Pugh's closing lemma

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In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows:

Let be a C¹ diffeomorphism of a compact smooth manifold M. Given a nonwandering point x of f, there exists a diffeomorphism g arbitrarily close to f in the C¹ topology of such that x is a periodic point of g.^[1]

[edit] Interpretation

Pugh's closing lemma means, for example, that any chaotic set in a bounded continuous dynamical system corresponds to a periodic orbit in a different but closely related dynamical system. As such, any set of conditions on a bounded continuous dynamical system that rules out periodic behaviour also implies that the system cannot behave chaotically.

[edit] References

1. ^ Charles C. Pugh, "An Improved Closing Lemma and a General Density Theorem", American Journal of Mathematics, 89(4):1010-1021, 1967

This article incorporates material from Pugh's closing lemma on PlanetMath, which is licensed under the GFDL.