# 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 diffeomorphism of a compact smooth manifold . Given a nonwandering point of , there exists a diffeomorphism arbitrarily close to in the topology of such that is a periodic point of .
^{[1]}

## 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, an open set of conditions on a bounded continuous dynamical system that rules out periodic behaviour also implies that the system cannot behave chaotically; this is the basis of some autonomous convergence theorems.

## See also

## References

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