A Morse energy function for topological flows with finite hyperbolic chain recurrent sets (Q2191967)

\textit{C. Conley} [Isolated invariant sets and the Morse index. Providence, RI: American Mathematical Society (AMS) (1978; Zbl 0397.34056)] showed that any flow generated by a continuous vector field has a Lyapunov function, i.e., a continuous function that decreases along orbits outside the chain recurrent set and is constant on each chain component. A Lyapunov function is also called an energy function if its set of critical points coincides with the chain recurrent set. In the work of \textit{S. Smale} [Ann. Math. (2) 74, 199--206 (1961; Zbl 0136.43702)] it was shown that any gradient flow on a manifold has a smooth Morse energy function. Here this direction is continued, and the main result is that a topological flow with a finite hyperbolic chain recurrent set on a closed surface has a continuous Morse energy function.