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

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Abstract | References | Similar Articles | Additional Information

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 .

**[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., Springer-Verlag, Berlin, 2009, arXiv:0903.3289v1.**[B]**F. Bracci,*Local dynamics of holomorphic diffeomorphisms*, Boll. UMI (8), 7-B (2004), 609-636. MR**2101654 (2005m:32034)****[Br]**A. D. Brjuno,*Analytic form of differential equations*, Trans. Moscow Math. Soc.**25**(1971), pp. 131-288;**26**(1972), pp. 199-239. MR**0377192 (51:13365)****[M]**S. Marmi,*An introduction to small divisors problems*, I.E.P.I., Pisa, 2003.**[R]**J. Raissy,*Linearization of holomorphic germs with quasi-Brjuno 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 CR-singularities*, 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:
https://doi.org/10.1090/S1088-4173-09-00199-4

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.