Poincare Recurrence Theorem states that: "If a flow preserves volume and has only bounded orbits then for each open set there exist orbits that intersect the set infinitely often." But it does not imply (does it?) that "In hamiltonian system with bounded phase space, all trajectories will eventually return arbitrarily close to the original starting point." Only some not all trajectories will do so. When we consider a small neighbourhood of the starting point, and by the theorem, there exist some orbits (not all) that intersect the set later.