Pugh's closing lemma

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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 $M$ .

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

Synonym:

closing lemma

Type of Math Object:

Theorem

Major Section:

Reference