Smooth linearization of nonautonomous coupled systems

In a joint work with Palmer [3] we have formulated sufficient conditions under which there exist continuous and invertible transformations of the form $ H_n(x, y) taking solutions coupled system \begin{document}$ \begin{equation*} x_{n+1} = A_nx_n+f_n(x_n, y_n), \quad y_{n+1} g_n( \end{equation*} $\end{document} onto associated partially linearized uncoupled A_nx_n, y_n). present go one step further provide H_n H_n^{-1} are smooth in variables x y $. We emphasise that our general do not involve any kind dichotomy, nonresonance or spectral gap assumptions for linear part on most related works.