Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points
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Smale [12,13] studied diffeomorphisms with transversal homoclinic points and showed that the dynamics are chaotic in the neighbourhood of the orbit of such a point, in the sense that there is a compact invariant set on which the action of some iterate of the diffeomorphism is topologically conjugate to the action of the Bernoulli shift. One immediate consequence of this is Birkhoff’s result that the diffeomorphism has infinitely many periodic points. It also turns out that nearby diffeomorphisms must also have transversal homoclinic points and hence also infinitely many periodic points. Thus the property of having infinitely many periodic points cannot, in general, be perturbed away.
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