Holomorphic linearization of commuting germs of holomorphic maps

Details Holomorphic linearization of commuting germs of holomorphic maps Holomorphic linearization of commuting germs of holomorphic maps

2010-05-19

arxiv.org/abs/1005.3434v2

Mathematics

Dynamical Systems

23 pages; simultaneous Brjuno condition improved, exposition improved, references updated

Scientific paper

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.

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