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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?

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    $\begingroup$ My guess is that there are some differences between 'topologically general' (or generic) and 'probability 1'. An example is that the set of Diophantine numbers: it has probability 1, while its complement is generic. $\endgroup$
    – Pengfei
    Feb 16, 2015 at 6:27

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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.

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