  We examine the evolution of highly eccentric, planet-crossing orbits in the
restricted three-body problem (Sun, planet, comet). We construct a simple
Keplerian map in which the comet energy changes instantaneously at perihelion,
by an amount depending only on the azimuthal angle between the planet and the
comet at the time of perihelion passage. This approximate but very fast mapping
allow us to explore the evolution of large ensembles of long-period comets. We
compare our results on comet evolution with those given by the diffusion
approximation and by direct integration of comet orbits. We find that at long
times the number of surviving comets is determined by resonance sticking rather
than a random walk.
