Simultaneous linearization of holomorphic germs in presence of resonances
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- by Jasmin Raissy
- Conform. Geom. Dyn. 13 (2009), 217-224
- DOI: https://doi.org/10.1090/S1088-4173-09-00199-4
- Published electronically: September 9, 2009
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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
- 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.
Bibliographic Information
- Jasmin Raissy
- Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
- Email: raissy@mail.dm.unipi.it
- Received by editor(s): February 13, 2009
- Received by editor(s) in revised form: July 27, 2009
- Published electronically: September 9, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2540705