Geodesic
On the representation of minimal sets of currents on two-dimensional manifolds by geodesics
Izvestiya. Mathematics, Tome 12 (1978) no. 1, pp. 103-124 
S. Kh. Aranson; V. Z. Grines. On the representation of minimal sets of currents on two-dimensional manifolds by geodesics. Izvestiya. Mathematics, Tome 12 (1978) no. 1, pp. 103-124. http://geodesic.mathdoc.fr/item/IM2_1978_12_1_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The authors have previously introduced a topological invariant of currents defined on a smooth connected orientable manifold $M$ of genus $n\geqslant2$, called the rotation homotopy class. With the help of this invariant necessary and sufficient conditions were established for the topological equivalence of minimal sets of currents containing a nonclosed recurrent trajectory. However, the question of classification (that is, the question of distinguishing all the equivalence classes and constructing standard currents in each class) remains open. The present article is devoted to the solution of the above problem, and standard minimal sets of currents are constructed in such a way that their trajectories are geodesics on $M$ in a metric of constant negative curvature. Bibliography: 15 titles.

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[2] Aranson S. X., Grines V. Z., “O topologicheskoi ekvivalentnosti minimalnykh mnozhestv dinamicheskikh sistem na dvumernykh mnogoobraziyakh”, Uspekhi matem. nauk, 28:4(172) (1973), 205–206 | MR | Zbl

[3] Aranson S. X., Grines V. Z., “O topologicheskikh invariantakh minimalnykh mnozhestv dinamicheskikh sistem na dvumernykh mnogoobraziyakh”, Uchenye zapiski GGU, 187 (1973), 3–28 | MR

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[5] Markley N. G., “Homeomorphisms of the circle without periodic points”, J. Lond. Math. Soc., 20 (1970), 688–698 | DOI | MR | Zbl

[6] Aranson S. X., Zhuzhoma E. V., “O topologicheskoi klassifikatsii singulyarnykh dinamicheskikh sistem na tore”, Izv. Vuzov, Matematika, 1976, no. 5(168), 104–107 | MR | Zbl

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[9] Maier A. G., “O traektoriyakh na orientiruemykh poverkhnostyakh”, Matem. sb., 12(54) (1943), 71–84

[10] Aranson S. X., “Ob otsutstvii nezamknutykh ustoichivykh po Puassonu polutraektorii i traektorii, dvoyakoasimptoticheskikh k dvoinomu predelnomu tsiklu u dinamicheskikh sistem pervoi stepeni negrubostk na orientiruemykh dvumernykh mnogoobraziyakh”, Matem. sb., 76(118):2 (1968), 214–230 | MR | Zbl

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[14] Hedlund G. A., “Two-dimensional manifolds and transitivity”, Ann. Math., 37:3 (1936), 534–542 | DOI | MR | Zbl

[15] Aranson S. Kh., “O nekotorykh arifmeticheskikh svoistvakh dinamicheskikh sistem na dvumernykh mnogoobraziyakh”, Dokl. AN SSSR, 222:2 (1975), 265–268 | MR | Zbl