Geodesic
On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces.~I
Izvestiya. Mathematics , Tome 30 (1988) no. 1, pp. 15-38

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Consider a flow on a surface $M$ of nonpositive Euler characteristic whose set of equilibrium points can be deformed, in $M$, to a point (this, for example, is the case if there are only finitely many equilibrium points). For such a flow, it is proved that the semitrajectory of the covering flow on the universal cover (the Euclidean or Lobachevsky plane) of $M$ is either bounded or tends to infinity in a definite direction. For analytic flows (but not for $C^\infty$-flows), this conclusion holds without any conditions on the equilibrium points. Bibliography: 21 titles. 
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     title = {On the behavior in the {Euclidean} or {Lobachevsky} plane of trajectories that cover trajectories of flows on closed {surfaces.~I}},
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D. V. Anosov. On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces.~I. Izvestiya. Mathematics , Tome 30 (1988) no. 1, pp. 15-38. http://geodesic.mathdoc.fr/item/IM2_1988_30_1_a1/