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

TL;DR

This paper introduces a new distinguishability-based metric derived from Wootters' statistical distance for analyzing states and invariant densities in discrete maps, aiding in identifying dissipative regions and wandering sets.

Contribution

It develops a novel metric for discrete maps based on Wootters' distance and characterizes wandering sets and dissipative regions in phase space.

Findings

Defined a new metric $ar{d}$ for discrete maps

Characterized wandering sets using the new metric

Applied the approach to logistic and circle maps

Abstract

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 distance in chaotic unidimensional maps. Based on that definition, we provide a metric $\overline{d}$ for an arbitrary discrete map. Moreover, from this 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 given metric 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 the metric and the wandering set for some characteristic values of their parameters.