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

Abstract

Consider a closed, smooth manifold M of non-positive curvature. Write $p:\textit{UM}\rightarrow M$ for the unit tangent bundle over M and let ${\mathcal R}_>$ 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 $\mathcal{R}_>$ which, essentially, is the dimension of the set $p(\mathcal{R}_>)$ of base points of $\mathcal{R}_>$.

The main result of this paper holds for manifolds with s-dim $\mathcal{R}_><\dim M/2$: for every $\epsilon>0$, there is an $\epsilon$-dense, flow invariant, closed subset $\Xi_\epsilon\subset \textit{UM}\backslash{\mathcal{R}}_>$ such that $p(\Xi_\epsilon)=M$.