---
tag: E1Z444F6TV
kind: statement
role: theorem
name: Closing lemma
msc: [37C20, 37C25]
uses: [EY2W34JZA4, T75CZ5H47N, 0Z1Y7JHGBR]
about: [EY2W34JZA4, T75CZ5H47N]
sources:
  - url: https://en.wikipedia.org/wiki/Pugh%27s_closing_lemma
same_as:
  wikidata: Q7258945
  wikipedia: Pugh's closing lemma
---

## Statement

Let $M$ be a compact smooth manifold and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
Let $p$ be a [[non-wandering point|EY2W34JZA4]] of $f$.
Give the space of $C^1$ diffeomorphisms of $M$ the $C^1$ topology, and let $\mathcal{U}$ be a neighbourhood of $f$ in it.
Then there is $g \in \mathcal{U}$ such that $p$ is a [[periodic point|T75CZ5H47N]] of $g$.

## Remarks

The result is due to Pugh, and it is often called Pugh's closing lemma.
Since $p$ is non-wandering, some orbit starting near $p$ returns near $p$.
The difficulty is to close this orbit by a perturbation that is small in $C^1$.
A naive push changes the derivative by about the size of the gap divided by the size of the region where one pushes.
Pugh's argument spreads the push over many returns of the orbit, so that each step is small in $C^1$.
A consequence is the general density theorem: for a $C^1$-generic diffeomorphism, the periodic points are dense in the non-wandering set.
The analogous statement with the $C^r$ topology for $r \ge 2$ is open in general; it is the tenth of Smale's problems.

## Explanations

### L1 human

A non-wandering point is a point near which some orbits come back again and again.
The closing lemma says that by changing the system only very slightly, one can make the point lie on a closed, periodic orbit.
So recurrence can always be turned into periodicity at small cost.

### L2 human

The closing lemma is the basic perturbation tool of $C^1$ generic dynamics.
Together with the Kupka–Smale theorem it shows that a $C^1$-generic diffeomorphism has dense hyperbolic periodic points in its non-wandering set.
It is a key ingredient in Mañé's proof of the stability conjecture.
Related tools are the connecting lemma of Hayashi and Franks' lemma, which also rely on $C^1$-small perturbations.