Distinguishability notion based on Wootters statistical distance: Application to discrete maps

GOMEZ, IGNACIO S.; PORTESI, M.; LAMBERTI, P. W.

CHAOS AN INTERDISCIPLINARY JR OF NONLINEAR SCIENCE

AMER INST PHYSICS

Año: 2017 vol. 27

1054-1500

We study the distinguishability notion given by Wootters for states represented by probability density functions. This presents the particularity that it can also be used for defining a statistical distance in chaotic unidimensional maps. Based on that definition, we provide a metric d for an arbitrary discrete map. Moreover, from d we associate a metric space to each invariant density of a given map, which results to be the set of all distinguished points when the number of iterations of the map tends to infinity. Also, we give a characterization of the wandering set of a map in terms of the metric d which allows to identify the dissipative regions in the phase space. We illustrate the results in the case of the logistic and the circle maps numerically and theoretically, and we obtain d and the wandering set for some characteristic values of their parameters.Finally, an extension of the metric space associated for arbitrary probability distributions (not necessarily invariant densities) is given along with some consequences. Statistical properties of distributions given by histograms are characterized in terms of the cardinal of the metric space associated. For two conjugated variables, the Uncertainty Principle is expressed in terms of the diameters of the metric spaces associated with those variables.