Pugh's closing lemma

In the mathematical field of dynamical systems theory, Pugh's closing lemma is a result that establishes a close relationship between chaotic behavior and periodic behavior. Broadly, the lemma states that any point that is "nonwandering" within a system can be turned into a periodic (or repeating) point by making a very small, carefully chosen change to the system's rules. ^[1]

This has significant implications. For example, it means that if a set of conditions on a bounded, continuous dynamical system rules out periodic orbits, that system cannot behave chaotically. This principle is the basis of some autonomous convergence theorems.

Formal statement

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 ${\displaystyle g}$. ^[2]

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.
2. 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.