Proceedings of the International Geometry Center

ISSN-print: 2072-9812
ISSN-online: 2409-8906
ISO: 26324:2012
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Trajectory equivalence of optimal Morse flows on closed surfaces

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Published Jun 10, 2018
DOI https://doi.org/10.15673/tmgc.v11i1.916

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Злата Кибалко
Taras Shevchenko University of Kyiv
Олександр Олегович Пришляк
Taras Shevchenko University of Kyiv
http://orcid.org/0000-0002-7164-807X
Roman Shchurko
Taras Shevchenko University of Kyiv

Abstract

We consider optimal Morse flows on closed surfaces. Up to topological trajectory equivalence such flows are determined by marked chord diagrams. We present list all such diagrams for flows on nonorientable surfaces of genus at most 4 and indicate pairs of diagrams corresponding to the flows and their inverses.

Keywords:
Morse, topological classification, flow, chord diagram

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How to Cite
Кибалко, З., Пришляк, О., & Shchurko, R. (2018). Trajectory equivalence of optimal Morse flows on closed surfaces. Proceedings of the International Geometry Center, 11(1). https://doi.org/10.15673/tmgc.v11i1.916
Issue
Vol 11 No 1 (2018)
Section
Papers

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Author Biography

Олександр Олегович Пришляк, Taras Shevchenko University of Kyiv

Професор кафедри геометрії, топології і динамічних систем

References

1. A. V. Bolsinov, A. T. Fomenko. Integrable Hamiltonian systems. Geometry, Topology, Classification. A CRC Press Company, Boca Raton London New York Washington, D.C., 2004. 724 p.

2. G. Fleitas. Classification of gradient-like flows on dimensions two and three. Bol. Soc. Brasil. Mat., 6(2):155 -183, 1975.

3. O. A. Giryk. Classification of polar Morse-Smale vector fields on two-dimensional manifolds. Methods Funct. Anal. Topology, 2(1):23 - 37, 1996.

4. O. A. Kadubovskyj. Classification of Morse-Smale vector fields on 2-manifolds. Visn., Mat. Mekh., Kyiv. Univ. Im. Tarasa Shevchenka, (14):85-88, 2005.

5. Y. Matsumoto. An introduction to Morse theory, volume 208 of Translations of Mathematical Monographs. American Mathematical Soc., 2002.

6. A. A. Oshemkov, V. V. Sharko. Classication of Morse-Smale flows on two-dimensional manifolds. Mat. Sbornik, 189(8):93-140, 1998.

7. Jacob Palis, Welington de Melo. Geometric theory of dynamical systems. An introduction. Springer-Verlag, New York-Berlin„ 1982. xii+198 p.

8. Jacob Palis, Stephen Smale. Structural stability theorems. Global Analysis (Proc. Sym-pos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), 1970.

9. M. M. Peixoto. On the classication of flows of 2-manifolds. Dynamical Systems (Proc. Symp. Univ. of Bahia, Salvador, Brasil, 1971), 389-419, 1973.

10. M.M. Peixoto. Structural stability on two-dimensional manifolds. i. Topology, 1(2):101-120, 1962.

11. Stephen Smale. On gradient dynamical systems. Ann. of Math., 74:199-206, 1961. Received: December 18, 2017, accepted: February, 20, 2018.