Pugh's closing lemma

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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 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 .[1]

Interpretation[edit]

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[edit]

References[edit]

  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. 

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