  This article treats chaotic scattering with three degrees of freedom, where
one of them is open and the other two are closed, as a first step toward a more
general understanding of chaotic scattering in higher dimensions.
  Despite of the strong restrictions it breaks the essential simplicity
implicit in any two-dimensional time-independent scattering problem.
Introducing the third degree of freedom by breaking a continuous symmetry, we
first explore the topological structure of the homoclinic/heteroclinic tangle
and the structures in the scattering functions. Then we work out implications
of these structures for the doubly differential cross section. The most
prominent structures in the cross section are rainbow singularities. They form
a fractal pattern which reflects the fractal structure of the chaotic invariant
set. This allows to determine structures in the cross section from the
invariant set and conversely, to obtain information about the topology of the
invariant set from the cross section. The latter is a contribution to the
inverse scattering problem for chaotic systems.
