Local structure of analytic transformations of two complex variables. I
type: Item
instance of: scholarly article
title: Local structure of analytic transformations of two complex variables. I
zbMATH Open document ID: 0611.32001
author: Tetsuo Ueda
published in: Journal of Mathematics of Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 32H99
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 3987654
zbMATH Keywords: iteration
zbMATH Keywords: fixed point
zbMATH Keywords: local analytic transformation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520921
OpenAlex ID: W4233967407
DOI: 10.1215/KJM/1250520921
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description: scientific article; zbMATH DE number 3987654