This talk is dedicated to the possibility to characterize the density function of the first Poincaré
returns in terms of unstable periodic orbits. I will present a conjecture on how
periodic orbits may be used to compute the density of the first returns
and some numerical evidences with well known dynamical systems. The conjecture is true
for linear Markov transformations if the subset of the phase space is an element or a
perfect union of elements of the Markov partition. I will present this result and discuss possible extensions.