Consider a closed, smooth manifold M of non-positive curvature. Write p:UM→M for the unit tangent bundle over M and let ℛ > denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow φ on UM. We define the structured dimension s-dim ℛ > which, essentially, is the dimension of the set p(ℛ > ) of base points of ℛ > .
The main result of this paper holds for manifolds with s-dim ℛ > <dim M/2: for every ε>0, there is an ε-dense, flow invariant, closed subset Ξ ε ⊂UM∖ℛ > such that p(Ξ ε )=M.