Simultaneous linearization of holomorphic germs in presence of resonances

Abstract

Let $f_1,...,f_m$ be $m\ge 2$ germs of biholomorphisms of $\C^n$, fixing the origin, with $(\d f_1)_O$ diagonalizable and such that $f_1$ commutes with $f_h$ for any $h=2,...,m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\d f_1)_O$ and some restrictions on their resonances, $f_1,...,f_m$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_1,...,f_m$.

7 pages, corrected misprints, to appear in Conformal Geometry and Dynamics

References in corpus (1)

• Discrete holomorphic local dynamical systems

Cited by in corpus (1)

• Torus actions in the normalization problem