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  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 .[1]


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]


  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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