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

Importance: Outstanding ✭✭✭✭
Subject: Topology
Recomm. for undergrads: no
Posted by: Jailton Viana
on: April 24th, 2013

\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 .

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.


Bonatti C, Diaz L.; Viana M.; Dynamics beyond uniform hyperbolicity , Springer[Encyclopaedia of Mathematics Sciences ], Volume 102, 2005

* indicates original appearance(s) of problem.