Sur la classification des germes d'applications holomorphes contractantes. (On the classification of germs of contracting holomorphic maps)
The author considers the problem of determining normal forms of germs of holomorphic maps F: (\({\mathbb{C}}^ 2,0)\to ({\mathbb{C}}^ 2,0)\) which satisfy the following conditions: (1) All the eigenvalues of the Jacobian matrix dF(0) have modulus strictly less than 1. (2) \(F=\Pi \cdot \sigma\), where \(\pi\) : \(B^{\Pi}\to B\) is a modification at the center 0 of the unit ball \(B\subset {\mathbb{C}}^ 2\), and \(\sigma\) : \(\bar B\to B^{\Pi}\) is a holomorphic embedding such that \(\sigma (0)\in \Pi^{-1}(0).\) If dF(0) is nonsingular, the normal forms are well-known. If dF(0) is singular, F defines a generalized Inoue-Hirzebruch surface. For each f, the author defines two kinds of invariants, a(F) and \(\sigma_ n(F)\). Both are given by the continued fraction associated to the self- intersection numbers of the curves on the surface. By these invariants, he characterizes the germs of holomorphic maps which define Inoue- Hirzebruch surfaces in the strict sense, and gives an algorithm of calculating their normal forms.
scientific article; zbMATH DE number 4109109