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

Abstract

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.

Spectral and ergodic properties of completely positive maps and decoherence

“…The following result will come in useful in the sequel. 4 Proposition 4.2. Let Φ ∶ A → A be a unital * -endomorphism of an abelian C * -algebra A, together with its natural extension…”

“…Concerning some relevant ergodic properties associated to the peripheral spectrum, the reader is referred to [9] and the reference cited therein. Very recent developments of ergodic theory in the quantum setting, instead, can be found in [4,5].…”

Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

“…For some examples of uniquely ergodic C * -dynamical systems, we get σ ph pp (Φ) σ ph pp (V ϕ,Φ ), see e.g. [5,11,16].…”

“…Some examples based on noncommutative 2-torus relative to the non convergence of M a,λ (n) for λ ∈ σ ph pp (V ϕ,Φ ) σ ph pp (Φ) are also exhibited in [5].…”

“…More complicated examples arising from the noncommutative geometry (i.e. the noncommutative 2-torus, see [5]) will be presented elsewhere.…”