  We examine the relationship between two different types of ranked data,
frequencies and magnitudes. We consider data that can be sorted out either way,
through numbers of occurrences or size of the measures, as it is the case, say,
of moon craters, earthquakes, billionaires, etc. We indicate that these two
types of distributions are functional inverses of each other, and specify this
link, first in terms of the assumed parent probability distribution that
generates the data samples, and then in terms of an analog (deterministic)
nonlinear iterated map that reproduces them. For the particular case of
hyperbolic decay with rank the distributions are identical, that is, the
classical Zipf plot, a pure power law. But their difference is largest when one
displays logarithmic decay and its counterpart shows the inverse exponential
decay, as it is the case of Benford law, or viceversa. For all intermediate
decay rates generic differences appear not only between the power-law exponents
for the midway rank decline but also for small and large rank. We extend the
theoretical framework to include thermodynamic and statistical-mechanical
concepts, such as entropies and configuration.
