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 $f:M\; mapsto\; M$ 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 $g$. [Charles C. Pugh, "An Improved Closing Lemma and a General Density Theorem", "American Journal of Mathematics", 89(4):1010-1021, 1967]

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, any set of conditions on a bounded continuous dynamical system that rules out periodic behaviour also implies that the system cannot behave chaotically.

References