Improved convergence estimates for the Schröder-Siegel problem

Abstract

We reconsider the Schröder-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 ${\lambda }_1,\dots ,{\lambda }_n$ of the linear part we show that the convergence radius $\rho$ of the conjugating transformation satisfies $\ln \rho (\lambda )\ge -C\Gamma (\lambda )+C^{\prime }$ with $\Gamma (\lambda )$ characterizing the eigenvalues $\lambda$, a constant $C^{\prime }$ 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 $n=1$.

21 pages

References in corpus (2)

• On the representation of maps by Lie transforms
• On a theorem of Lyapounov