In 1995, Jacob Palis stated an ambitious conjecture on denseness of finitude of attractors for
diffeomorphisms in arbitrary dimensions. Before that, Newhouse proved the existence of residual sets of
surface diffeomorphisms (in certain nonempty open sets) in the C² topology displaying infinitely many sinks
(hyperbolic periodic attractors), and Bonatti and Diaz did the same in the C¹ topology in higher dimensions.
We will discuss the problem of proving the denseness (in the C¹ topology) of diffeomorphisms (in arbitrary
dimensions) displaying only a finite number of sinks. This is a joint work with Fernando Lenarduzzi and Jacob
Palis. | 
