Newhouse proved that in the space of C^r smooth diffeomorphisms r > 2, a topologically general dynamical system can have an infinite number of attractors (he goes even further, actually in showing the “abundance” of hyperbolic sets with this property). 


Doesn't this invalidate Palis’ conjecture that a dynamical system has a finite number of attractors with probability 1?



Palis' conjecture and Newhouse's results
The Palis conjectures are carefully formulated to avoid the issue with the Newhouse phenomenon (in its original form). 


For a discussion, see the following paper by Berger, who **does** disprove one aspect of Palis's conjectures using this kind of phenomenon: <http://arxiv.org/abs/1411.6441>.


