%0 Journal Article
%T Multifractality of the Feigenbaum attractor and fractional derivatives
%A U. Frisch
%A K. Khanin
%A T. Matsumoto
%J Statistics
%D 2003
%I arXiv
%R 10.1007/s10955-005-7011-4
%X 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 $f(\alpha)$. 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.
%U http://arxiv.org/abs/nlin/0309068v2