Geodesic
A Morse Energy Function for Topological Flows with Finite Hyperbolic Chain Recurrent Sets
Matematičeskie zametki, Tome 107 (2020) no. 2, pp. 276-285.

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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. 
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O. V. Pochinka; S. Kh. Zinina. A Morse Energy Function for Topological Flows with Finite Hyperbolic Chain Recurrent Sets. Matematičeskie zametki, Tome 107 (2020) no. 2, pp. 276-285. http://geodesic.mathdoc.fr/item/MZM_2020_107_2_a9/

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