Ergodic properties of the Anzai skew- product for 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 $\text{F}$ on the noncommutative $2$-torus ${\text{ba}}_{\text{a}}$, $\text{a}\in \text{br}$, we investigate the pointwise limit, $\underset{n\to +\mathrm{\infty }}{lim}\frac{1}{n}\sum _{k=0}^{n-1}{\text{l}}^{-k}{\text{F}}^k(x)$, for $x\in {\text{ba}}_{\text{a}}$ and $\text{l}$ a point in the unit circle, and show that there exist examples for which the limit does not exist even in the weak topology.