Crossed Products by Minimal Homeomorphisms

Let X be an infinite compact metric space with finite covering dimension and let h : X → X be a minimal homeomorphism. We show that the associated crossed product C*-algebra A = C∗(Z, X, h) has tracial rank zero whenever the image of K0(A) in Aff(T (A)) is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the classification result for nuclear simple C*-algebras with tracial rank zero, we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered K-theory.