Extraction of underlying multiplicative processes from multifractals via the thermodynamic formalism

This paper describes various procedures for extracting an underlying multiplicative process from the thermodynamic description of multifractals (i.e., its $D_q$ curve), and points out the associated pitfalls of such procedures. We extend previous work by Feigenbaum, Jensen, and Procaccia [Phys. Rev. Lett. 57, 1567 (1986)] using transfer matrices to the case of singular measures, and develop the corresponding thermodynamic formalism. We find that the extraction procedure based solely on information from the $D_q$ curves allows for an infinity of cascade processes and that additional dynamical information is required to remove this degeneracy. In addition, we find that different multiplicative processes with only three free parameters produce excellent fits to all the $D_q$ curves studied in this paper, which further confuses the inversion process when it is applied to experimental data. We then examine the application of these procedures to a variety of computer and laboratory experiments, such as the period-doubling attractor, the golden-mean circle-map attractor, and a reanalysis of Rayleigh-Bénard experiments corresponding to these examples. Finally we consider laboratory experiments on open flows in two different circumstances. The first deals with velocity measurements in the wake of an oscillating cylinder and has dynamics closely related to that of the circle map. The second case corresponds to the spatial distribution of turbulent energy dissipation in several flows (grid turbulence, wakes, and boundary layers in the laboratory and atmosphere), where the underlying dynamics are presently not well understood. In each of these examples we highlight the above-mentioned ambiguities and, in cases where additional information is available, apply the procedure to extract basic underlying length scales of the phenomena.