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

Nondense orbits on homogeneous spaces and applications to geometry and number theory

“…Theorem B2 is motivated by the unpublished work of Burns and Pollicott [6] and subsequent papers [8,35,7,32], where hyperbolic manifolds and more general manifolds of nonpositive curvature are considered. However, in all the aforementioned papers the set Ξ consisted of a single point.…”