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.

中文翻译：

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非交换环面 Anzai 斜积的遍历性质

我们系统地研究了经典 Anzai 斜积的非交换扩展，将单位圆的两个副本的笛卡尔乘积扩展到非交换 2-tori。特别是，证明了这些量子动力学系统的一些相关遍历性质，扩展了经典 Anzai 偏积所享有的相应性质。作为一个应用程序，对于一个独特的遍历 Anzai 斜积$\unicode[STIX]{x1D6F7}$关于不可交换的$2$-环面$\mathbb{A}_{\unicode[STIX]{x1D6FC}}$,$\unicode[STIX]{x1D6FC}\in \mathbb{R}$，我们研究逐点限制，$\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{ x1D6F7}^{k}(x)$， 为了$x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$和$\unicode[STIX]{x1D706}$单位圆中的一个点，并表明存在不存在极限的示例，即使在弱拓扑中也是如此。