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Ergodic properties of the Anzai skew-product for the non-commutative torus
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 4 / April 2021
- Published online by Cambridge University Press:
- 14 January 2020, pp. 1064-1085
- Print publication:
- April 2021
-
- Article
- Export citation
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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.
