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 {\displaystyle g} 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]

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

References

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. JSTOR 2373414.