Improved convergence estimates for the Schr\"oder-Siegel problem

TL;DR

This paper improves the bounds on the convergence radius for conjugating analytic maps to their linear parts in higher dimensions, under Bruno's condition, refining previous estimates from a factor of 2 to 1.

Contribution

It extends convergence estimates for the Schr"oder-Siegel problem to higher dimensions, achieving a sharper bound with a constant C=1, improving upon the previous C=2.

Findings

Convergence radius bound improved to ' with C=1 for n>1.

Extension of Schrf6der-Siegel problem to higher dimensions.

Refinement of previous estimates for conjugation in complex dynamics.

Abstract

We reconsider the Schr\"oder-Siegel problem of conjugating an analytic map in $\mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Bruno's one on the eigenvalues $\lambda_1,\ldots,\lambda_n$ of the linear part we show that the convergence radius $\rho$ of the conjugating transformation satisfies $\ln \rho(\lambda )\geq -C\Gamma(\lambda)+C'$ with $\Gamma(\lambda)$ characterizing the eigenvalues $\lambda$, a constant $C'$ not depending on $\lambda$ and $C=1$. This improves the previous results for $n>1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for $n=1$.