Linear fractional maps of the unit ball: A geometric study
A linear fractional map of the unit ball \(\mathbb{B}^n\) of \(\mathbb{C}^n\) is a map of the form \(f(z)={Az+B \over\langle z,C\rangle +d}\) such that \(f(\mathbb{B}^n)\subset \mathbb{B}^n\); here \(A\) is an \(n\times n\)-matrix, \(B\) and \(C\) are vectors in \(\mathbb{C}^n\), and \(d\in\mathbb{C}\). This class of maps was introduced and studied by \textit{C. C. Cowen} and \textit{B. D. MacCluer} [Acta Sci. Math. 66, No. 1/2, 351-376 (2000; Zbl 0970.47011)] via their representations in a space with indefinite metric (a Kreĭn space). In the present paper, they are considered from a geometric point of view, which leads to their classification. There are basically four classes of the maps according to the number of boundary fixed points: zero, one, two, and more than two. Further subdivision of these classes is obtained, which gives a representation \(f=T\circ g\circ T^{-1}\) with \(T\) an automorphism of the ball and \(g\) the corresponding normal form depending only on ``geometry'' of \(f\). Some applications to composition operators are given.
scientific article; zbMATH DE number 1800227
scientific article