Multifractality of the Feigenbaum attractor and fractional derivatives

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 char-acterizing 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.