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

October 25, 2019

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 \Phi on the noncommutative 2-torus
\AA_\a, \a\in\TT, we investigate the pointwise limit,
\lim_{n\to+\infty}\frac1{n}\sum_{k=0}^{n-1}\l^{-k}\Phi^k(x), for x\in\AA_\a
and \lambda a point in the unit circle, and show that there exist examples for
which the limit does not exist even in the weak topology.

Keywords:

operator algebras, infinite dimensional dynamics, classical ergodic theory