The Kupka-Smale Theorem



Let M be a compact manifold of dimension m and X r (M) the space of C r vector fields on M, r ≥ 1, with a C r norm. In Chapter 2 we showed that the set G1X r (M), consisting of fields whose singularities are hyperbolic, is open and dense in X r (M). This is an example of a generic property, i.e. a property that is satisfied by almost all vector fields. In this chapter we shall analyse other generic properties in X r (M). The original proof of the results dealt with here can be found in [44], [82] and [107].


Vector Field Periodic Point Invariant Manifold Compact Manifold Unstable Manifold 
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© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  1. 1.Instituto de Matemática Pura e AplicadaJardim BotânicoBrazil

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