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

IGNACIO GÓMEZ; M. PORTESI; P.W. LAMBERTI

CHAOS AN INTERDISCIPLINARY JR OF NONLINEAR SCIENCE

AMER INST PHYSICS

Lugar: New York; Año: 2017 vol. 27 p. 831121 - 8311211

1054-1500

We study the distinguishability notion given by Wootters for states represented by probabilitydensity functions. This presents the particularity that it can also be used for defining a statisticaldistance in chaotic unidimensional maps. Based on that definition, we provide a metric d for anarbitrary discrete map. Moreover, from d, we associate a metric space with each invariant densityof a given map, which results to be the set of all distinguished points when the number of iterationsof the map tends to infinity. Also, we give a characterization of the wandering set of a map in termsof the metric d, which allows us to identify the dissipative regions in the phase space. We illustratethe results in the case of the logistic and the circle maps numerically and analytically, and weobtain d and the wandering set for some characteristic values of their parameters. Finally, anextension of the metric space associated for arbitrary probability distributions (not necessarilyinvariant densities) is given along with some consequences. The statistical properties ofdistributions given by histograms are characterized in terms of the cardinal of the associated metricspace. For two conjugate variables, the uncertainty principle is expressed in terms of the diametersof the associated metric space with those variables. Pub