Trivial centralizers for axiom A diffeomorphisms

We show there is a residual set of non-Anosov C-infinity axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If M is a surface and 2 <= r <= infinity, then we will show there exists an open and dense set of C-r axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. Additionally, we examine commuting diffeomorphisms preserving a compact invariant set Lambda where Lambda is a hyperbolic chain recurrent class for one of the diffeomorphisms.