Geodesic
A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces
Sbornik. Mathematics, Tome 205 (2014) no. 10, pp. 1387-1412 Cet article a éte moissonné depuis la source Math-Net.Ru

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In a paper of Oshemkov and Sharko, three-colour graphs were used to make the topological equivalence of Morse-Smale flows on surfaces obtained by Peixoto more precise. In the present paper, in the language of three-colour graphs equipped with automorphisms, we obtain a complete (including realization) topological classification of gradient-like cascades on surfaces. Bibliography: 25 titles.
Keywords: Morse-Smale diffeomorphism, gradient-like diffeomorphism, three-colour graph, topological classification. 
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V. Z. Grines; S. H. Kapkaeva; O. V. Pochinka. A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces. Sbornik. Mathematics, Tome 205 (2014) no. 10, pp. 1387-1412. http://geodesic.mathdoc.fr/item/SM_2014_205_10_a1/

[1] S. Smale, “Morse inequalities for a dynamical systems”, Bull. Amer. Math. Soc., 66 (1960), 43–49 | DOI | MR | Zbl

[2] A. A. Andponov, L. S. Pontpyagin, “Gpubye sistemy”, Dokl. AN SSSP, 14:5 (1937), 247–250 | Zbl

[3] V. Z. Grines, O. V. Pochinka, Vvedenie v topologicheskuyu klassifikatsiyu kaskadov na mnogoobraziyakh razmernosti dva i tri, RKhD, Izhevsk, 2011, 424 pp.

[4] V. Z. Grines, O. V. Pochinka, “Morse–Smale cascades on 3-manifolds”, Russian Math. Surveys, 68:1 (2013), 117–173 | DOI | DOI | MR | Zbl

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

[6] M. M. Peixoto, “On the classification of flows on 2-manifolds”, Dynamical systems, Proc. Sympos. (Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 389–419 | MR | Zbl

[7] A. A. Oshemkov, V. V. Sharko, “Classification of Morse–Smale flows on two-dimensional manifolds”, Sb. Math., 189:8 (1998), 1205–1250 | DOI | DOI | MR | Zbl

[8] A. N. Bezdenezhnykh, V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I”, Selecta Math. Soviet, 11:1 (1992), 1–11 | MR | MR | Zbl

[9] A. N. Bezdenezhnykh, V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. II”, Selecta Math. Soviet, 11:1 (1992), 13–17 | MR | MR | Zbl

[10] V. Z. Grines, “Topological classification of Morse–Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces”, Math. Notes, 54:3 (1993), 881–889 | DOI | MR | Zbl

[11] R. Langevin, “Quelques nouveaux invariants des difféomorphismes Morse–Smale d'une surface”, Ann. Inst. Fourier (Grenoble), 43:1 (1993), 265–278 | DOI | MR | Zbl

[12] T. M. Mitryakova, O. V. Pochinka, “K voprosu o klassifikatsii diffeomorfizmov poverkhnostei s konechnym chislom modulei topologicheskoi sopryazhennosti”, Nelineinaya dinam., 6:1 (2010), 91–105

[13] C. Bonatti, R. Langevin, Difféomorphismes de Smale des surfaces, Astérisque, 250, Soc. Math. France, Paris, 1998, viii+235 pp. | MR | Zbl

[14] A. N. Bezdenezhnykh, V. Z. Grines, “Realization of gradient-like diffeomorphisms of two-dimensional manifolds”, Selecta Math. Soviet, 11:1 (1992), 19–23 | MR | MR | Zbl

[15] T. M. Mitryakova, O. V. Pochinka, “Realization of cascades on surfaces with finitely many moduli of topological conjugacy”, Math. Notes, 93:6 (2013), 890–905 | DOI | DOI | MR | Zbl

[16] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maĭer, Qualitative theory of second-order dynamic systems, Halsted Press (A division of John Wiley Sons), New York–Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem–London, 1973, xxiii+524 pp. | MR | MR | Zbl | Zbl

[17] J. Palis, Jr., W. de Melo, Geometric theory of dynamical systems. An introduction, Springer-Verlag, New York–Berlin, 1982, xii+198 pp. | MR | MR | Zbl

[18] D. Rolfsen, Knots and links, Math. Lecture Ser., 7, Corrected reprint of the 1976 original, Publish or Perish, Inc., Houston, TX, 1990, xiv+439 pp. | MR | Zbl

[19] D. P. Demuner, M. Federson, C. Gutierrez, “The Poincaré–Bendixson theorem on the Klein bottle for continuous vector fields”, Discrete Contin. Dyn. Syst., 25:2 (2009), 495–509 | DOI | MR | Zbl

[20] W. Hurewicz, H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, 1948, vii+165 pp. | MR | Zbl

[21] M. Postnikov, Smooth manifolds, Lectures in geometry. Semester III, Mir, Moscow, 1989, 511 pp. | MR | MR | Zbl | Zbl

[22] V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, O. V. Pochinka, “Embedding in a flow of Morse–Smale diffeomorphisms on manifolds of dimension higher than two”, Math. Notes, 91:5 (2012), 742–745 | DOI | DOI | MR | Zbl

[23] J. Palis, “On Morse–Smale dynamical systems”, Topology, 8:4 (1969), 385–404 | DOI | MR | Zbl

[24] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747–817 | MR | MR

[25] L. S. Pontryagin, Foundations of combinatorial topology, Graylock Press, Rochester, N. Y., 1952, xii+99 pp. | MR | MR | Zbl