Local structure of analytic transformations of two complex variables. I
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).
scientific article; zbMATH DE number 3987654