WIAS Preprint No. 298, (1996)

On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity


  • Barreira, Luis
  • Pesin, Yakov
  • Schmeling, Jörg
    ORCID: 0000-0001-6956-9463

2010 Mathematics Subject Classification

  • 58Fll 58F12 28D99 28C99


  • Conformal repeller, Gibbs measure, local entropy, Lyapunov exponents, multifractal analysis, multifractal rigidity, pointwise dimension




We introduce the mathematical concept of multifracfality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by providing some physical motivation and describing several non-trivial examples. Among them are subshifts of finite type and one-dimensional Markov maps. An essential part of the paper is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a "physical" classification of dynamical systems. For a class of Markov maps, we show that if the multifractal spectra for dimensions of two maps coincide, then the maps are differentiably equivalent.

Appeared in

  • Chaos 7 (1997) no. 1, pp. 27--38.

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