Renyi entropies of aperiodic dynamical systems

Floris Takens, Evgeny Verbitskiy

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    Abstract

    In this paper we continue the study of Renyi entropies of measure-preserving transformations started in [22]. We have established there that for ergodic transformations with positive entropy, the Renyi entropies of order q, qis an element ofR, are equal to either plus infinity (q1 are equal to the essential infimum of the measure-theoretic entropies of measures forming the decomposition into ergodic components. Thus, it is possible that the Renyi entropies of order q>1 are strictly smaller than the measure-theoretic entropy, which is the average value of entropies of ergodic components.

    This result is a bit surprising: the Renyi entropies are metric invariants, which are sensitive to ergodicity.

    The proof of the described result is based on the construction of partitions with independent iterates. However, these partitions are obtained in different ways depending on q: for q>1 we use a version of the well-known Sinai theorem on Bernoulli factors for the non-ergodic transformations; for q

    Original languageEnglish
    Pages (from-to)279-302
    Number of pages24
    JournalIsrael journal of mathematics
    Volume127
    Publication statusPublished - 2002

    Keywords

    • STRANGE ATTRACTORS
    • DIMENSIONS

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