Abstract
In this paper, we consider regular topological flows on closed \(n\)-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence of the Morse – Bott energy function for the Morse – Smale flows. It is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds, on which dynamical systems can be considered only in a continuous category. The existence of continuous analogs of regular flows on any topological manifolds is an open question, as is the existence of energy functions for such flows. In this paper, we study the dynamics of regular topological flows, investigate the topology of the embedding and the asymptotic behavior of invariant manifolds of fixed points and periodic orbits. The main result is the construction of the Morse – Bott energy function for such flows, which ensures their close connection with the topology of the ambient manifold.
Similar content being viewed by others
References
Conley, Ch., Isolated Invariant Sets and the Morse Index, Providence, R.I.: AMS, 1978.
Grines, V., Medvedev, T., and Pochinka, O., Dynamical Systems on \(2\)- and \(3\)-Manifolds, New York: Springer, 2016.
Grines, V. and Pochinka, O., The Constructing of Energy Functions for \(\Omega\)-Stable Diffeomorphisms on \(2\)- and \(3\)-Manifolds, J. Math. Sci., 2020, vol. 250, pp. 537–568.
Grines, V. Z., Gurevich, E. Ya., Medvedev, V. S., and Pochinka, O. V., An Analogue of Smale’s Theorem for Homeomorphisms with Regular Dynamics, Math. Notes, 2017, vol. 102, no. 3–4, pp. 569–574; see also: Mat. Zametki, 2017, vol. 102, no. 4, pp. 613-618.
Guest, M., Morse Theory in the 1990’s, 0 (2001).
Irwin, M. C., A Classification of Elementary Cycles, Topology, 1970, vol. 9, pp. 35–47.
Medvedev, T. V., Pochinka, O. V., and Zinina, S. Kh., On Existence of Morse Energy Function for Topological Flows, Adv. Math., 2021, vol. 378, 107518, 15 pp.
Meyer, K. R., Energy Functions for Morse Smale Systems, Amer. J. Math., 1968, vol. 90, pp. 1031–1040.
Morse, M., Topologically Non-Degenerate Functions on a Compact \(n\)-Manifold \(M\), J. Anal. Math., 1959, no. 7, pp. 189–208.
Pochinka, O. V. and Zinina, S. Kh., A Morse Energy Function for Topological Flows with Finite Hyperbolic Chain Recurrent Sets, Math. Notes, 2020, vol. 107, no. 2, pp. 313–321; see also: Mat. Zametki, 2020, vol. 107, no. 2, pp. 276-285.
Robinson, C., Dynamical Systems: Stability, Symbolic Dynamics, Chaos, Boca Raton, Fla.: CRC, 1998.
Smale, S., On Gradient Dynamical Systems, Ann. of Math. (2), 1961, vol. 74, pp. 199–206.
Wilson, F. W., Jr., Smoothing Derivatives of Functions and Applications, Trans. Amer. Math. Soc., 1969, vol. 139, pp. 413–428.
Funding
The work on Section 3 was partially supported by the Laboratory of Dynamical Systems and Applications NRU HSE, by the Ministry of Science and Higher Education of the Russian Federation (ag. 075-15-2019-1931) and by the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS (project 19-7-1-15-1); the work on Section 4 was funded by RFBR, project number 20-31-90069.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
MSC2010
37D05, 37B20, 37B35
Rights and permissions
About this article
Cite this article
Pochinka, O.V., Zinina, S.K. Construction of the Morse – Bott Energy Function for Regular Topological Flows. Regul. Chaot. Dyn. 26, 350–369 (2021). https://doi.org/10.1134/S1560354721040031
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354721040031