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Geodesics avoiding open subsets in surfaces of negative curvature. (English) Zbl 1039.53039

From the introduction: This paper is motivated by a question posed by F. Labourie: Suppose that \(M\) is a compact surface equipped with a Riemannian metric of pinched negative curvature \(-b^2\leq K\leq-a^2<0\). Does there exist a proper closed subset \(W\) of the unit tangent bundle \(SM\) invariant under the geodesic flow \(\varphi^t: SM\to SM\) such that \(\pi(W ) = M\) , where \(\pi: SM\to M\) is the footpoint projection? We prove existence and nonexistence results for geodesics avoiding \(a\)-separated sets.

MSC:

53C22 Geodesics in global differential geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
53Cxx Global differential geometry
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