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Posted	by:	Jailton Viana
	on:	April 24th, 2013

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 .

Definition : A set $\Lambda \subset M$ is an attractor for a Diffeomorphism (or a flow ) if it is invariant , transitive and the basin of attraction $B(\Lambda) := \{p\in M / \omega(p)\subset \Lambda \}$ has positive Lebesgue Measure.