Robustly Transitive Dynamics

SET OF QUESTIONS A: Is every robustly transitive (RT) diffeomorphism topologically mixing? Or at least does topological mixing hold for diffeos in some open-and-dense subset of the set of RT diffeos?

(Remark: I and S. Crovisier have shown that *generically* RT diffeos are topologically mixing. Bonatti-Diaz-Ures have answered the second question affirmatively in the context of RT diffeos which are strongly partially hyperbolic with 1-dimensional center bundle.)

SET OF QUESTIONS B: Does every RT attractor support some physical measure? If so, is there a unique physical measure whose basin of attraction contains a full-Lebesgue-measure subset of the topological basin of attraction? Do these properties hold at least C1-open-and-densely among RT diffeos? Or at least Cˆ1-generically?

(Remark: we seem to be quite far from answering this second set of questions. Only recently has it been announced that Cˆ1-generic transitive *Anosov* diffeos suppport unique physical measures)

(F. Abdenur)