Multifractality and nonextensivity at the edge of chaos of unimodal maps

We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity z>1. First we determine analytically the sensitivity to initial conditions ξt. Then we consider a renormalization group operation on the partition function Z of the multifractal attractor that eliminates one half of the multifractal points each time it is applied. Invariance of Z fixes a length-scale transformation factor 2−η in terms of the generalized dimensions Dβ. There exists a gap Δη in the values of η equal to λq=1/(1−q)=D∞−1−D−∞−1 where λq is the q-generalized Lyapunov exponent and q is the nonextensive entropic index. We provide an interpretation for this relationship—previously derived by Lyra and Tsallis—between dynamical and geometrical properties.