Abstract
We consider a class of gradient-like flows on three-dimensional closed manifolds whose attractors and repellers belongs to a finite union of embedded surfaces and find conditions when the ambient manifold is Seifert.
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Notes
Let us recall that a surface \(S_{g}\subset M^{3}\) is locally flat if for any point \(x\in S_{g}\) there exist a neighborhood \(U_{x}\subset M^{3}\) and a homeomorphism \(h_{x}:U_{x}\to\mathbb{R}^{3}\) such that \(h_{x}(S_{g}\cap U_{x})=Oxy\).
Invariant set \(A\) is called an attractor of a flow \(f^{t}\) if there exists a closed neighborhood (trapping neighborhood) \(V\subset M^{3}\) such that all trajectories of the flow \(f^{t}\) intersect the boundary of \(V\) transversally, and \(A=\bigcap\limits_{t>0}f^{t}(V)\). The set \(R\) is called a repeller of \(f^{t}\) if it is an attractor for \(f^{-t}\)., and the restriction of the flows \(f^{t}\) on this component is topologically equivalent to a gradient-like flow.
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Funding
This work was performed with support of the Laboratory of Dynamical Systems and Applications NRU HSE, grant of the Ministry of science and higher education of the RF no. 075-15-2019-1931.
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(Submitted by A. B. Muravnik)
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Grines, V.Z., Gurevich, E.Y. & Kevlia, S.S. On Gradient-Like Flows on Seifert Manifolds. Lobachevskii J Math 42, 901–910 (2021). https://doi.org/10.1134/S1995080221050061
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DOI: https://doi.org/10.1134/S1995080221050061