Pugh’s general density theorem
Let $M$ be a compact smooth manifold^{}. There is a residual subset of ${\mathrm{Diff}}^{1}(M)$ in which every element $f$ satisfies $\overline{\mathrm{Per}(f)}=\mathrm{\Omega}(f)$. In other words: Generically, the set of periodic points of a ${\mathcal{C}}^{1}$ diffeomorphism is dense in its nonwandering set.
Here, ${\mathrm{Diff}}^{1}(M)$ denotes the set of all ${\mathcal{C}}^{1}$ difeomorphisms from $M$ to itself, endowed with the (strong) ${\mathcal{C}}^{1}$ topology.
References
- 1 Pugh, C., An improved closing lemma^{} and a general density theorem, Amer. J. Math. 89 (1967).
Title | Pugh’s general density theorem |
---|---|
Canonical name | PughsGeneralDensityTheorem |
Date of creation | 2013-03-22 13:40:42 |
Last modified on | 2013-03-22 13:40:42 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 8 |
Author | Koro (127) |
Entry type | Theorem |
Classification | msc 37C20 |
Classification | msc 37C25 |
Synonym | general density theorem |