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

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity 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 Ȣ q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. <Emphasis Type="Bold">80, 53 (1998)) for such maps the scaling law 1/(1 − q)=1/α <Subscript>min</Subscript> − 1/α <Subscript>max</Subscript>, where α <Subscript>min</Subscript> and α <Subscript>max</Subscript> are the extreme values appearing in the multifractal f(α) function. We generalize herein the usual circular map by considering inflexions 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 <Subscript> f </Subscript> equals unity for all z in contrast with q which does depend on z, it becomes clear that d <Subscript> f </Subscript> plays no major role in the sensitivity to the initial conditions. Copyright Società Italiana di Fisica, Springer-Verlag 1999