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