Logistic map trajectory distributions: Renormalization-group, entropy, and criticality at the transition to chaos

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