Anomalous dynamics and the choice of Poincaré recurrence set

We investigate the dependence of Poincaré recurrence-time 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 a procedure, which is shown to be quite effective in the detection of tiny regions of regular motion, allows us to explain why similar recurrence sets have very different distributions and how to modify them in order to enhance their return probabilities. Applied to data, this enables us to understand the coexistence of extremely long, transient powerlike decays whose anomalous exponent depends on the chosen recurrence set.