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

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 C∗-algebra of such a system is an A-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product C∗-algebra has real rank zero, which tells us that such C∗-algebras are classifiable by K-theory. The associated crossed product C∗-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).