  Starting from a discrete $C^*$-dynamical system $(\mathfrak{A}, \theta,
\omega_o)$, we define and study most of the main ergodic properties of the
crossed product $C^*$-dynamical system $(\mathfrak{A}\rtimes_\alpha\mathbb{Z},
\Phi_{\theta, u},\om_o\circ E)$,
$E:\mathfrak{A}\rtimes_\alpha\mathbb{Z}\rightarrow\ga$ being the canonical
conditional expectation of $\mathfrak{A}\rtimes_\alpha\mathbb{Z}$ onto
$\mathfrak{A}$, provided $\a\in\aut(\ga)$ commute with the $*$-automorphism
$\th$ up tu a unitary $u\in\ga$. Here, $\Phi_{\theta,
u}\in\aut(\mathfrak{A}\rtimes_\alpha\mathbb{Z})$ can be considered as the fully
noncommutative generalisation of the celebrated skew-product defined by H.
Anzai for the product of two tori in the classical case.
