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

Let ϕ:X×R→X be a continuous flow on compact metric space (X,d). In this article we constructively prove the existence of Lyapunov function for ϕ which is strictly decreasing outside SCRd(ϕ). Such result generalizes Conley's Fundamental Theorem Dynamical Systems strong chain recurrent set.