**Conjecture**Let be the space of Diffeomorphisms on the connected , compact and boundaryles manifold M and the space of vector fields. There is a dense set ( ) such that exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space

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

Definition : A set is an attractor for a Diffeomorphism (or a flow ) if it is invariant , transitive and the basin of attraction has positive Lebesgue Measure.

## Bibliography

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

* indicates original appearance(s) of problem.