  We investigate the topological properties of invariant sets associated with
the dynamics of scattering systems with three or more degrees of freedom. We
show that the asymptotic separation of one degree of freedom from the rest in
the asymptotic regime, a common property in a large class of scattering models,
defines a dynamical object with phase space separating invariant manifolds and
an invariant set with larger dimension than that of the set defined by bounded
orbits. In particular, the set of typical periodic orbits involving all the
degrees of freedom of the system form a nowhere dense subset of the large
invariant set.
