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Flow invariant subsets for geodesic flows of manifolds with non-positive curvature. (English) Zbl 1127.53070

Summary: Consider a closed, smooth manifold \(M\) of non-positive curvature. Write \(p: UM\to M\) 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 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 \(\varepsilon>0\), there is an \(\varepsilon\)-dense, flow invariant, closed subset \(\Xi_\varepsilon\subset \text{UM}\backslash{\mathcal{R}}_>\) such that \(p(\Xi_\varepsilon)=M\).

MSC:

53D25 Geodesic flows in symplectic geometry and contact geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C22 Geodesics in global differential geometry
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