Open Problem Garden

Author(s):

Conjecture Let $Diff^{r}(M)$ be the space of $C^{r}$ Diffeomorphisms on the connected , compact and boundaryles manifold M and $\chi^{r}(M)$ the space of $C^{r}$ vector fields. There is a dense set $D\subset Diff^{r}(M)$ ( $D\subset \chi^{r}(M)$ ) such that $\forall f\in D$ exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space $M$

This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .

Keywords: Attractors , basins, Finite

Posted by Jailton Viana
updated April 24th, 2013

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Attractors , basins, Finite

Dynamics , Pertubation

Topology

Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★

Author(s): Charles Pugh

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 $r > 1$