Decompositions and Liapounov functions

From the work of C. Conley [Isolated invariant sets and the Morse index. in: Conference Board Math. Sci., No. 38. Providence: AMS; 1978], it is known that a compact metric space with a fixed flow can be decomposed into a finite collection of mutually disjoint compact invariant subsets (an attractor–repeller pair, or more generally, a Morse decomposition is such a decomposition). Here we shall display the intrinsic relation between the existence of these decompositions and that of Liapounov functions.