Flow invariant subsets for geodesic flows of manifolds with non-positive curvature

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 phi on UM. We will 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 epsilon > 0, there is an epsilon-dense, flow invariant, closed subset Xi(epsilon) subset of UM\R-> such that p(Xi(epsilon)) = M.