Remarks on the article "A note on horospheric points"

An improvement of the main result

There is now a shorter, cleaner proof, in french, of the main theorem of the article. The assumption that the relation induced by the stable sets has been removed. This new version can be found in my habilitation à diriger les recherches, in chapter III, 1.2, which is devoted to horospheric points.

Horospheric points on negatively curved manifolds

We didn't prove in the article that the definition we give of an horospheric point actually coincides with the usual definition in the context of negatively curved manifolds. This is obvious if the non-wandering set of the geodesic flow is everything, and the general case was left to the reader. Here is the solution.

Let X=T¹D be the unitary bundle of a connected, simply connected, complete negatively curved manifold D, and Γ be a group of isometries acting properly discontinuously on D without fixed points. Let v ∈ X be the lift of an horospheric point which belongs to the lift Ω' ⊂ X of the non-wandering set of the geodesic flow on X/Γ. Then there exist γᵢ ∈ Γ, vᵢ ∈ Ω', tᵢ → ∞, C>0 such that d(vᵢ,γᵢ0) < C and vᵢ belongs to the strong stable leaf of g_tᵢ(v).

Let tᵢ, γᵢ such that γᵢ belongs to the horosphere associated to g_tᵢ(v). Assuming the theorem does not hold then for all R>0, the ball B(γᵢo,R), the horosphere associated to g_tᵢ(v) and Ω' have empty intersection.

Let v+ ∈ ∂D the end point of the ray generated by v and O_{v+}(γᵢo,R) the shadow of the ball B(γᵢo,R) seen from v+. This shadow is disjoint from the limit set ΛΓ. So the limit set is also disjoint from the shadow O_{γᵢ⁻¹v+}(o,R).

Let us extract a subsequence from γᵢ⁻¹v+, converging to ξ. The shadow O_ξ(o,R) is disjoint from the limit set ΛΓ for all R > 0, an absurdity.

The locally compact case

If the metric space X is locally compact, the main theorem may be rephrased by saying that the following dichotomy holds :

-- either the strong stable set is dense,
-- or the strong stable set goes to infinity under the action of the flow (aka it leaves all compact subsets of X).

The visibility property

Recall that a riemannian manifold M is a Hadamard manifold if it is connected, simply connected, complete for the riemannian metric, and with sectional curvatures <=0. Such a manifold is diffeomorphic to R^n, and there is a natural way to compactify it by adding a sphere at infinity.

A Hadamard manifold satisfies the visibility property if any two distinct points on its boundary at infinity are the two ends of a geodesic in M. This property is satisfied in the negatively curved case, and is a necessary condition for the existence of a global product structure on T^1M, for the geodesic flow. If such a product structure exists, then it induces a local product on any quotient of M by a group of isometry acting properly discontinuously without fixed points.

For a visibility manifold with transitive horosphere foliation, there is the following caracterization of horospheric points (Eberlein , "Geodesic flow on negatively curved manifolds II" th 5.5):

A geodesic c(t) does not end on a horospheric point iff the geodesic is almost minimizing. (almost minimizing : there exists C such that d(c(t),c(0))>t-C for all t>0)

If the curvature is strictly negative, then being almost minimizing is equivalent to being non recurrent with respect to the horosphere foliation (aka the stable set of the vector does not accumulate on itself). (Ledrappier, "horospheres on abelian covers") So, in that case, the following are equivalent :

-- the stable set does not go to infinity under the action of the flow
-- the stable set is recurrent
-- the stable set is dense
-- the geodesic is not almost minimizing

These results are true in restriction to the non-wandering set.

There are also some results for manifolds with no conjuguate points (see the book of Eberlein, Geometry of nonpositively curved manifolds ; and also MJ Druetta). Note that the visibility property for the universal cover of a compact real analytic manifold M with K<=0 can be caracterized in term of the fundamental group of M : it amounts to saying that the only abelian subgroups of the fundamental group of M is infinite cyclic (Bangert Schroeder 91).

Yves Coudene 17/10/2011