  We investigate the dependence of Poincar\'e recurrence-times statistics on
the choice of recurrence-set, by sampling the dynamics of two- and
four-dimensional Hamiltonian maps. We derive a method that allows us to
visualize the direct relation between the shape of a recurrence-set and the
values of its return probability distribution in arbitrary phase-space
dimensions. Such procedure, which is shown to be quite effective in the
detection of tiny regions of regular motion, allows to explain it why similar
recurrence-sets have very different distributions and how to modify them in
order to enhance their return probabilities. Applied on data, this permits to
understand the co-existence of extremely long, transient power-like decays
whose anomalous exponent depends on the chosen recurrence-set.

Chaotic Dynamics