Generalizations of linear fractional maps for classical symmetric domains and related fixed point theorems for generalized balls

TL;DR

This paper generalizes the study of linear fractional self maps from unit balls to broader symmetric domains called generalized type-I domains, establishing fundamental properties and fixed point theorems in this more general setting.

Contribution

It extends the theory of linear fractional maps to generalized type-I domains, including classical symmetric domains and generalized balls, with new results on their structure and fixed points.

Findings

Almost every linear self map can be represented by a matrix satisfying an expansion property.

Results on boundary extension, normal forms, and fixed points of automorphisms.

Generalization of known fixed point theorems for unit balls to broader domains.

Abstract

We extended the study of the linear fractional self maps (e.g. by Cowen-MacCluer and Bisi-Bracci on the unit balls) to a much more general class of domains, called generalized type-I domains, which includes in particular the classical bounded symmetric domains of type-I and the generalized balls. Since the linear fractional maps on the unit balls are simply the restrictions of the linear maps of the ambient projective space (in which the unit ball is embedded) on a Euclidean chart with inhomogeneous coordinates, and in this article we always worked with homogeneous coordinates, here the term linear map was used in this more general context. After establishing the fundamental result which essentially says that almost every linear self map of a generalized type-I domain can be represented by a matrix satisfying the "expansion property" with respect to some indefinite Hermitian form, we gave a variety of results for the linear self maps on the generalized balls, such as the holomorphic extension across the boundary, the normal form and partial double transitivity on the boundary for automorphisms, the existence and the behavior of the fixed points, etc. Our results generalize a number of known statements for the unit balls, including, for example, a theorem of Bisi-Bracci saying that any linear fractional map of the unit ball with more than two boundary fixed points must have an interior fixed point.