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

Simone Del Vecchio, Francesco Fidaleo, Luca Giorgetti, Stefano Rossi
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.

operator algebras, infinite dimensional dynamics, classical ergodic theory