Phase transition on Exel crossed products associated to dilation matrices

An integer matrix A∈Md(Z) induces a covering σA of Td and an endomorphism αA:f↦f∘σA of C(Td) for which there is a natural transfer operator L. In this paper, we compute the KMS states on the Exel crossed product C(Td)⋊αA,LN and its Toeplitz extension. We find that C(Td)⋊αA,LN has a unique KMS state, which has inverse temperature . Its Toeplitz extension, on the other hand, exhibits a phase transition at , and for larger β the simplex of KMSβ states is isomorphic to the simplex of probability measures on Td.