Ergodic Properties of the Relaxation Phase in Nonchaotic Unimodal Maps

The convergence to the mean values of observables is studied for nonlinear dynamical systems in the period-doubling bifurcation regime. The phase space convergence to the mean values is studied numerically; it reveals a characteristic behaviour induced by several special points in phase space. The convergence to the mean value for these points is exponential as opposed to the power-law convergence of the majority of the phase space. The issue of universality of these results which characterize the period doubling bifurcation behaviour is discussed.