ON STRONG ERGODIC PROPERTIES OF QUANTUM DYNAMICAL SYSTEMS

We show that the shift on the reduced C*-algebras of RD-groups, including the free group on infinitely many generators, and the amalgamated free product C*-algebras, enjoys the very strong ergodic property of the convergence to the equilibrium. Namely, the free shift converges, pointwise in the weak topology, to the conditional expectation onto the fixed-point subalgebra. Provided the invariant state is unique, we also show that such an ergodic property cannot be fulfilled by any classical dynamical system, unless it is conjugate to the trivial one-point dynamical system.

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Logical entropy of quantum dynamical systems

Open Physics ◽

10.1515/phys-2015-0058 ◽

2016 ◽

Vol 14 (1) ◽

pp. 1-5 ◽

Cited By ~ 14

Author(s):

Abolfazl Ebrahimzadeh

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Quantum Logic ◽

Quantum Dynamical ◽

Quantum Dynamical System ◽

Ergodic Properties ◽

Quantum Dynamical Systems ◽

Logical Entropy

AbstractThis paper introduces the concepts of logical entropy and conditional logical entropy of hnite partitions on a quantum logic. Some of their ergodic properties are presented. Also logical entropy of a quantum dynamical system is dehned and ergodic properties of dynamical systems on a quantum logic are investigated. Finally, the version of Kolmogorov-Sinai theorem is proved.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.116 ◽

2020 ◽

pp. 1-22 ◽

Cited By ~ 1

Author(s):

SIMONE DEL VECCHIO ◽

FRANCESCO FIDALEO ◽

LUCA GIORGETTI ◽

STEFANO ROSSI

Keyword(s):

Dynamical Systems ◽

Unit Circle ◽

Systematic Study ◽

Cartesian Product ◽

Skew Product ◽

Quantum Dynamical ◽

Ergodic Properties ◽

Pointwise Limit ◽

Quantum Dynamical Systems ◽

Uniquely Ergodic

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.

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Spectrum of quantum dynamical systems: Subsystems and entropy

Advances in Pure and Applied Mathematics ◽

10.1515/apam-2013-0034 ◽

2014 ◽

Vol 5 (2) ◽

Author(s):

Mona Khare ◽

Anurag Shukla

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Orthomodular Lattices ◽

Quantum Dynamical ◽

Quantum Dynamical System ◽

Quantum Dynamical Systems

Abstract.In the present paper, we have studied quantum dynamical systems of difference posets, their equivalence, subsystems and spectra. It is shown that every subsystem of a mixing quantum dynamical system is mixing, and also a bounded spectrum of quantum dynamical systems has its supremum, and all such suprema are equivalent. Entropy of subsystems of a quantum dynamical system of orthomodular lattices is also investigated.

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BACH FLOWS OF PRODUCT MANIFOLDS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500399 ◽

2012 ◽

Vol 09 (05) ◽

pp. 1250039 ◽

Cited By ~ 2

Author(s):

SANJIT DAS ◽

SAYAN KAR

Keyword(s):

Dynamical System ◽

Fixed Point ◽

Ricci Flow ◽

Flow Characteristics ◽

Geometric Flow ◽

Flow Equations ◽

First Order ◽

Bach Tensor ◽

Point Condition ◽

Future Work

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S2 × S2, R2 × S2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.

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Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418300112 ◽

2018 ◽

Vol 28 (04) ◽

pp. 1830011

Author(s):

Mio Kobayashi ◽

Tetsuya Yoshinaga

Keyword(s):

Dynamical System ◽

Fixed Point ◽

Chaotic Behavior ◽

Periodic Points ◽

Two Dimensional ◽

Stable Fixed Point ◽

Bifurcation Structure ◽

Unstable Fixed Point ◽

Gaussian Map ◽

Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

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