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Quantum geometry of noncommutative Bernoulli shifts

Robert Alicki

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 25-29
  • ISSN: 0137-6934

Abstract

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We construct an example of a noncommutative dynamical system defined over a two dimensional noncommutative differential manifold with two positive Lyapunov exponents equal to ln d each. This dynamical system is isomorphic to the quantum Bernoulli shift on the half-chain with the quantum dynamical entropy equal to 2 ln d. This result can be interpreted as a noncommutative analog of the isomorphism between the classical one-sided Bernoulli shift and the expanding map of the circle and moreover as an example of the noncommutative Pesin theorem.

How to cite

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Alicki, Robert. "Quantum geometry of noncommutative Bernoulli shifts." Banach Center Publications 43.1 (1998): 25-29. <http://eudml.org/doc/208847>.

@article{Alicki1998,
abstract = {We construct an example of a noncommutative dynamical system defined over a two dimensional noncommutative differential manifold with two positive Lyapunov exponents equal to ln d each. This dynamical system is isomorphic to the quantum Bernoulli shift on the half-chain with the quantum dynamical entropy equal to 2 ln d. This result can be interpreted as a noncommutative analog of the isomorphism between the classical one-sided Bernoulli shift and the expanding map of the circle and moreover as an example of the noncommutative Pesin theorem.},
author = {Alicki, Robert},
journal = {Banach Center Publications},
keywords = {noncommutative dynamical system; two-dimensional noncommutative differential manifold; positive Lyapunov exponents; quantum Bernoulli shift; noncommutative Pesin theorem},
language = {eng},
number = {1},
pages = {25-29},
title = {Quantum geometry of noncommutative Bernoulli shifts},
url = {http://eudml.org/doc/208847},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Alicki, Robert
TI - Quantum geometry of noncommutative Bernoulli shifts
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 25
EP - 29
AB - We construct an example of a noncommutative dynamical system defined over a two dimensional noncommutative differential manifold with two positive Lyapunov exponents equal to ln d each. This dynamical system is isomorphic to the quantum Bernoulli shift on the half-chain with the quantum dynamical entropy equal to 2 ln d. This result can be interpreted as a noncommutative analog of the isomorphism between the classical one-sided Bernoulli shift and the expanding map of the circle and moreover as an example of the noncommutative Pesin theorem.
LA - eng
KW - noncommutative dynamical system; two-dimensional noncommutative differential manifold; positive Lyapunov exponents; quantum Bernoulli shift; noncommutative Pesin theorem
UR - http://eudml.org/doc/208847
ER -

References

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  9. [9] T. Hudetz, Quantum dynamical entropy revised, this volume. 
  10. [10] G. Lindblad, Dynamical Entropy for Quantum Systems, in: Quantum Probability and Applications, Vol.III, L. Accardi and W. von Waldenfels (eds.), Springer LNM 1303, Berlin, 1988, 183-191. 
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  12. [12] P. Tuyls, Towards Quantum Kolmogorov-Sinai Entropy, Ph.D. Thesis, Leuven, 1997. 
  13. [13] D. Voiculescu, Dynamical approximation entropies and topological entropy in operator algebras, Commun. Math. Phys. 144, (1992) 443-490. 

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