Lyapunov 1-forms for flows

Abstract

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space X admits a Lyapunov 1-form $\omega$ lying in a prescribed Čech 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 $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem by Conley about the existence of Lyapunov functions.