Pugh's closing lemma

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 $f:M \to M$ be a $C^{1}$ 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 $C^{1}$ topology of $\operatorname{Diff}^1(M)$ such that $x$ is a periodic point of $g$.^[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

• Smale's problems

References

1. Pugh, Charles C. (1967). "An Improved Closing Lemma and a General Density Theorem". American Journal of Mathematics. 89 (4): 1010–1021. doi:10.2307/2373414.

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