Remote Access Conformal Geometry and Dynamics
Green Open Access

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
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 $f_{1}, \dots , f_{m}$.

References [Enhancements On Off] (What's this?)

  • 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),, Online First.
  • L. Stolovitch, Family of intersecting totally real manifolds of $(\mathbb {C}^{n},0)$ and CR-singularities, preprint 2005, arXiv: math/0506052v2.

Similar Articles

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

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.