We study the scaling of the density of iterates of the logistic map at a representative set of dynamical regimes: From the periodic motion at superstable orbits of the period-doubling route to chaos and its corresponding, aperiodic, accumulation point; to the chaotic motion at Misiurewicz points along the band-splitting chaotic cascade. We restate the problem of studying individual orbits of the logistic map and change the picture to ensembles of positions evolving under the action of a linear operator acting on the density of iterations, in close resemblance to the classical normal diffusion, on which the problem of tracking the dynamics of single particles via the set Langevin equations for each particle, can be recast into a single partial differential equation for the evolution of the probability density of finding a particle at a specific position and time, i.e., the Fokker-Planck equation. As a novelty, we have characterized in detail the evolution of the density of iterates in the phase space originated by the action of the Frobenius-Perron (FP) operator. It is found how a separation of time scales emerges; the dynamics underlying the evolution of individual trajectories on the time scale of single iterations of the map becomes, asymptotically, a larger time scale, if the condition that the density be invariant under the action of the FP operator is imposed. This time scales are exponentially separated: iterations follow the scale t, while τ=2^t are the times for which the action of the FP dynamics is invariant independently of the initial distribution of points in phase space. To be clear, the asymptotic distribution will almost always be different for different initial distributions but it will be invariant under the FP operator at times τ. The scaling in the support of the density over this induced time scale τ is found to follow a renormalization-group (RG) structure. The entropy associated with the evolving and invariant densities via the Shannon expression has extrema at the fixed points of the proposed RG operation. This formalism summarizes in a novel and compact fashion many of the findings exposed in [1-3] and opens the way to new insights by allowing the direct application of our RG-entropy approach.
