Multifractality of the feigenbaum attractor and fractional derivatives

U. Frisch, K. Khanin, T. Matsumoto

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities (f(a). This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions [Frisch and Matsumoto, J. Stat. Phys. 108:1181, 2002]. The relation between the thermodynamic approach [Vul, Sinai and Khanin, Russian Math. Surveys 39:1, 1984] and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations. © 2005 Springer Science+Business Media, Inc.

Original languageEnglish
Pages (from-to)671-695
Number of pages25
JournalJournal of Statistical Physics
Issue number5-6
Publication statusPublished - Dec 2005


  • Chaotic dynamics
  • Multifractals
  • Thermodynamic formalism


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