Geodesic
On the behaviour in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces.~III
Izvestiya. Mathematics , Tome 59 (1995) no. 2, pp. 287-320.

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This paper is related to the previous papers [1] and [2]. We fill a gap in the proof in [1] of the following alternative: under assumptions mentioned there, a semi-trajectory $\widetilde L$ of the covering flow on the universal covering plane is either bounded or tends to infinity with an asymptotic direction. For the torus, we prove under the same assumptions that in the second case the deviation of $\widetilde L$ from the line corresponding to this direction is bounded. We prove that for every (semi-)infinite non-self-intersecting $L$ on a closed surface and every $r>0$ there is a $C^\infty$-flow with an invariant measure having a specified $C^\infty$-smooth everywhere-positive density such that some positive semi-trajectory of the flow approximates $L$ up to $r$. (In [2] an analogous approximation assertion was proved, with no mention of an invariant measure.) 
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D. V. Anosov. On the behaviour in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces.~III. Izvestiya. Mathematics , Tome 59 (1995) no. 2, pp. 287-320. http://geodesic.mathdoc.fr/item/IM2_1995_59_2_a2/

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[2] Anosov D. V., “O povedenii traektorii na ploskosti Evklida ili Lobachevskogo, nakryvayuschikh traektorii potokov na zamknutykh poverkhnostyakh, II”, Izv. AN SSSR. Ser. matem., 52:3 (1988), 451–478 | MR | Zbl

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[6] Anosov D. V., “O beskonechnykh krivykh na butylke Kleina”, Matem. sb., 180:1 (1989), 39–56 | MR | Zbl

[7] Anosov D. V., “Potoki na poverkhnostyakh”, Topologiya i ee primeneniya, Trudy Mezhdunarodnoi topologicheskoi konferentsii (Baku, 3–8 oktyabrya 1987 g.), Tr. Matem. in-ta im. V. A. Steklova RAN, 193, 1992, 10–14 | MR | Zbl