A Morse Energy Function for Topological Flows with Finite Hyperbolic Chain Recurrent Sets

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.