Let φ:X → X be a homeomorphism of a compact metric space X. For any continuous function F:X →R there is a one-parameter group αF C(X) ×φ Z of automorphisms (or a flow) on the crossed product C∗-algebra defined such that αFt (fU) = fUe-itf when f ∈ C(X) and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo - Martin - Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when C(X) ×φ Z is simple this set is either {0} or the whole line R.
