Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum

One considers a system on \begin{document}$ \mathbb{C}^2 $\end{document} close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of \begin{document}$ d\alpha $\end{document}, where \begin{document}$ d\in \mathbb{N}^* $\end{document} and \begin{document}$ \alpha $\end{document} is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.