  We study the statistical distribution of the closest encounter between
observations computed along different trajectories of a mixing dynamical
system. At the limit of large trajectories, the distribution is of Gumbel type
and depends on the length of the trajectories and on the Generalized Dimensions
of the image measure. It is also modulated by an Extremal Index, for which we
give a formula in the case of expanding maps of the interval and regular
observations. We discuss the implications of these results for the study of
physical systems.
