Caustic ring singularity - art. no. 063501.

I investigate the caustics produced by the fall of collisionless dark matte r in and out of a galaxy in the limit of negligible velocity dispersion. Th e outer caustics are spherical shells enveloping the galaxy. The inner caus tics are rings. These are located near where the particles with the most an gular momentum are at their distance of closest approach to the galactic ce nter. 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 part icles is axially and reflection symmetric and where the transverse dimensio ns of the ring are small compared to the ring radius. Five parameters descr ibe the caustic in that limit. The relations between these parameters and t he initial velocity distribution of the particles are derived. The structur e 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. [S 0556-2821(99)05916-0].