  I investigate the caustics produced by the fall of collisionless dark matter
in and out of a galaxy in the limit of negligible velocity dispersion. The
outer caustics are spherical shells enveloping the galaxy. The inner caustics
are rings. These are located near where the particles with the most angular
momentum are at their distance of closest approach to the galactic center. The
surface of a caustic ring is a closed tube whose cross-section is a $D_{-4}$
catastrophe. It has three cusps amongst which exists a discrete $Z_3$ symmetry.
A detailed analysis is given in the limit where the flow of particles is
axially and reflection symmetric and where the transverse dimensions of the
ring are small compared to the ring radius. Five parameters describe the
caustic in that limit. The relations between these parameters and the initial
velocity distribution of the particles are derived. The structure of the
caustic ring is used to predict the shape of the bump produced in a galactic
rotation curve by a caustic ring lying in the galactic plane.
