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 amanifold.
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References
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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 2, pp. 276–285.
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This work was supported by the Russian Science Foundation under grant 17-11-01041 in the framework of the 2019 project of the Center for Basic Research, National Research University Higher School of Economics, and by the “BASIS” Foundation for the Advancement of Theoretical Physics and Mathematics.
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Pochinka, O.V., Zinina, S.K. A Morse Energy Function for Topological Flows with Finite Hyperbolic Chain Recurrent Sets. Math Notes 107, 313–321 (2020). https://doi.org/10.1134/S0001434620010319
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DOI: https://doi.org/10.1134/S0001434620010319