  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 that system is rigid. In the case that cocycles take values in the
rotation group, it is also shown that rigidity 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 $C(X\times \T)$ onto $C(X\times \T)$.
