AT-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", and we prove that the associated crossed product $C^*$-algebra of such a system is an AT-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. We show that the definition of fiberwise essentially minimality allows one to produce many nontrivial examples of such systems (ones that are neither minimal nor essentially minimal). The associated crossed product $C^*$-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).