In the mathematical field of dynamical systems theory, Pugh's closing lemma is a result that establishes a close relationship between chaotic behavior and periodic behavior. Broadly, the lemma states that any point that is "nonwandering" within a system can be turned into a periodic (or repeating) point by making a very small, carefully chosen change to the system's rules. This has significant implications. For example, it means that if a set of conditions on a bounded, continuous dynamical system rules out periodic orbits, that system cannot behave chaotically. This principle is the basis of some autonomous convergence theorems.
Formal statement Let f : M → M {\displaystyle f:M\to M} be a C 1 {\displaystyle C^{1}} diffeomorphism of a compact smooth manifold M {\displaystyle M} . Given a nonwandering point x {\displaystyle x} of f {\displaystyle f} , there exists a diffeomorphism g {\displaystyle g} arbitrarily close to f {\displaystyle f} in the C 1 {\displaystyle C^{1}} topology of Diff 1 ( M ) {\displaystyle \operatorname {Diff} ^{1}(M)} such that x {\displaystyle x} is a periodic point of g {\displaystyle g} .
See also Smale's problems
References
Further reading Araújo, Vítor; Pacifico, Maria José (2010). Three-Dimensional Flows. Berlin: Springer. ISBN 978-3-642-11414-4. This article incorporates material from Pugh's closing lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.