Ergodic properties of the Anzai skew-product for the non-commutative torus

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 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 on the non-commutative -torus , , we investigate the pointwise limit, , for and a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.