The Kupka-Smale Theorem
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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 G1 ⊂ X 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 ,  and .
KeywordsVector Field Periodic Point Invariant Manifold Compact Manifold Unstable Manifold
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