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This article is cited in 1 scientific paper (total in 1 paper)
A Morse Energy Function for Topological Flows with Finite Hyperbolic Chain Recurrent Sets
O. V. Pochinkaa, S. Kh. Zininaba a National Research University "Higher School of Economics", Nizhny Novgorod Branch
b Ogarev Mordovia State University
Abstract:
A Lyapunov function for a flow on a manifold is a continuous function which decreases along orbits outside the chain recurrent set and is constant on each chain component. By virtue of C. Conley's results, such a function exists for any flow generated by a continuous vector field; the very fact of its existence is known as the fundamental theorem of dynamical systems. If the set of critical points of a Lyapunov function coincides with the chain recurrent set of the flow, then this function is called an energy function. The paper considers topological flows with a finite hyperbolic (in the topological sense) chain recurrent set on closed surfaces. It is proved that any such flow has a (continuous) Morse energy function. The work is a conceptual continuation of that of S. Smale and K. Meyer, who proved the existence of a smooth Morse energy function for any gradient flow on a manifold.
Keywords:
Lyapunov function, energy function, chain recurrent set.
Received: 17.02.2019 Revised: 12.04.2019
Citation:
O. V. Pochinka, S. Kh. Zinina, “A Morse Energy Function for Topological Flows with Finite Hyperbolic Chain Recurrent Sets”, Mat. Zametki, 107:2 (2020), 276–285; Math. Notes, 107:2 (2020), 313–321
Linking options:
https://www.mathnet.ru/eng/mzm12360https://doi.org/10.4213/mzm12360 https://www.mathnet.ru/eng/mzm/v107/i2/p276
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Abstract page: | 312 | Full-text PDF : | 31 | References: | 32 | First page: | 10 |
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