Topological structure of simple pro-Hamiltonian flows on the Möbius strip

Authors

  • O. Prishlyak Taras Shevchenko National University of Kyiv
  • S. Stas’ Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv

DOI:

https://doi.org/10.3842/umzh.v77i9.9029

Keywords:

Morse function, Hamiltonian flow, topological equivalence, Reeb graph

Abstract

UDC 515.1

We investigate the topological properties of flows on the Möbius strip, whose lift to a double cover, which is a cylinder, consists of Hamiltonian flows with a Hamiltonian that is a Morse function, constant on the boundary components. We construct a topological classification of such simple flows using distinguishing graphs made up of rooted trees, which are Reeb graphs. The resulting recursive formula calculates the number of topologically non-equivalent flows with a given number of saddles.

References

1. A. Bolsinov, A. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, CRC Press (2004); DOI: 10.1201/9780203643426. DOI: https://doi.org/10.1201/9780203643426

2. K. de Rezende, R. Franzosa, Lyapunov graphs and ows on surfaces, Trans. Amer. Math. Soc., 340, № 2, 767–784 (1993); DOI: 10.1090/S0002-9947-1993-1127155-9. DOI: https://doi.org/10.1090/S0002-9947-1993-1127155-9

3. C. Gutierrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic

Theory and Dynamical Systems, 6, № 1, 17–44 (1986); DOI: 10.1017/S0143385700003278. DOI: https://doi.org/10.1017/S0143385700003278

4. B. Hladysh, A. Prishlyak, Simple Morse functions on an oriented surface with boundary, J. Math. Phys., Anal., Geom., 15, № 3, 354–368 (2019); DOI: 10.15407/mag15.03.354. DOI: https://doi.org/10.15407/mag15.03.354

5. A. Kravchenko, S. Maksymenko, Automorphisms of Kronrod–Reeb graphs of Morse functions on compact surfaces, Eur. J. Math., 6, № 1, 114–131 (2019); DOI: 10.1007/s40879-019-00379-8. DOI: https://doi.org/10.1007/s40879-019-00379-8

6. S. Maksymenko, Deformations of functions on surfaces by isotopic to the identity diffeomorphisms, Topology and Appl., 282, Article~107312 (2020); DOI: 10.1016/j.topol.2020.107312. DOI: https://doi.org/10.1016/j.topol.2020.107312

7. D. Neumann, T. O'Brien, Global structure of continuous flows on 2-manifolds, J. Different. Equat., 22, № 1, 89–110 (1976); DOI: 10.1016/0022-0396(76)90006-1. DOI: https://doi.org/10.1016/0022-0396(76)90006-1

8. A. Oshemkov, V. Sharko, Classication of Morse–Smale flows on two-dimensional manifolds, Mat. Sb., 189, № 8, 93–140 (1998). DOI: https://doi.org/10.4213/sm341

9. A. Prishlyak, Conjugacy of Morse functions on surfaces with values on a straight line and circle, Ukr. Math. J., 52, № 10, 1623–1627 (2000); DOI: 10.1023/A:1010461319703. DOI: https://doi.org/10.1023/A:1010461319703

10. A. Prishlyak, M. Loseva, Topology of optimal flows with collective dynamics on closed orientable surfaces, Proc. Intern. Geom. Center, 13, № 2, 50–67 (2020); DOI: 10.15673/tmgc.v13i2.1731. DOI: https://doi.org/10.15673/tmgc.v13i2.1731

11. G. Reeb, Sur les points singuliers d'une forme de Pfaff complétement intégrable ou d'une fonction numérique, Compt. Rend. Acad. Sci., 222, 847–849 (1946).

12. V. Sharko, Smooth and topological equivalence of functions on surfaces, Ukr. Math. J., 55, № 5, 832–846 (2003); DOI: 10.1023/B:UKMA.0000010259.21815.d7. DOI: https://doi.org/10.1023/B:UKMA.0000010259.21815.d7

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Published

06.11.2025

Issue

Vol. 77 No. 9 (2025)

Section

Research articles

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Copyright (c) 2025 Олександр Пришляк, Сергій Стась

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Prishlyak, O., and S. Stas’. “Topological Structure of Simple Pro-Hamiltonian Flows on the Möbius Strip”. Ukrains’kyi Matematychnyi Zhurnal, vol. 77, no. 9, Nov. 2025, pp. 584–589, https://doi.org/10.3842/umzh.v77i9.9029.