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 
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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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.)

[1] Anosov D. V., “O povedenii traektorii na ploskosti Evklida ili Lobachevskogo, nakryvayuschikh traektorii potokov na zamknutykh poverkhnostyakh, I”, Izv. AN SSSR. Ser. matem., 51:1 (1987), 16–43 | MR | Zbl

[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

[3] Nemytskii V. V., Stepanov V. V., Kachestvennaya teoriya differentsialnykh uravnenii, Gostekhizdat, M.-L., 1949, 550 pp.

[4] Anosov D. V., “O beskonechnykh krivykh na tore i zamknutykh poverkhnostyakh otritsatelnoi eilerovoi kharakteristiki”, Optimizatsiya i differentsialnye igry, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 185, 1988, 30–53 | MR

[5] Anosov D. V., “Kak mogut ukhodit v beskonechnost krivye na universalnoi nakryvayuschei ploskosti, nakryvayuschie nesamoperesekayuschiesya krivye na zamknutoi poverkhnosti”, Statisticheskaya mekhanika i teoriya dinamicheskikh sistem, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 191, 1989, 34–44 | MR | Zbl

[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