  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 $\Delta \eta $ in the values of $\eta $ equal to $\lambda
_{q}=1/(1-q)=D_{\infty}^{-1}-D_{-\infty}^{-1}$ where $\lambda_{q}$ 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. Key Words: Edge of
chaos, multifractal attractor, nonextensivity
