Pugh's closing lemma

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.^[1]

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 ${\displaystyle f:M\to M}$ be a ${\displaystyle C^{1}}$ diffeomorphism of a compact smooth manifold ${\displaystyle M}$. Given a nonwandering point ${\displaystyle x}$ of ${\displaystyle f}$, there exists a diffeomorphism ${\displaystyle g}$ arbitrarily close to ${\displaystyle f}$ in the ${\displaystyle C^{1}}$ topology of ${\displaystyle \operatorname {Diff} ^{1}(M)}$ such that ${\displaystyle x}$ is a periodic point of ${\displaystyle g}$.^[2]

See also

• Smale's problems