  We study the evolution of the probability density of ensembles of iterates of
the logistic map that advance towards and finally remain at attractors of
representative dynamical regimes. We consider the mirror families of
superstable attractors along the period-doubling cascade, and of chaotic-band
attractors along the inverse band-splitting cascade. We examine also their
common aperiodic accumulation point. The iteration time progress of the
densities of trajectories is determined via the action of the Frobenius-Perron
(FP) operator. As a difference with the study of individual orbits, the
analysis of ensembles of positions offers a viewpoint from which the nonlinear
dynamical features of this iconic model can be better characterized in
statistical-mechanical terms. The scaling of the densities along the considered
families of attractors conforms to a renormalization-group (RG) structure,
while their entropies are seen to attain extrema at the fixed points of the RG
flows. Additionally, this entropy as a function of the map control parameter
displays the characteristic features of an equation of state of a thermal
system undergoing a second-order phase transition. We discuss our results.
