# On existence of a Morse energy function for topological flows with finite chain recurrent sets

## TL;DR

This paper proves that topological manifolds with certain flows have continuous Morse energy functions, advancing the understanding of Morse functions' existence on topological manifolds.

## Contribution

It establishes the existence of Morse energy functions for flows with finite hyperbolic chain recurrent sets on topological manifolds, partially solving the Morse problem.

## Key findings

- Existence of Morse energy functions for specified flows
- Applicable to manifolds of any dimension
- Progress towards Morse function existence on topological manifolds

## Abstract

We prove the existence of a continuous Morse energy function for an arbitrary topological flow with finite hyperbolic (in topological sense) chain recurrent set on a topological manifold of any dimension. This result is a partial solution of the Morse problem of existence of continuous Morse functions on any topological manifolds. Namely, we prove that a topological manifold admits a continuous Morse function if it admits a topological flow with finite hyperbolic chain recurrent set.