Linearization of holomorphic germs with quasi-Brjuno fixed points

TL;DR

This paper establishes conditions under which a holomorphic germ with a quasi-Brjuno fixed point can be linearized, linking the problem to the existence of an invariant complex manifold, thus unifying classical results.

Contribution

It provides a new criterion for holomorphic linearization of germs based on arithmetic conditions and invariant manifolds, extending classical linearization theorems.

Findings

Linearization characterized by invariant complex manifolds

Arithmetic conditions on eigenvalues are crucial

Most classical results are corollaries of this framework

Abstract

Let $f$ be a germ of holomorphic diffeomorphism of $\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is locally holomorphically linearizable if and only if there exists a particular $f$-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.