Horosphere-invariant mappings of the unit ball in [formula omitted

Abstract

Horospheres, particular ellipsoids internally tangent to the boundary of the Euclidean unit ball B of Cn, play a pivotal role in the Denjoy–Wolff theory of holomorphic self-mappings of B. We use horospheres to introduce a new family of transforms on the class of normalized locally biholomorphic mappings of B into Cn in a vein similar to the Koebe transforms with respect to automorphisms of B. Like the Koebe transforms, these horosphere transforms preserve the property of convexity of a mapping. After studying the transforms with regard to convex mappings in particular, we give a broad characterization of all normalized holomorphic mappings that are invariant under the horosphere transforms. We also show that the notion of horosphere invariance for locally biholomorphic mappings is preserved by several commonly studied extension operators. We then apply our technique to provide a similar, simplified characterization of Koebe-invariant mappings. With these characterizations in hand, we show that, despite the existence of many mappings satisfying either type of invariance separately, the only normalized holomorphic mappings that enjoy both types of invariance are those mappings of the formf(z)=(z11−z1+Q(zˆ)(1−z1)2,zˆ(1−z1)) for z=(z1,zˆ)∈B, where zˆ=(z2,…,zn) and Q:Cn−1→C is a homogeneous polynomial of degree 2. For certain Q, these mappings are extreme points of the closed convex hull of the family of convex mappings of B.