Nonsingular smale flows on S3

THE QUALITATIVE description of the dynamical behavior of a smooth flow is often divided into two parts: the “gradient-like” behavior reflected in the existence of a Lyapunov function, and the “chain recurrent” behavior (see $1 for definition). In studying a class of dynamical systems it is important to understand two things. First, for each of these two parts what kinds of behavior are possible, and second how do these two aspects of dynamical behavior interact? In this spirit we will associate to a flow 4,: M + M, with a Lyapunov functionf: M + R, a rather natural object called the Lyapunov graph. The idea is to form a topological space by identifying to a point each component off -l(c) for each c E R. This often forms a finite graph oriented by the flow. A vertex on this graph will correspond to a component of a critical level off which always contains an indecomposable piece of the chain recurrent set: so we label each vertex with this chain recurrent flow on a compact set. Each edge of the graph corresponds to a component of the space obtained by removing from M any component of a level set off if that component contains a critical point of I: This component is diffeomorphic to N x (0, 1) where N is some codimension 1 submanifold of M, transverse to the flow. To further enhance the picture of the flow one might label the edges of the graph with topological properties of the embeddings N + M; e.g. the diffeomorphism type of N. However, we will not make this part of our definition. We also restrict our attention to the case where M is compact and the graph is finite.