Holomorphic linearization of commuting germs of holomorphic maps

Abstract

Let $f_1,...,f_h$ be $h\ge 2$ germs of biholomorphisms of $\C^n$ fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1,...,f_h$ commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.

24 pages; final version with erratum (My original paper failed to cite the work of L. Stolovitch [ArXiv:math/0506052v2]); J. Geom. Anal. 2012

References in corpus (1)

• Linearization of holomorphic germs with quasi-Brjuno fixed points

Cited by in corpus (1)

• Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step