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Multifractality of the Feigenbaum attractor and fractional derivatives

U. Frisch·K. Khanin·T. Matsumoto
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Abstract

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 $f(α)$. 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.

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U. Frisch, K. Khanin, T. Matsumoto. 2004-04-10. Multifractality of the Feigenbaum attractor and fractional derivatives. https://doi.org/10.1007/s10955-005-7011-4

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