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

Published online by Cambridge University Press:  14 January 2020

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma00133, Italy email,,
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma00133, Italy email,,
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN37240, USA email
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma00133, Italy email,,


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.

Original Article
© The Author(s) 2020. Published by Cambridge University Press

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