Simultaneous linearization of holomorphic germs in presence of resonances

Author:
Jasmin Raissy

Journal:
Conform. Geom. Dyn. **13** (2009), 217-224

MSC (2010):
Primary 37F50; Secondary 32H50

DOI:
https://doi.org/10.1090/S1088-4173-09-00199-4

Published electronically:
September 9, 2009

MathSciNet review:
2540705

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $f_{1}, \dots , f_{m}$ be $m\ge 2$ germs of biholomorphisms of $\mathbb {C}^{n}$, fixing the origin, with $(\mathrm {d}f_{1})_{O}$ diagonalizable and such that $f_{1}$ commutes with $f_{h}$ for any $h=2,\dots , m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\mathrm {d}f_{1})_{O}$ and some restrictions on their resonances, $f_{1}, \dots , f_{m}$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_{1}, \dots , f_{m}$.

- M. Abate,
*Discrete holomorphic local dynamical systems*, to appear in “Holomorphic Dynamical Systems”, Eds. G. Gentili, J. Guenot, G. Patrizio, Lecture notes in Math., Springer-Verlag, Berlin, 2009, arXiv:0903.3289v1. - Filippo Bracci,
*Local dynamics of holomorphic diffeomorphisms*, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)**7**(2004), no. 3, 609–636 (English, with English and Italian summaries). MR**2101654** - A. D. Brjuno,
*Analytic form of differential equations. I, II*, Trudy Moskov. Mat. Obšč.**25**(1971), 119–262; ibid. 26 (1972), 199–239 (Russian). MR**0377192** - S. Marmi,
*An introduction to small divisors problems*, I.E.P.I., Pisa, 2003. - J. Raissy,
*Linearization of holomorphic germs with quasi-Brjuno fixed points*, Math. Z. (2009), http://www.springerlink.com/content/3853667627008057/fulltext.pdf, Online First. - L. Stolovitch,
*Family of intersecting totally real manifolds of $(\mathbb {C}^{n},0)$ and CR-singularities*, preprint 2005, arXiv: math/0506052v2.

Retrieve articles in *Conformal Geometry and Dynamics of the American Mathematical Society*
with MSC (2010):
37F50,
32H50

Retrieve articles in all journals with MSC (2010): 37F50, 32H50

Additional Information

**Jasmin Raissy**

Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Email:
raissy@mail.dm.unipi.it

Keywords:
Linearization problem,
commuting holomorphic maps,
resonances,
small divisors,
Brjuno condition

Received by editor(s):
February 13, 2009

Received by editor(s) in revised form:
July 27, 2009

Published electronically:
September 9, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.