Smale flows on the three-sphere
Transactions of the American Mathematical Society
In this paper, a complete classification of Smale flows on S3 is obtained. This classification is presented by means of establishing a concise set of properties that must be satisfied by an (abstract) Lyapunov graph associated to a Smale flow and a Lyapunov function. We show that these properties are necessary, that is, given a Smale flow and a Lyapunov function, its Lyapunov graph satisfies this set of properties. We also show that these properties are sufficient, that is, given an abstract
... iven an abstract Lyapunov graph V satisfying this set of properties, it is possible to realize a Smale flow on S3 that has a graph L as its Lyapunov graph where L is equal to V up to topological equivalence. The techniques employed in proving that the conditions imposed on the graph are necessary involve some use of homology theory. Geometrical methods are used to construct the flow on S3 associated to the given graph and therefore establish the sufficiency of the above conditions. The main theorem in this paper generalizes a result of Franks  who classified nonsingular Smale flows onS3.