Rank ordered beta distributions of nonlinear map symbolic dynamics families with a first-order transition between...

Rank-ordered distributions have been a matter of intense study. Often Zipf type invariant scaling is invoked; however, in the last decade the ubiquity of a Discrete Generalized Beta Distribution, DGBD, with two scaling exponents has been established. This distribution incorporates deviations from the power law at the extremes. A proper understanding of the meaning of these exponents is still lacking. Here, using two families of unimodal maps on the [0,1] interval, we construct binary sequences via standard symbolic dynamics. In both cases, the tent map, which is at the convex-concave border of the mapping families, separates intermittent regimes from chaotic dynamics. We show that the frequencies of n-tuples of the generated symbolic sequences are remarkably well fitted by the DGBD. We argue that in the underlying dynamics an order-disorder competition takes place and that one of the exponents is related to multiple range correlations, while the other is sensitive to disorder. In our study, we implement thermodynamic formalisms with which we can readily calculate n-tuple frequencies, in some particular cases, analytically. We show that for the convex mappings there is a first-order thermodynamic phase transition, while concave mappings have smooth free energy densities. Within our DGBD study, the transition between these two regimes coincides with a zero value for both exponents; in this sense, they may even be considered as indicators of the transition. An analysis of the difference between the exponents reinforces the interpretation we have assigned to them. Furthermore, the two regimes can be identified by the sign of such a difference. We also show that divergences in the invariant densities are responsible for the first order phase transitions observed in a range of the rank-frequency distributions. Our findings give further support to previous studies based on expansion-modification algorithms, birth-death processes, and random variable subtraction dynamics.