Topological aspects of chaotic scattering in higher dimensions

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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.