Invariant subsets of rank 1 manifolds
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 \textit{K. Burns} and \textit{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.
scientific article; zbMATH DE number 1747279