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Information Meaning of Entropy of Nonergodic Measures

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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).

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Authors and Affiliations

  1. John Paul II Catholic University of Lublin, Lublin, 20-950, Poland

    V. I. Bakhtin

  2. Belarusian State University, Minsk, 220030, Belarus

    V. I. Bakhtin

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  1. V. I. Bakhtin
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Correspondence to V. I. Bakhtin.

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Bakhtin, V.I. Information Meaning of Entropy of Nonergodic Measures. Diff Equat 55, 294–302 (2019). https://doi.org/10.1134/S0012266119030029

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  • DOI: https://doi.org/10.1134/S0012266119030029

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