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It is shown that if  F  is a hyperbolic contraction of      R n {R^n}      , coordinates may be chosen so that not only is  F  a polynomial mapping, but so is any diffeomorphism which commutes with  F . This implies an identity principle for diffeomorphisms      G 1 {G_1}       and      G 2 {G_2}       commuting with an arbitrary Morse-Smale diffeomorphism  F  of a compact manifold  M :  if       G 1 , G 2 ∈  Z ( F ) {G_1},{G_2} \in Z(F)      , then      G 1 = G 2 {G_1} = {G_2}       on an open subset of      M ⇒  G 1 ≡  G 2 M \Rightarrow {G_1} \equiv {G_2}       on  M . Finally it is shown that under a certain linearisability condition at the saddles of  F ,      Z ( F ) Z(F)       is in fact a Lie group in its induced topology.

Centralisers of 𝐶^{∞} diffeomorphisms