Geodesic
Functional equations and local conjugacy of mappings of class~$C^\infty$
Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 587-602

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Theorems are proved on conjugacy of $C^\infty$ mappings in a neighborhood of a fixed point, under the assumption of formal conjugacy. In constrast to a well-known theorem of Sternberg, we assume the existence of a linear approximation of points of the spectrum on the unit circle and at zero. We establish theorems on conjugacy in a subgroup of the group of diffeomorphisms, and give conditions for the existence of local solutions of more general functional equations. A fixed-point principle is used in the proof. Bibliography: 14 titles. 
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     author = {G. R. Belitskii},
     title = {Functional equations and local conjugacy of mappings of class~$C^\infty$},
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G. R. Belitskii. Functional equations and local conjugacy of mappings of class~$C^\infty$. Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 587-602. http://geodesic.mathdoc.fr/item/SM_1973_20_4_a6/