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Starting from a discrete $C^*$-dynamical system $(\mathfrak{A}, \theta,\omega_o)$, we define and study most of the main ergodic properties of thecrossed 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 canonicalconditional 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 fullynoncommutative generalisation of the celebrated skew-product defined by H.Anzai for the product of two tori in the classical case.

Skew-product dynamical systems for crossed product $C^*$-algebras and their ergodic properties