Pugh’s closing lemma

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

The analogous theorem holds when $x$ is a nonwandering point of a $\mathcal{C}^{1}$ flow on $M$ .

References

1 Pugh, C., An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967).

Title	Pugh’s closing lemma
Canonical name	PughsClosingLemma
Date of creation	2013-03-22 14:07:13
Last modified on	2013-03-22 14:07:13
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	8
Author	Koro (127)
Entry type	Theorem
Classification	msc 37C20
Classification	msc 37C25
Synonym	closing lemma