Skip to main content Accessibility help

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


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.



Hide All
[1]Abadie, B. and Dykema, K.. Unique ergodicity of free shifts and some other automorphisms of C -algebras. J. Operator Theory 61 (2009), 279294.
[2]Anzai, H.. Ergodic Anzai skew-product transformations on the torus. Osaka Math. J. 3 (1951), 8399.
[3]Barreto, S. D. and Fidaleo, F.. Disordered Fermions on lattices and their spectral properties. J. Stat. Phys. 143 (2011), 657684.
[4]Boca, F.-P. Rotation C -Algebras and almost Mathieu Operators. Theta, Bucharest, 2001.
[5]Carey, A., Phillips, J. and Rennie, A.. Spectral triples: examples and index theory. Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory (ESI Lectures in Mathematics and Physics). Ed. Carey, A.. European Mathematical Society, Zurich, 2011, pp. 175265.
[6]Connes, A.. Noncommutative Geometry. Academic Press, San Diego. CA, 1994.
[7]Connes, A. and Moscovici, H.. Modular curvature for noncommutative two-tori. J. Amer. Math. Soc. 27 (2014), 639684.
[8]Cornfeld, I. P., Fomin, S. V. and Sinai, Ya G.. Ergodic Theory. Springer, Berlin, 1982.
[9]Crismale, V. and Fidaleo, F.. De Finetti theorem on the CAR algebra. Commun. Math. Phys. 315 (2012), 135152.
[10]Crismale, V. and Fidaleo, F.. Exchangeable stochastic processes and symmetric states in quantum probability. Ann. Mat. Pura Appl. (4) 194 (2015), 969993.
[11]Davidson, K. R.. C -Algebras by Example (Fields Institute Monographs, 6). American Mathematical Society, Providence, RI, 1996.
[12]Fidaleo, F.. KMS states and the chemical potential for disordered systems. Commun. Math. Phys. 262 (2006), 373391.
[13]Fidaleo, F.. On the entangled ergodic theorem. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), 6777.
[14]Fidaleo, F.. On strong ergodic properties of quantum dynamical systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 551564.
[15]Fidaleo, F.. The entangled ergodic theorem in the almost periodic case. Linear Algebra Appl. 432 (2010), 526535.
[16]Fidaleo, F.. Nonconventional ergodic theorems for quantum dynamical systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), 1450009.
[17]Fidaleo, F.. Uniform convergence of Cesaro averages for uniquely ergodic C -dynamical systems. Entropy 20 (2018), 987.
[18]Fidaleo, F.. Fourier analysis for type III representations of the noncommutative torus. J. Fourier Anal. Appl. 25 (2019), 28012835.
[19]Fidaleo, F. and Mukhamedov, F.. Strict weak mixing of some C -dynamical systems based on free shifts. J. Math. Anal. Appl. 336 (2007), 180187.
[20]Fidaleo, F. and Mukhamedov, F.. Ergodic properties of Bogoliubov automorphisms in free probability. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), 393411.
[21]Fidaleo, F. and Suriano, L.. Type III representations and modular spectral triples for the noncommutative torus. J. Funct. Anal. 275(2018) 14841531.
[22]Furstenberg, H.. Strict ergodicity and transformation on the torus. Amer. J. Math. 83(1961) 573601.
[23]Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31(1977) 204256.
[24]Longo, R. and Peligrad, C.. Noncommutative topological dynamics and compact actions on C -algebras. J. Funct. Anal. 58(1984) 157174.
[25]Niculescu, C. P., Ströh, A. and Zsidó, L.. Noncommutative extensions of classical and multiple recurrence theorems. J. Operator Theory 50 (2003), 352.
[26]Osaka, H. and Phillips, N. C.. Furstenberg transformations on irrational rotation algebras. Ergod. Th. & Dynam. Sys. 26 (2006), 16231651.
[27]Rieffel, M. A.. The cancellation theorem for projective modules over irrational rotation C -algebras. Proc. Lond. Math. Soc. 47 (1983), 285302.
[28]Robinson, E. A. Jr. On uniform convergence in the Wiener–Wintner theorem. J. Lond. Math. Soc. 49 (1994), 493501.
[29]Sakai, S.. C -Algebras and W -Algebras. Springer, Berlin, 1971.


MSC classification

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


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed