The dynamics of comets and other solar system objects which have a three-body energy close to that of the collinear libration points are known to exhibit a complicated array of behaviors such as rapid transition between the interior and exterior Hillls regions, temporary capture, and collision. The invariant manifold structures of the collinear libration points for the restricted three-body problem, which exist for a range of energies, provide the framework for understanding these complex dynamical phenomena from a geometric point of view. In particular, the stable and unstable invariant manifold tubes associated to libration point orbits are the phase space structures that provide a conduit for orbits travelling to and from the smaller primary body (e.g., Jupiter). Using the structures around libration points, a statistical theory for the probability of transition between the interior and exterior Hill's regions and collision with the smaller primary body can be developed. Comparisons with observations of Jupiter family comets are made.