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 ${\xi }_t$. Then we consider a renormalization group (RG) 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 $2^{-\eta }$ in terms of the generalized dimensions $D_{\beta }$. There exists a gap $\mathrm{\Delta }\eta$ in the values of $\eta$ equal to $\lambda {q}=1/(1-q)=D{\infty}^{-1}-D_{-\infty}^{-1}$where$\lambda_{q}$isthe$q$-generalizedLyapunovexponentand$q$ is the nonextensive entropic index. We provide an interpretation for this relationship - previously derived by Lyra and Tsallis - between dynamical and geometrical properties. Key Words: Edge of chaos, multifractal attractor, nonextensivity