Multifractal Analysis of Local Entropies for Gibbs Measures

Floris Takens, Evgeni Verbitski

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    Abstract

    Recently a complete multifractal analysis of local dimensions, entropies and Lyapunov exponents of conformal expanding maps and surface Axion A diffeomorphisms for Gibbs measures was performed. The main goal of this was primarily the analysis of the local (pointwise) dimensions. This is an extremely difficult problem and, for example, similar results for hyperbolic systems in dimensions 3 and higher have not been yet obtained. In the present work we concentrate our attention on the multifractal analysis of the local (pointwise) entropies. We are able to obtain results, which are similar to those mentioned above, for Gibbs measures of the expansive homeomorphisms with specification property. Note that such homeomorphisms may not have Markov partitions, which is a crucial condition in the previous works. However, due to the fact that less is known about thermodynamical properties of these dynamical systems we were able to obtain only the continuous differentiability of the multifractal spectrum of local entropies (compare: the same spectra for the dynamical systems with Markov partitions are analytic). We believe that the smoothness of the multifractal spectrum in our case can be improved. We have related the multifractal spectrum of the local entropies to the the spectrum of correlation entropies. These correlation entropies serve as entopy-like analogs of the Hentshel-Procaccia and Rényi spectra of generalized dimensions. This allows us to complete the duality between the multifractal analyses of local dimensions and entropies.
    Original languageEnglish
    Pages (from-to)143-151
    Number of pages9
    JournalPeriodica Mathematica Hungarica
    Volume37
    Issue number1-3
    Publication statusPublished - 1998

    Keywords

    • thermodynamical formalism
    • local entropies
    • multifractal analysis

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