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. 
@article{MZM_2020_107_2_a9,
     author = {O. V. Pochinka and S. Kh. Zinina},
     title = {A {Morse} {Energy} {Function} for {Topological} {Flows} with {Finite} {Hyperbolic} {Chain} {Recurrent} {Sets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {276--285},
     publisher = {mathdoc},
     volume = {107},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_2_a9/}
}
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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/