Classification of Morse–Smale flows on two-dimensional manifolds
Sbornik. Mathematics, Tome 189 (1998) no. 8, pp. 1205-1250 Cet article a éte moissonné depuis la source Math-Net.Ru

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The problem of topological trajectory classification of Morse–Smale flows on closed two-dimensional surfaces is considered. Important results in this direction have been obtained by Peixoto and his school. However, the complete solution of this problem has not yet been accurately presented. The new topological invariants constructed in our work have a simpler form than those in the works of Peixoto. In particular, a list of Morse–Smale flows of small complexity is given which has been obtained by the authors by means of the invariants constructed by them.
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A. A. Oshemkov; V. V. Sharko. Classification of Morse–Smale flows on two-dimensional manifolds. Sbornik. Mathematics, Tome 189 (1998) no. 8, pp. 1205-1250. http://geodesic.mathdoc.fr/item/SM_1998_189_8_a5/

[1] Andronov A. A., Pontryagin L. S., “Grubye sistemy”, Dokl. AN SSSR, 14:5 (1937), 247–250 | MR | Zbl

[2] Leontovich E. A., Maier A. G., “O traektoriyakh, opredelyayuschikh kachestvennuyu strukturu razbieniya sfery na traektorii”, Dokl. AN SSSR, 14:5 (1937), 251–257

[3] Leontovich E. A., Maier A. G., “O skheme, opredelyayuschei topologicheskuyu strukturu razbieniya na traektorii”, Dokl. AN SSSR, 103:4 (1955), 557–560 | MR | Zbl

[4] Maier A. G., “O traektoriyakh na orientiruemykh poverkhnostyakh”, Matem. sb., 12 (54):1 (1943), 71–84 | MR

[5] Anosov D. V., “Grubye sistemy”, Tr. MIAN, 169, Nauka, M., 1985, 59–93 | MR | Zbl

[6] Aranson S. Kh., Grines V. Z., “Topologicheskaya klassifikatsiya potokov na zamknutykh dvumernykh mnogoobraziyakh”, UMN, 41:1 (247) (1986), 149–169 | MR | Zbl

[7] Palis Zh., di Melu V., Geometricheskaya teoriya dinamicheskikh sistem. Vvedenie, Mir, M., 1986 | MR

[8] Smeil S., “Neravenstva Morsa dlya dinamicheskikh sistem”, Matematika. Sb. per., 11:4 (1967), 79–87 | MR

[9] Smeil S., “Differentsiruemye dinamicheskie sistemy”, UMN, 25:1 (1970), 113–185 | MR

[10] Peixoto M. M., “On the classification of flows on $2$-manifolds”, Dynamical systems, Academic Press, New York, 1973, 389–419 | MR

[11] Peixoto M. C., Peixoto M. M., “Structural stability in the plane with enlarged boundary conditions”, An. Acad. Brasil Ciênc., 31:2 (1959), 135–160 | MR | Zbl

[12] Fleitas G., “Classification of gradient-like flows on dimensions two and three”, Bol. Soc. Brasil Mat., 6 (1975), 155–183 | DOI | MR | Zbl

[13] Wang X., “The $C^*$-algebras of Morse–Smale flows on two-manifolds”, Ergodic Theory Dynam. Systems, 10 (1990), 565–597 | DOI | MR | Zbl

[14] Fomenko A. T., “Topologiya poverkhnostei postoyannoi energii integriruemykh gamiltonovykh sistem i prepyatstviya k integriruemosti”, Izv. AN SSSR. Ser. matem., 50:6 (1986), 1276–1307 | MR | Zbl

[15] Fomenko A. T., “Teoriya Morsa integriruemykh gamiltonovykh sistem”, Dokl. AN SSSR, 287:5 (1986), 1071–1075 | MR | Zbl

[16] Bolsinov A. V., Matveev S. V., Fomenko A. T., “Topologicheskaya klassifikatsiya integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody. Spisok sistem maloi slozhnosti”, UMN, 45:2 (272) (1990), 49–77 | MR

[17] Oshemkov A. A., “Funktsii Morsa na dvumernykh poverkhnostyakh. Kodirovanie osobennostei”, Trudy MIRAN, 205, Nauka, M., 1994, 131–140 | Zbl

[18] Bolsinov A. V., Oshemkov A. A., Sharko V. V., “On classification of flows on manifolds, I”, Methods Funct. Anal. Topology, 2:2 (1996), 190–204 | MR

[19] Peixoto M. M., “Structural stability on two-dimensional manifolds”, Topology, 1:2 (1962), 101–120 ; “Structural stability on two-dimensional manifolds – a further remark”, Topology, 2:2 (1963), 179–180 | DOI | MR | Zbl | DOI | MR | Zbl

[20] Smale S., “On gradient dynamical systems”, Ann. of Math. (2), 74 (1961), 199–206 | DOI | MR | Zbl

[21] Fomenko A. T., Fuks D. B., Kurs gomotopicheskoi topologii, Nauka, M., 1989 | MR

[22] Meyer K. R., “Energy functions for Morse–Smale systems”, Amer. J. Math., 90:4 (1968), 1031–1040 | DOI | MR | Zbl

[23] Bar-Natan D., “On the Vassiliev knot invariants”, Topology, 34:2 (1995), 423–472 | DOI | MR | Zbl