On the convergence of coordinate transformations in the KAM procedure (Q5938491)

The author proves that, for every sequence \(\varepsilon_n\) of positive numbers, there exists a sequence of mappings \(U_n: \varphi \to \varphi + u_n(\varphi)\) of the unit circle onto itself such that \(|u_n(\varphi)|< \varepsilon_n\), and the sequence of mappings \(W_n = U_1 U_2 \cdots U_n\) does not converge, even pointwise. It is worthy of mention that if the sequence \(U_n\) of \(C^r\)-mappings converges to the identity mapping in the \(C^r\)-topology with \(r \geq 1\), then the sequence \(W_n\) converges in the \(C^r\)-topology, and this property is used in KAM theory.