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

Importance: Outstanding ✭✭✭✭
Subject: Topology
Recomm. for undergrads: no
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.


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

* indicates original appearance(s) of problem.