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Charles Pugh
Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★
Author(s): Charles Pugh
Conjecture Let
and
. Then for any neighborhood
there is
such that
is periodic point of
![$ f\in Diff^{r}(M) $](/files/tex/d45758dc65c33e4e1d18e889906d62017e3ad4b8.png)
![$ p\in\omega_{f} $](/files/tex/6895bfe444c3ed2420744f8fa4a2244580943023.png)
![$ V_{f}\subset Diff^{r}(M) $](/files/tex/5068fd3c71b626b7da0dc70cd04a36131d00be24.png)
![$ g\in V_{f} $](/files/tex/b17a7c9db99af8d718bac041b728beda5e26c111.png)
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for
. In the case of Flows this has been solved by Sushei Hayahshy for
. But in the two cases the problem is wide open for
Keywords: Dynamics , Pertubation
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