Email this article Classification of nonsingular four-dimensional flows with a untwisted saddle orbit
- Authors: Galkin V.D.1, Pochinka O.V.1, Shubin D.D.1
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Affiliations:
- National Research University – Higher School of Economics in Nizhny Novgorod
- Issue: Vol 215, No 11 (2024)
- Pages: 65-91
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/279063
- DOI: https://doi.org/10.4213/sm10091
- ID: 279063
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Abstract
The topological equivalence of low-dimensional Morse–Smale flows without fixed point (NMS-flows) under assumptions of various generality is the subject of a number of publications. Starting from dimension 4, there are only few results on classification. However, it is known that there exists nonsingular flows with wildly embedded invariant saddle manifolds. In this paper the class of nonsingular Morse–Smale flows on closed orientable 4-manifolds with a unique saddle orbit which is, moreover, nontwisted, is considered. It is shown that the equivalence class of a certain knot embedded in $\mathbb S^2\times\mathbb S^1$ is a complete invariant of such a flow. Given a knot in $\mathbb S^2\times\mathbb S^1$, a standard representative in the class of flows under consideration is constructed. The supporting manifold of all such flows is shown to be the manifold $\mathbb S^3\times\mathbb S^1$.Bibliography: 24 titles.
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About the authors
Vladislav Dmitrievich Galkin
National Research University – Higher School of Economics in Nizhny Novgorod
Email: vgalkin@hse.ru
Olga Vital'evna Pochinka
National Research University – Higher School of Economics in Nizhny Novgorod
Email: olga-pochinka@yandex.ru
ORCID iD: 0000-0002-6587-5305
Doctor of physico-mathematical sciences, no status
Danila Denisovich Shubin
National Research University – Higher School of Economics in Nizhny Novgorod
Email: schub.danil@yandex.ru
References
- P. M. Akhmet'ev, T. V. Medvedev, O. V. Pochinka, “On the number of the classes of topological conjugacy of Pixton diffeomorphisms”, Qual. Theory Dyn. Syst., 20:3 (2021), 76, 15 pp.
- F. Bonahon, J.-P. Otal, “Scindements de Heegaard des espaces lenticulaires”, Ann. Sci. Ecole Norm. Sup. (4), 16:3 (1983), 451–466
- C. Bonatti, V. Z. Grines, “Knots as topological invariants for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dynam. Control Systems, 6:4 (2000), 579–602
- C. Bonatti, V. Grines, O. Pochinka, “Topological classification of Morse–Smale diffeomorphisms on 3-manifolds”, Duke Math. J., 168:13 (2019), 2507–2558
- J. Franks, “Nonsingular Smale flows on $S^3$”, Topology, 24:3 (1985), 265–282
- D. Gabai, “Foliations and the topology of 3-manifolds. III”, J. Differential Geom., 26:3 (1987), 479–536
- C. McA. Gordon, J. Luecke, “Knots are determined by their complements”, J. Amer. Math. Soc., 2:2 (1989), 371–415
- V. Grines, Yu. Levchenko, V. Medvedev, O. Pochinka, “The topological classification of structurally stable 3-diffeomorphisms with two-dimensional basic sets”, Nonlinearity, 28:11 (2015), 4081–4102
- В. З. Гринес, Е. В. Жужома, В. С. Медведев, О. В. Починка, “Глобальные аттрактор и репеллер диффеоморфизмов Морса–Смейла”, Дифференциальные уравнения и топология. II, Сборник статей. К 100-летию со дня рождения академика Льва Семеновича Понтрягина, Труды МИАН, 271, МАИК «Наука/Интерпериодика», М., 2010, 111–133
- В. Гуревич, Г. Волмэн, Теория размерности, ИЛ, М., 1948, 232 с.
- M. C. Irwin, “A classification of elementary cycles”, Topology, 9:1 (1970), 35–47
- N. L. Max, “Homeomorphisms of $S^{n}times S^{1}$”, Bull. Amer. Math. Soc., 73:6 (1967), 939–942
- W. D. Neumann, Notes on geometry and 3-manifolds, Citeseer, 1996
- E. M. Osenkov, O. V. Pochinka, Morse–Smale 3-diffeomorphisms with saddles of the same unstable manifold dimension
- D. Pixton, “Wild unstable manifolds”, Topology, 16:2 (1977), 167–172
- O. V. Pochinka, D. D. Shubin, “On 4-dimensional flows with wildly embedded invariant manifolds of a periodic orbit”, Appl. Math. Nonlinear Sci., 5:2 (2020), 261–266
- O. V. Pochinka, D. D. Shubin, “Non-singular Morse–Smale flows on $n$-manifolds with attractor-repeller dynamics”, Nonlinearity, 35:3 (2022), 1485–1499
- D. Rolfsen, Knots and links, AMS Chelsea Publ. Ser., 346, Reprint with corr. of the 1976 ed., Amer. Math. Soc., Providence, RI, 2003, ix+439 pp.
- С. Смейл, “Дифференцируемые динамические системы”, УМН, 25:1(151) (1970), 113–185
- Я. Л. Уманский, “Необходимые и достаточные условия топологической эквивалентности трехмерных динамических систем Морса–Смейла с конечным числом особых траекторий”, Матем. сб., 181:2 (1990), 212–239
- V. Galkin, O. Pochinka, D. Shubin, Classification of NMS-flows with unique twisted saddle orbit on orientable 4-manifolds
- Bin Yu, “Behavior $0$ nonsingular Morse Smale flows on $S^3$”, Discrete Contin. Dyn. Syst., 36:1 (2016), 509–540
- О. В. Починка, Д. Д. Шубин, “Неособые потоки Морса-Смейла с тремя периодическими орбитами на ориентируемых $3$-многообразиях”, Матем. заметки, 112:3 (2022), 426–443
- А. О. Пришляк, “Полный топологический инвариант потоков Морса–Смейла и разложений на ручки трeхмерных многообразий”, Фундамент. и прикл. матем., 11:4 (2005), 185–196
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