Classification of semigroups of linear fractional maps in the unit ball (Q854105)

\textit{C.\,C.\thinspace Cowen} and \textit{B.\,D.\thinspace MacCluer} [Acta Sci.\ Math.\ 66, No.\,1--2, 351--376 (2000; Zbl 0970.47011)] introduced a class of holomorphic maps of the unit ball \(B^n\) of \({\mathbb{{C}}}^n\), called the linear fractional self-maps, which generalize the automorphisms of \(B^n\) and the linear fractional maps in one variable. In the paper under review, the authors deal with continuous semigroups of linear fractional maps of the unit ball. They give a complete classification of these analytic objects up to conjugation with injective linear fractional maps. Essentially, they prove that semigroups of linear fractional maps of \(B^n\) are linearizable. Their classification is based on the presence or not of (common) fixed points in \(B^n\). In the case of common fixed points (the elliptic case), the semigroup is essentially given by a matrix semigroup of the type \(Z\mapsto e^{tM}Z\) with \(M\) being dissipative and asymptotically stable. In the case where the semigroup has no common fixed points in \(B^n\), all the iterates have a common fixed point on \(\partial B^n\). In this case the semigroup is hyperbolic or parabolic, depending on the boundary dilatation coefficient. For the hyperbolic and parabolic case, the paper provides a general form and several simpler forms, depending on the geometrical invariants which the semigroup might have.