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

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 $(mathfrak{A}rtimes_alphamathbb{Z}, Phi_{theta, u},om_ocirc E)$, $E:mathfrak{A}rtimes_alphamathbb{Z}rightarrowga$ being the canonical conditional expectation of $mathfrak{A}rtimes_alphamathbb{Z}$ onto $mathfrak{A}$, provided $ainaut(ga)$ commute with the $*$-automorphism $th$ up tu a unitary $uinga$. Here, $Phi_{theta, u}inaut(mathfrak{A}rtimes_alphamathbb{Z})$ 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.