TY - JOUR

T1 - Generalizations of linear fractional maps for classical symmetric domains and related fixed point theorems for generalized balls

AU - Gao, Yun

AU - Ng, Sui Chung

AU - Seo, Aeryeong

N1 - Publisher Copyright:
© 2020, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/2

Y1 - 2021/2

N2 - 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.

AB - 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.

KW - Automorphisms

KW - Fixed points

KW - Generalized balls

KW - Linear fractional maps

UR - http://www.scopus.com/inward/record.url?scp=85085132009&partnerID=8YFLogxK

U2 - 10.1007/s10231-020-00984-5

DO - 10.1007/s10231-020-00984-5

M3 - Article

AN - SCOPUS:85085132009

SN - 0373-3114

VL - 200

SP - 81

EP - 99

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

IS - 1

ER -