Functional equations and local conjugacy of mappings of class $C^\infty$

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.