A𝕋-algebras from fiberwise essentially minimal zero-dimensional dynamical systems

Abstract

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product [Formula: see text]-algebra of such a system is an A[Formula: see text]-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product [Formula: see text]-algebra has real rank zero, which tells us that such [Formula: see text]-algebras are classifiable by [Formula: see text]-theory. The associated crossed product [Formula: see text]-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).