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Closing Lemma for Diffeomorphism (Dynamical Systems)

Importance: Outstanding ✭✭✭✭
Author(s): Charles Pugh
Subject: Topology
Keywords: Dynamics , Pertubation
Recomm. for undergrads: no
Posted by: Jailton Viana
on: April 24th, 2013
Conjecture   Let $ f\in Diff^{r}(M) $ and $ p\in\omega_{f} $. Then for any neighborhood $ V_{f}\subset Diff^{r}(M) $ there is $ g\in V_{f} $ such that $ p $ is periodic point of $ g $

There is an analogous conjecture for flows ( $ C^{r} $ vector fields . In the case of diffeos this was proved by Charles Pugh for $ r = 1 $. In the case of Flows this has been solved by Sushei Hayahshy for $ r = 1 $ . But in the two cases the problem is wide open for $ r > 1 $


Bibliography

Dynamics beyond uniform hyperbolicity :\Springer [Encyclopaedia of Mathematical Sciences Volume 102, Mathematical Phisics,2005]


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