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We consider nonequilibrium probabilistic dynamics in logistic-like maps$x_{t+1}=1-a|x_t|^z$, $(z>1)$ at their chaos threshold: We first introduce manyinitial conditions within one among $W>>1$ intervals partitioning the phasespace and focus on the unique value $q_{sen}<1$ for which the entropic form$S_q \equiv \frac{1-\sum_{i=1}^{W} p_i^q}{q-1}$ {\it linearly} increases withtime. We then verify that $S_{q_{sen}}(t) - S_{q_{sen}}(\infty)$ vanishes like$t^{-1/[q_{rel}(W)-1]}$ [$q_{rel}(W)>1$]. We finally exhibit a new finite-sizescaling, $q_{rel}(\infty) - q_{rel}(W) \propto W^{-|q_{sen}|}$. Thisestablishes quantitatively, for the first time, a long pursued relation betweensensitivity to the initial conditions and relaxation, concepts which playcentral roles in nonextensive statistical mechanics.Comment: Final version with new Title and small modifications. REVTeX, 8 pages  and 4 eps figure

Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos