Ergodic properties of the Anzai skew-product on the noncommutative torus

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 Φ on the noncommutative 2-torus $mathbb{A}_alpha$, $alphainmathbb{R}$, we investigate the pointwise limit, $lim_{n ightarrowinfty}rac{1}{n}sum_{k=0}^{n-1}lambda^{-k}Phi^k(x)$, for $xinmathbb{A}_alpha$ and $lambda$ apoint in the unit circle, and show that there exist examples for which the limit does not exist even in the weak topology.