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

Part of: Infinite-dimensional manifolds Partial differential equations on manifolds; differential operators Selfadjoint operator algebras Ergodic theory

Published online by Cambridge University Press:  14 January 2020

SIMONE DEL VECCHIO Open the ORCID record for SIMONE DEL VECCHIO [Opens in a new window] ,
FRANCESCO FIDALEO Open the ORCID record for FRANCESCO FIDALEO [Opens in a new window] ,
LUCA GIORGETTI Open the ORCID record for LUCA GIORGETTI [Opens in a new window]  and
STEFANO ROSSI Open the ORCID record for STEFANO ROSSI [Opens in a new window]
SIMONE DEL VECCHIO
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma00133, Italy email delvecch@mat.uniroma2.it, fidaleo@mat.uniroma2.it, rossis@mat.uniroma2.it
FRANCESCO FIDALEO
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma00133, Italy email delvecch@mat.uniroma2.it, fidaleo@mat.uniroma2.it, rossis@mat.uniroma2.it
LUCA GIORGETTI
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN37240, USA email luca.giorgetti@vanderbilt.edu
STEFANO ROSSI
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma00133, Italy email delvecch@mat.uniroma2.it, fidaleo@mat.uniroma2.it, rossis@mat.uniroma2.it
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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.

Keywords

operator algebrasnon-commutative torusskew-productergodic dynamical systemsunique ergodicity

MSC classification

Primary: 46L55: Noncommutative dynamical systems 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity
Secondary: 37A55: Relations with the theory of $C^*$-algebras 58B34: Noncommutative geometry (à la Connes) 46L65: Quantizations, deformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1064 - 1085
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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