Geodesic
On the number of invartiant measures for flows on orientable surfaces
Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 813-830.

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The following theorem is proved. For any natural numbers $n$ and $k$, $n\geqslant k$, on a two-dimensional orientable compact manifold without boundary of class $C^\infty$ and genus there exists a topologically transitive flow of class $C^\infty$ having $2n-2$ fixed points and exactly $k$ invariant ergodic normalized measures such that the measure of each trajectory is equal to zero. Bibliography: 3 items. 
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E. A. Sataev. On the number of invartiant measures for flows on orientable surfaces. Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 813-830. http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a5/

[1] Katok A. B., Stepin A. M., “Approksimatsii v ergodicheskoi teorii”, Uspekhi matem. nauk, XXII:5 (1967), 81–106 | MR

[2] Katok A. B., “Invariantnye mery na orientiruemykh poverkhnostyakh”, Dokl. AN SSSR, 211:4 (1973), 775–778 | MR | Zbl

[3] Blokhinn A. A., “Gladkie ergodicheskie potoki na poverkhnostyakh”, Tr. Mosk. matem. ob-va, 27, 1972, 113–128