Circular-like maps: sensitivity to the initial conditions, multifractalityand nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos th reshold) at which the Lyapunov exponent vanishes. Consistently, the sensiti vity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1 /(1 - q), where q characterizes the nonextensivity of a generalized entropi c form currently used to extend standard, Boltzmann-Gibbs statistical mecha nics in order to cover a variety of anomalous situations. It has been recen tly proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such ma ps the scaling law 1/(1 - q) = 1/alpha(min) - 1/(alpha max), where alpha(mi n) and alpha(max) are the extreme values appearing in the multifractal f(al pha) function. We generalize herein the usual circular map by considering i nflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d(f) equals unity for all z in contrast with q which does depend on z, it b ecomes clear that d(f) plays no major role in the sensitivity to the initia l conditions.