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

• Published in: Ergodic theory and dynamical systems Vol. 41; no. 4; pp. 1064 - 1085
• Main Authors: DEL VECCHIO, SIMONE, FIDALEO, FRANCESCO, GIORGETTI, LUCA, ROSSI, STEFANO
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.04.2021
• Subjects: Cartesian coordinates; Ergodic processes; Original Article; Topology; Toruses; non-commutative torus; skew-product; 37A55; 58B34; 58J51; operator algebras; ergodic dynamical systems; unique ergodicity; 46L55; 46L65
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2019.116

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.