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

For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly 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 oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor’s fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of $W(t)$ for the circle map whose critical attractor is dense. In this case, we found $W(t)$ to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of non-extensive Tsallis entropies.