A Conley-type Lyapunov function for the strong chain recurrent set

Abstract

Let $\varphi :X\times \mathbb{R}\to X$ be a continuous flow on a compact metric space $(X,d)$. In this article we constructively prove the existence of a continuous Lyapunov function for $\varphi$ which is strictly decreasing outside ${\mathcal{SC}\mathcal{R}}_d(\varphi )$. Such a result generalizes Conley's Fundamental Theorem of Dynamical Systems for the strong chain recurrent set.