Horospheres and iterates of holomorpic maps (Q1093060)

Let \(D\) be a bounded strictly convex \(C^ 2\) domain of \({\mathbb{C}}^ n\), and \(f: D\to D\) a holomorphic map. The aim of this paper is to desribe the behavior of the sequence of iterates of \(f\). We shall prove that (a) if \(f\) has a fixed point \(z_ 0\in D\), then the sequence of iterates converges iff \(df_{zo}\) has no eigenvalues \(\lambda\) \(\neq 1\) with \(| \lambda | =1\); (b) if \(f\) has no fixed points in \(D\), then the sequence of iterates converges to a point of the boundary. The main tool in the proof is a generalization of the classical notion of horospheres obtained by means of the Kobayashi distance.