Preface The qualitative study of the Solutions to ordinary differential equations has had a long and varied history. In recent years much attention has been paid to the connections between the theory of these smooth "dynamical Systems" and two other areas of mathematics: er- godic theory and Statistical mechanics on the one hand and algebraic and differential topol- ogy on the other. It is the relationships between dynamical Systems and topology which these lectures address. This particular part of dynamical Systems dates from the work of Poincare, espe- cially his beautiful theorem equating the Euler characteristic of a surface with the sum of the indices of rest points of a flow on the surface. In this Century the most important contributions to this area of investigation have been made by Marston Morse and Steve Smale. It is not possible to survey all their contributions in a meaningful way in this brief introduction. However, the importance of their contribu- tions can perhaps be gauged by the number of times their names occur in the chapter titles of these lectures. It would be remiss however not to mention the special importance of Smale's paper Differentiable dynamical Systems [Sl], not only for investigations of the type we consider here, but all qualitative investigation of smooth dynamical Systems. In addition to proving important new results, this article had a major influence on the direction of the whole field of dynamical Systems. Two influences merit special mention. First it emphasized ciassifying dynamical Systems according to the complexity of their qualitative dynamical be= havior, rather than, for example, the more traditional way of Classification by complexity of the algebraic form of the differential equation. Secondly Smale drew attention to structur- ally stable Systems as particularly worthy of investigation and conjectured a characterization of them, which has subsequently been proven correct in many cases. The relationship between qualitative dynamics and topology is much too large an area to consider fruitfully in its entirety within the framework of these lectures and accordingly we will narrow our attention to a collection of results with a particularly homological flavor. The theme of these lectures is illustrated in the following diagram. Differential -^ topology Chain COmplex algebra ,, , Dynamics * ^ , . . • Homology ~ descnption EsL (basic sets and unstable manifolds)

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