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.