Symplectic Geometry and its dynamics originated from classical mechanics as the geometry of physical phase space, in particular from celestial mechanics, and one of the most driving questions is up to today that of stability for such systems. One of the most fundamental features of phase space dynamics is Poincare Recurrence which derives from conservation of phase space volume. However, where is the particular symplectic quality left? In this talk, I will present an answer which connects to the features of symplectic rigidity. Those have their roots in the Arnold conjectures and Gromov's pseudoholomorphic curves and found one among many expressions in Lagrangian intersection rigidity. In this talk I will introduce a new type of Lagrangian intersection phenomenon which can be viewed as Symplectic Poincare Recurrence. The talk will be addressing a general mathematical audience.