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 $$ g $$.

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.