Pugh's closing lemma

Short description: Mathematical result

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

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

Further reading

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