# Pugh’s general density theorem

Let $M$ be a compact smooth manifold. There is a residual subset of $\operatorname{Diff}^{1}(M)$ in which every element $f$ satisfies $\overline{\operatorname{Per}(f)}=\Omega(f)$. In other words: Generically, the set of periodic points of a $\mathcal{C}^{1}$ diffeomorphism is dense in its nonwandering set.

Here, $\operatorname{Diff}^{1}(M)$ denotes the set of all $\mathcal{C}^{1}$ difeomorphisms from $M$ to itself, endowed with the (strong) $\mathcal{C}^{1}$ topology.

## References

Title Pugh’s general density theorem PughsGeneralDensityTheorem 2013-03-22 13:40:42 2013-03-22 13:40:42 Koro (127) Koro (127) 8 Koro (127) Theorem msc 37C20 msc 37C25 general density theorem