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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Further reductions of Poincaré-Dulac normal forms in $ {\mathbf{C}}^{n+1}$


Author: Adrian Jenkins
Journal: Proc. Amer. Math. Soc. 136 (2008), 1671-1680
MSC (2000): Primary 32A05, 32H50; Secondary 30D05
Published electronically: January 30, 2008
MathSciNet review: 2373596
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Abstract: In this paper, we will consider (germs of) holomorphic mappings of the form $ (f(z),\lambda _{1} w_{1}(1+g_{1}(z)),\ldots ,\lambda_{n}w_{n}(1+g_{n}(z)))$, defined in a neighborhood of the origin in $ {\mathbf{C}}^{n+1}$. Most of our interest is in those mappings where $ f(z)=z+a_{m}z^{m}+\cdots $ is a germ tangent to the identity and $ g_{i}(0)=0$ for $ i=1,\ldots ,n$, and $ \lambda _{i}\in {\mathbf{C}}$ possess no resonances, for these are the so-called Poincaré-Dulac normal forms of the mappings $ (z+O(2), \lambda _{1}w+O(2),\ldots ,\lambda _{n}w+O(2))$. We construct formal normal forms for these mappings and discuss a condition which tests for the convergence or divergence of the conjugating maps, giving specific examples.


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Additional Information

Adrian Jenkins
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
Email: majenkin@math.purdue.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09041-2
PII: S 0002-9939(08)09041-2
Keywords: Holomorphic mappings, conjugacy, equivalence
Received by editor(s): August 28, 2006
Received by editor(s) in revised form: December 11, 2006
Published electronically: January 30, 2008
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.