  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.
