  Several versions of approximate conjugacy for minimal dynamical systems are
introduced. Relation between approximate conjugacy and corresponding crossed
product $C^*$-algebras is discussed. For the Cantor minimal systems, a complete
description is given for these relations via $K$-theory and $C^*$-algebras. For
example, it is shown that two Cantor minimal systems are approximately
$\tau$-conjugate if and only if they are orbit equivalent and have the same
periodic spectrum. It is also shown that two such systems are approximately
$K$-conjugate if and only if the corresponding crossed product $C^*$-algebras
have the same scaled ordered $K$-theory. Consequently, two Cantor minimal
systems are approximately $K$-conjugate if and only if the associated
transformation $C^*$-algebras are isomorphic. Incidentally, this approximate
$K$-conjugacy coincides with Giordano, Putnam and Skau's strong orbit
equivalence for the Cantor minimal systems.
