DOI 10.15507/2079-6900.23.202104.379–393

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 37B35

On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves

V. Z. Grines, E. Y. Gurevich, E. I. Yakovlev

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. We consider a class $GSD(M^3)$ of gradient-like diffeomorphisms with surface dynamics given on closed oriented manifold $M^3$ of dimension three. Earlier it was proved that manifolds admitting such diffeomorphisms are mapping tori under closed orientable surface of genus $g$, and the number of non-compact heteroclinic curves of such diffeomorphisms is not less than $12g$. In this paper, we determine a class of diffeomorphisms $GSDR (M^3) \subset GSD(M^3)$ that have the minimum number of heteroclinic curves for a given number of periodic points, and prove that the supporting manifold of such diffeomorphisms is a Seifert manifold. The separatrices of periodic points of diffeomorphisms from the class $ GSDR (M^3) $ have regular asymptotic behavior, in particular, their closures are locally flat. We provide sufficient conditions (independent on dynamics) for mapping torus to be Seifert. At the same time, the paper establishes that for any fixed $g \ geq 1$, fixed number of periodic points, and any integer $ n \geq 12g$, there exists a manifold $M^3$ and a diffeomorphism $f \in GSD (M^3)$ having exactly $ n $ non-compact heteroclinic curves.

Key Words: gradient-like diffeomorphism, surface dynamics, topological classification, non-compact heteroclinic curve, Seifert manifolds

For citation: V. Z. Grines, E. Y. Gurevich, E. I. Yakovlev. On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:4(2021), 379–393. DOI: https://doi.org/10.15507/2079-6900.23.202104.379–393

Submitted: 02.09.2021; Revised: 28.10.2021; Accepted: 16.11.2021

Information about the authors:

Vyacheslav Z. Grines, Professor of the Department of Fundamental Mathematics, National Research University «High School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), D. Sci. (Physics and Mathematics), ORCID: https://orcid.org/0000-0003-4709-6858, vgrines@yandex.ru

Elena Ya. Gurevich, Associate Professor of the Department of Fundamental Mathematics, National Research University «High School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), Ph. D. (Physics and Mathematics), ORCID: https://orcid.org/0000-0003-1815-3120, egurevich@hse.ru

Evgenii Iv. Yakovlev, Professor of the Department of Fundamental Mathematics, National Research University «High School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), D. Sci. (Physics and Mathematics), ORCID: https://orcid.org/0000-0001-6501- 353X, eyakovlev@hse.ru.hse.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.