Minimal Dynamical Systems on the Product of the Cantor Set and the Circle

We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if the set of invariant measures of the system come from the associated Cantor minimal system. In the case that cocycles take values in the rotation group, it is also shown that this condition implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately K-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps [inline-graphic not available: see fulltext]