  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).
