Simultaneous linearization of holomorphic germs in presence of resonances
Author:
Jasmin Raissy
Journal:
Conform. Geom. Dyn. 13 (2009), 217224
MSC (2010):
Primary 37F50; Secondary 32H50
Published electronically:
September 9, 2009
MathSciNet review:
2540705
Fulltext PDF Free Access
Abstract 
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Abstract: Let be germs of biholomorphisms of , fixing the origin, with diagonalizable and such that commutes with for any . We prove that, under certain arithmetic conditions on the eigenvalues of and some restrictions on their resonances, are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under .
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M. Abate, Discrete holomorphic local dynamical systems, to appear in ``Holomorphic Dynamical Systems'', Eds. G. Gentili, J. Guenot, G. Patrizio, Lecture notes in Math., SpringerVerlag, Berlin, 2009, arXiv:0903.3289v1.
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Filippo
Bracci, Local dynamics of holomorphic diffeomorphisms, Boll.
Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7 (2004),
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(2005m:32034)
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D. Brjuno, Analytic form of differential equations. I, II,
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S. Marmi, An introduction to small divisors problems, I.E.P.I., Pisa, 2003.
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J. Raissy, Linearization of holomorphic germs with quasiBrjuno fixed points, Math. Z. (2009), http://www.springerlink.com/content/3853667627008057/fulltext.pdf, Online First.
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L. Stolovitch, Family of intersecting totally real manifolds of and CRsingularities, preprint 2005, arXiv: math/0506052v2.
 [A]
 M. Abate, Discrete holomorphic local dynamical systems, to appear in ``Holomorphic Dynamical Systems'', Eds. G. Gentili, J. Guenot, G. Patrizio, Lecture notes in Math., SpringerVerlag, Berlin, 2009, arXiv:0903.3289v1.
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 S. Marmi, An introduction to small divisors problems, I.E.P.I., Pisa, 2003.
 [R]
 J. Raissy, Linearization of holomorphic germs with quasiBrjuno fixed points, Math. Z. (2009), http://www.springerlink.com/content/3853667627008057/fulltext.pdf, Online First.
 [S]
 L. Stolovitch, Family of intersecting totally real manifolds of and CRsingularities, preprint 2005, arXiv: math/0506052v2.
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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/S1088417309001994
PII:
S 10884173(09)001994
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.
