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
$\unicode[STIX]{x1D6F7}$
on the non-commutative
$2$
-torus
$\mathbb{A}_{\unicode[STIX]{x1D6FC}}$
,
$\unicode[STIX]{x1D6FC}\in \mathbb{R}$
, we investigate the pointwise limit,
$\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$
, for
$x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$
and
$\unicode[STIX]{x1D706}$
a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.
SKEW-PRODUCT DYNAMICAL SYSTEMS, ELLIS GROUPS AND TOPOLOGICAL CENTRE