Asymmetrical diffeomorphisms.
Abstract. Which diffeomorphisms of a compact manifold M commute with no other diffeomorphisms
(except their own powers)? Smale asked if such highly asymmetrical diffeomorphisms are typical, in
that they are dense in the Cr topology on the space of Cr diffeomorphisms
Diffr(M). In this talk I will explain the recent (positive) solution to Smale's question
for C1 symplectomorphisms and volume-preserving diffeomorphisms. I will also discuss
progress on the general case. This is joint work with Christian Bonatti and Sylvain Crovisier.