Attractors , basins, Finite

Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems) ★★★★

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\begin{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 $M$ \end{conjecture} This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .

Keywords: Attractors , basins, Finite