Non-wandering sets of topological dynamical systems and C^*-algebras

Abstract

Let $\Sigma = (X, \sigma)$ be a topological dynamical system where $\sigma$ is a homeomorphism in a compact metric space X. Denote by $A(\Sigma)$ the transformation group C^*-algebra associated with this system. We describe the shrinking steps of the non-wandering set $\Omega(\sigma)$ down to the Birkhoff center (depth of the center) in terms of a composition series of the particular ideal of type 1 in $A(\Sigma)$, which corresponds to the center. The result implies C^*-algebraic characterizations of the cases where the depth is 0 and 1. We also give the structure of dynamical systems for which the associated C^*-algebras become algebras with continuous traces.