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

Power-law sensitivity to initial conditions, characterizing the behavior of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear one-dimensional logisticlike maps x(t+1)= 1 - a\x(t)\(z) (z > 1; 0 < a less than or equal to 2; t = 0,1,2,...). The main ingredient of our approach is the generalized deviation law lim (Delta x(0)-->0)[Delta x(t)/Delta x(0)] = [1 + (1-q)lambda(q)t](1/(l - q)) (equal to e(lambda lt) for q = 1, and proportional, for large t, to t(1/(1-q)) for q not equal 1; q is an element of R is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension d(f) of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to -infinity when d(f) varies from 1 (nonfractal, ergodiclike, limit) to zero.