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 transition between the interior and exterior Hill's regions, temporary capture, and collision. The invariant manifold structures of the collinear libration points for the planar, circular 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. The stable and unstable invariant manifold tubes associated to libration point orbits are the phase space structures that provide a conduit for particles travelling to and from the secondary body (e.g., Jupiter). Using the structures around libration points, a statistical theory of the probability of interior-exterior transition and the probability of collision with the secondary body can be developed. Comparisons with observations of Jupiter family comets are made.