Ergodic properties of the Anzai skew-product for the noncommutative torus

TL;DR

This paper explores the ergodic properties of a noncommutative extension of the classical Anzai skew-product on noncommutative 2-tori, revealing new behaviors and limits in quantum dynamical systems.

Contribution

It extends classical ergodic results to noncommutative tori and investigates the convergence of certain averages in quantum dynamical systems.

Findings

Proved ergodic properties for noncommutative Anzai skew-products.

Identified cases where the pointwise limit does not exist.

Extended classical ergodic theory to noncommutative geometry.

Abstract

We provide a systematic study of a noncommutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the noncommutative 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 $\F$ on the noncommutative $2$-torus $\ba_\a$, $\a\in\br$, we investigate the pointwise limit, $\lim_{n\to+\infty}\frac1{n}\sum_{k=0}^{n-1}\l^{-k}\F^k(x)$, for $x\in\ba_\a$ and $\l$ a point in the unit circle, and show that there exist examples for which the limit does not exist even in the weak topology.