On existence of Morse energy function for topological flows

Abstract

It is a well known fact that any smooth manifold admits a Morse function, whereas the problem of existence of a Morse function for a topological manifold stated by Marston Morse in 1959 ([6]) is still open. In the present paper we prove that a topological manifold admits a continuous Morse function if it admits a topological flow with a finite hyperbolic chain recurrent set. We construct this function as a Lyapunov function whose set of the critical points coincides with the chain recurrent set of the flow.