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

Abstract

One considers a system on ${\mathbb{C}}^2$ 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 from below by $\exp (-\frac{2}{d}B(d\alpha )-C)$, where $C\ge 0$ does not depend on $\alpha$, $d\in {\mathbb{N}}^{\ast }$ and $\alpha$ is the frequency of the linear part. For a class of trigonometric polynomials, it is also bounded from above by a similar function. The error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.