  For dynamical systems defined by a covering map of a compact Hausdorff space
and the corresponding transfer operator, the associated crossed product
$C^*$-algebras $\cros$ introduced by Exel and Vershik are considered. An
important property for homeomorphism dynamical systems is topological freeness.
It can be extended in a natural way to in general non-invertible dynamical
systems generated by covering maps. In this article, it is shown that the
following four properties are equivalent: the dynamical system generated by a
covering map is topologically free; the canonical imbedding of $C(X)$ into
$\cros$ is a maximal abelian $C^*$-subalgebra of $\cros$; any nontrivial two
sided ideal of $\cros$ has non-zero intersection with the imbedded copy of
$C(X)$; a certain natural representation of $\cros$ is faithful. This result is
a generalization to non-invertible dynamics of the corresponding results for
crossed product $C^*$-algebras of homeomorphism dynamical systems.
