On the Convergence of Coordinate Transformations in the KAM Procedure

We study the $C^r$-convergence of the compositions $W_n = U_1 U_2 \cdots U_n$ where mappings $U_k$ tend to the identity transformation in the $C^r$-topology as $k \to \infty$. The cases $r = 0$ and $1 \geqslant r < +\infty$ turn out to be drastically different.