Dimension, entropy and Lyapunov exponents

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We consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

Original languageEnglish (US)
Pages (from-to)109-124
Number of pages16
JournalErgodic Theory and Dynamical Systems
Issue number1
StatePublished - Mar 1982


ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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