  We show there is a residual set of non-Anosov $C^{\infty}$ Axiom A
diffeomorphisms with the no cycles property whose elements have trivial
centralizer. If $M$ is a surface and $2\leq r\leq \infty$, then we will show
there exists an open and dense set of 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.
