Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality, and nonextensivity

For a family of logisticlike maps, we investigate the rate of convergence t o the critical attractor when an ensemble of initial conditions is uniforml y spread over the entire phase space. We found that the phase-space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic o scillations reflecting the multifractal character of the critical attractor . We explore the parametric dependence of the power-law exponent and the am plitude of the log-periodic oscillations with the attractor's fractal dimen sion governed by the inflection of the map near its extremal point. Further , we investigate the temporal evolution of W(t) for the circle map whose cr itical attractor is dense. In this case, we found W(t) to exhibit a rich pa ttern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.