Lyapunov 1-forms for flows

In this paper we find conditions which guarantee that a given flow $\mathrm{\Phi }$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi \in {\check{H}}^1(X;\mathbb{R})$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\mathrm{\Phi }$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.