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Invariant subsets of rank 1 manifolds. (English) Zbl 1003.53031

Let \(M\) be a complete Riemannian manifold of nonpositive sectional curvature. The authors prove that if \(\dim M\geq 3\) and \(v_0\) is a rank one unit vector, then there exists an \(\varepsilon>0\) such that for every point \(o\) in \(M\) there is a unit vector \(v\) in \(T_oM\) with the property that the orbit of the geodesic flow through \(v\) does not enter an \(\varepsilon\)-neighborhood around \(v_0\) in the unit tangent bundle. This result generalizes previous ones of K. Burns and M. Pollicott [Self-intersection of geodesics and projecting flow invariant sets. Preprint 1994] and of the second author [Math. Z. 235, 817-828 (2000; Zbl 0990.53038)]. The authors also prove that for a finite volume \(M\), there is an \(\varepsilon>0\) such that for every \(o\) in \(M\), the set \(C_\varepsilon\) of unit vectors \(v\) in \(T_oM\), such that the orbits do not enter an \(\varepsilon\)-neighborhood around \(v_0\) in the unit tangent bundle, is nonempty, compact, nowhere dense and with topological dimension \(n-2\), where \(n\) is the dimension of \(M\). Furthermore, they prove that the homotopy group \(\pi_{n-2}(C_\varepsilon)\) contains a free infinitely generated subgroup.

MSC:

53C22 Geodesics in global differential geometry
53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0990.53038
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