Centralizers of partially hyperbolic diffeomorphisms

The centralizer of a Cr diffeomorphism f, denoted $Cf$, is the set of all Cr diffeomorphisms that commute with f. The centralizer is closed under the operation of composition, and forms a topological group in the C r topology. We say that the centralizer is trivial if f commutes only with its integer powers.; We show that in the set of all partially hyperbolic diffeomorphisms on a compact manifold M, there is a C 1-open and dense subset with discrete centralizers. In particular, these diffeomorphisms do not embed as the time-t map of a flow.; Moreover, there is a large class of partially hyperbolic diffeomorphisms with trivial centralizer. We define an open set $U$ of partially hyperbolic diffeomorphisms, containing all time- t maps of Anosov flows, and all partially hyperbolic skew products on $Tn$ × S1 that cover an Anosov diffeomorphism on $Tn$. We show that within $U$ there is a residual subset of diffeomorphisms with trivial centralizer (in the C∞ topology).; We also study centralizers of partially hyperbolic skew products with higher dimensional center bundles. We show that in the set of partially hyperbolic skew products on $Tn$ × M, where the dimension of M is larger than one, there is an open, dense subset with centralizers of the following form: if G commutes with F, then Gk = Fj for some integers k and j.