# 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

Title Pugh’s closing lemma PughsClosingLemma 2013-03-22 14:07:13 2013-03-22 14:07:13 Koro (127) Koro (127) 8 Koro (127) Theorem msc 37C20 msc 37C25 closing lemma