  We derive, and discuss the properties of, a symplectic map for the dynamics
of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a
circular, coplanar orbit interior to the pericenter of the parabolic orbit. The
map shows excellent agreement with direct numerical integrations and elucidates
how the dynamics depends on perturber mass and pericenter distance. We also use
the map to explore the onset of chaos, statistical descriptions of chaotic
transport, and sticking in mean-motion resonances. We discuss implications of
our mapping model for the dynamical evolution of the solar system's scattered
disk and other highly eccentric trans-Neptunian objects.
