Minimal dynamical systems and approximate conjugacy

Abstract

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 τ-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.