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Local structure of analytic transformations of two complex variables. I. (English) Zbl 0611.32001

Let T be a local analytic transformation at \(0\in {\mathbb{C}}^ 2\) defined by T: (x,y)\(\to (x_ 1,y_ 1)\) with \[ x_ 1=x+a_ 2x^ 2+...+a_ ix^ i+a_{i+1}(y)x^{i+1}+... \]
\[ y_ 1=by+b_ 1yx+...+b_ jyx^ j+b_{j+1}(y)x^{j+1}+... \] where \(a_ i\) and \(b_ j\) are constants and \(a_ i(y)\) and \(b_ j(y)\) are holomorphic functions of y on a neighbourhood of \(y=0.\)
The main part of this paper is to investigate transformation T of type \((1,b)_ 1\) i.e. transformation T such that \(a_ 2\neq 0\). The set of points P such that the sequence \(\{T^ n(P)\}\) converges (pointwise or uniformly) is studied.
Then a global application of this local investigation is given, namely, the author constructs a proper subdomain D of \({\mathbb{C}}^ 2\) which is biholomorphic to \({\mathbb{C}}^ 2\) (known as Bieberbach example).
Reviewer: Vo Van Tan

MSC:

32A10 Holomorphic functions of several complex variables
32A30 Other generalizations of function theory of one complex variable
32H99 Holomorphic mappings and correspondences
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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