exel crossed products assocaited to dilation matrices

An integer matrix $A\in M_d(\mathbb{Z})$ induces a covering ${\sigma }_A$ of ${\text{T}}^d$ and an endomorphism ${\alpha }_A:f\mapsto f\circ {\sigma }_A$ of $C({\text{T}}^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C({\text{T}}^d){⋊}_{{\alpha }_A,L}\mathbb{N}$ and its Toeplitz extension. We find that $C({\text{T}}^d){⋊}_{{\alpha }_A,L}\mathbb{N}$ has a unique KMS state, which has inverse temperature $\beta =\log \mathrm{\mid }\det A\mathrm{\mid }$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $\beta=\log|\det A|$,andforlarger$\beta$thesimplexofKMS$_\beta$ states is isomorphic to the simplex of probability measures on ${\text{T}}^d$.