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 ${\displaystyle f:M\to M}$ be a ${\displaystyle C^{1}}$ diffeomorphism of a compact smooth manifold ${\displaystyle M}$. Given a nonwandering point ${\displaystyle x}$ of ${\displaystyle f}$, there exists a diffeomorphism ${\displaystyle g}$ arbitrarily close to ${\displaystyle f}$ in the ${\displaystyle C^{1}}$ topology of ${\displaystyle \operatorname {Diff} ^{1}(M)}$ such that ${\displaystyle x}$ is a periodic point of ${\displaystyle g}$.[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. JSTOR 2373414.

Further reading

• Araújo, Vítor; Pacifico, Maria José (2010). Three-Dimensional Flows. Berlin: Springer. ISBN 978-3-642-11414-4.

This article incorporates material from Pugh's closing lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.