On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces.~I

D. V. Anosov

Izvestiya. Mathematics , Tome 30 (1988) no. 1, pp. 15-38.

Résumé

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.

Bibliographie

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

[2] Lyusternik L. A., Shnirelman L. G., “Topologicheskie metody v variatsionnykh zadachakh i ikh prilozheniya k differentsialnoi geometrii poverkhnostei”, Uspekhi matem. nauk, 2:1 (1947), 166–217 | MR

[3] Aranson S. Kh., Grines V. Z., “Potoki na dvumernykh mnogoobraziyakh”, Dinamicheskie sistemy – 1, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 1, VINITI, M., 1985, 229–240

[4] Aranson S. X., Grines V. Z., “Topologicheskaya klassifikatsiya potokov na zamknutykh dvumernykh mnogoobraziyakh”, Uspekhi matem. nauk, 41:1 (1986), 149–170 | MR

[5] Vershik A. M., Kornfeld I. P., “Periodicheskie approksimatsii i ikh prilozheniya. Ergodicheskie teoremy, spektralnaya i entropiinaya teoriya dlya deistvii obschikh grupp”, Dinamicheskie sistemy – 2, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 2, VINITI, M., 1985, 70–89

[6] Weil A., “On systems of curves on a ring-shaped surface”, J. Indian Math. Soc., 19:5 (1931), 109–112 ; 6, 113–114 | Zbl | Zbl

[7] Weil A., “Les families de courbes sur le tore”, Matem. sb., 1:5 (1936), 779–781 | Zbl

[8] Kneseer H., “Reguläre Kurvenscharen auf den Ringflächen”, Math. Ann., 91:1,2 (1923), 135–154 | DOI | MR

[9] Tamura I., Topologiya sloenii, Mir, M., 1979, 320 pp. | MR | Zbl

[10] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970, 720 pp. | MR | Zbl

[11] Maltsev A. A., Polyakov V. Z., “Simpozium po obschei topologii”, Uspekhi matem. nauk, 21:4 (1966), 279–282

[12] Pupko V. I., “O nesamoperesekayuschikhsya krivykh na zamknutykh poverkhnostyakh”, Dokl. AN SSSR, 177:2 (1967), 272–274 | MR | Zbl

[13] Aranson S. X., Grines V. Z., “O nekotorykh invariantakh dinamicheskikh sistem na dvumernykh mnogoobraziyakh (neobkhodimye i dostatochnye usloviya topologicheskoi ekvivalentnosti tranzitivnykh sistem)”, Matem. sb., 90:3 (1973), 372–402 | MR | Zbl

[14] Maier A. T., “O traektoriyakh na orientiruemykh poverkhnostyakh”, Matem. sb., 12:1 (1943), 71–84 | MR

[15] Cherry T. M., “Analytic quasi-periodic curves of discontinuous type on a torus”, Proc. Lond. Math. Soc. Ser. 2, 44:2 (1938), 175–215 | DOI | Zbl

[16] Palis Zh., di Melu V., Geometricheskaya teoriya dinamicheskikh sistem, Mir, M., 1986, 296 pp. | MR

[17] Sternberg S., Lektsii po differentsialnoi geometrii, Mir, M., 1970, 412 pp. | MR | Zbl

[18] Whitney H., “Regular families of curves”, Ann. Math., 34:1,2 (1933), 244–270 | DOI | MR | Zbl

[19] Nemytskii V. V., “Struktura odnomernykh predelnykh integralnykh mnogoobrazii na ploskosti i v trekhmernom prostranstve”, Vestnik MGU, 1948, no. 10, 49–61

[20] Khausdorf F., Teoriya mnozhestv, Gostekhizdat, M., L., 1937, 304 pp.

[21] Zigel K. A., Avtomorfnye funktsii neskolkikh kompleksnykh peremennykh, IL, M., 1954, 167 pp.