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 $g$ .{{cite journal |first=Charles C. |last=Pugh |title=An Improved Closing Lemma and a General Density Theorem |journal=American Journal of Mathematics |volume=89 |issue=4 |pages=1010–1021 |year=1967 |doi=10.2307/2373414 |jstor=2373414 }}

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.

Pugh's closing lemma

Interpretation

See also

References

Further reading