Pugh's closing lemma
- Let be a diffeomorphism of a compact smooth manifold . Given a nonwandering point of , there exists a diffeomorphism arbitrarily close to in the topology of such that is a periodic point of .
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.