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

TL;DR

This paper investigates the analytic linearization of a generalized semi-standard map near an invariant curve in complex two-dimensional space, establishing bounds on the radius of convergence related to Brjuno sums and polynomial coefficients.

Contribution

It extends the analysis of linearization radius bounds to a broader class of trigonometric polynomial systems, generalizing previous semi-standard map results.

Findings

Lower bound on the radius of convergence involving Brjuno sums.

Upper bound on the radius for certain polynomial classes.

Error function's monotonicity with respect to polynomial coefficients.

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 $C\geq 0$ does not depend on $\alpha$, $d\in \mathbb{N}^*$ 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.