On Sternberg’s approximation for linearization of local contractions

Linearization theorems are very useful tools for the local classification of diffeomorphisms in the neighbourhood of a fixed point. The history of the theory of topological and smooth linearization is going back at least to Poincaré. Sternberg proved in 1957 that any local L
^∞
-contraction is equivalent via a L
^∞
-change of coordinates to a linear transformation, if certain diophantine inequalities are assumed to be satisfied by the eigenvalues of the Jacobian. His original proof used an interesting approximation process. Later Sternberg's result has been generalized by many authors in the last 60 years and applied to the development of the theory of normal forms of differentiable maps.