Abstract :
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,L N and its Toeplitz extension. We find that C(Td ) αA,L N has a
unique KMS state, which has inverse temperature β = log|detA|. Its Toeplitz extension, on the other hand,
exhibits a phase transition at β = log|detA|, and for larger β the simplex of KMSβ states is isomorphic to
the simplex of probability measures on Td .
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