Ueda, Tetsuo Local structure of analytic transformations of two complex variables. I. (English) Zbl 0611.32001 J. Math. Kyoto Univ. 26, 233-261 (1986). 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 Cited in 2 ReviewsCited in 34 Documents 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) Keywords:iteration; fixed point; local analytic transformation PDFBibTeX XMLCite \textit{T. Ueda}, J. Math. Kyoto Univ. 26, 233--261 (1986; Zbl 0611.32001) Full Text: DOI