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Power-law sensitivity to initial conditions, characterizing the behaviour ofdynamical systems at their critical points (where the standard Liapunovexponent vanishes), is studied in connection with the family of nonlinear 1Dlogistic-like maps $x_{t+1} = 1 - a | x_t |^z, (z > 1; 0 < a \le 2;t=0,1,2,...)$ The main ingredient of our approach is the generalized deviationlaw $\lim_{\Delta x(0) -> 0} \Delta x(t) / \Delta x(0)} = [1+(1-q)\lambda_qt]^{1/(1-q)}$ (equal to $e^{\lambda_1 t}$ for q=1, and proportional, for larget, to $t^{1/(1-q)}$ for $q \ne 1; q \in R$ is the entropic index appearing inthe recently introduced nonextensive generalized statistics). The relationbetween the parameter q and the fractal dimension d_f of the onset-to-chaosattractor is revealed: q appears to monotonically decrease from 1(Boltzmann-Gibbs, extensive, limit) to -infinity when d_f varies from 1(nonfractal, ergodic-like, limit) to zero.Comment: LaTeX, 6 pages , 5 figure

Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity