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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

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
Published electronically: September 9, 2009
MathSciNet review: 2540705
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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}$.


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Additional Information

Jasmin Raissy
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: raissy@mail.dm.unipi.it

DOI: http://dx.doi.org/10.1090/S1088-4173-09-00199-4
PII: S 1088-4173(09)00199-4
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.