  We provide a systematic study of a noncommutative extension of the classical
Anzai skew-product for the cartesian product of two copies of the unit circle
to the noncommutative 2-tori. In particular, some relevant ergodic properties
are proved for these quantum dynamical systems, extending the corresponding
ones enjoyed by the classical Anzai skew-product. As an application, for a
uniquely ergodic Anzai skew-product $\F$ on the noncommutative $2$-torus
$\ba_\a$, $\a\in\br$, we investigate the pointwise limit,
$\lim_{n\to+\infty}\frac1{n}\sum_{k=0}^{n-1}\l^{-k}\F^k(x)$, for $x\in\ba_\a$
and $\l$ a point in the unit circle, and show that there exist examples for
which the limit does not exist even in the weak topology.
