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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables


Authors: Patrick Ahern and Jean-Pierre Rosay
Journal: Trans. Amer. Math. Soc. 347 (1995), 543-572
MSC: Primary 30D05; Secondary 26A18, 26E05, 34A20
DOI: https://doi.org/10.1090/S0002-9947-1995-1276933-6
MathSciNet review: 1276933
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Abstract: This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in $ \mathbb{R},\mathbb{C},$ or $ {\mathbb{R}^2}$), (maps $ f$ defined near 0, such that $ f(0) = 0$, and $ f'(0)$ is the identity). Proofs are mostly original.

The paper is mostly oriented towards precise examples and the questions of descriptions of members in the conjugacy class, flows, $ k$th root.

It happened that entire functions provide clear and easy examples. However they should be considered just as a tool, not as the main topic. For example in Proposition $ 2$ the function $ z \mapsto z + {z^2}$ should be better thought of as the map $ (x,y) \to (x + {x^2} - {y^2},y + 2xy)$.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1276933-6
Article copyright: © Copyright 1995 American Mathematical Society