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Information Meaning of Entropy of Nonergodic Measures

Information Meaning of Entropy of Nonergodic Measures The limit frequency properties of trajectories of the simplest dynamical system generated by the left shift on the space of sequences of letters from a finite alphabet are studied. The following modification of the Shannon-McMillan-Breiman theorem is proved: for any invariant (not necessarily ergodic) probability measure μ on the sequence space, the logarithm of the cardinality of the set of all μ-typical sequences of length n is equivalent to nh(μ), where h(μ) is the entropy of the measure μ. Here a typical finite sequence of letters is understood as a sequence such that the empirical measure generated by it is close to μ (in the weak topology). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Information Meaning of Entropy of Nonergodic Measures

Differential Equations , Volume 55 (3) – Apr 24, 2019

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Pleiades Publishing, Ltd.
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266119030029
Publisher site
See Article on Publisher Site

Abstract

The limit frequency properties of trajectories of the simplest dynamical system generated by the left shift on the space of sequences of letters from a finite alphabet are studied. The following modification of the Shannon-McMillan-Breiman theorem is proved: for any invariant (not necessarily ergodic) probability measure μ on the sequence space, the logarithm of the cardinality of the set of all μ-typical sequences of length n is equivalent to nh(μ), where h(μ) is the entropy of the measure μ. Here a typical finite sequence of letters is understood as a sequence such that the empirical measure generated by it is close to μ (in the weak topology).

Journal

Differential EquationsSpringer Journals

Published: Apr 24, 2019

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