In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class So（Bn） of biholomorphic mappings on ...In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class So（Bn） of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f（-, O） of a Loewner chain f（x,t） = etz ＋ ... such that {e-tf（.,t）}t≥o is a normal family on Bn. We show that if f（.,O） is an extreme point （respectively a support point） of So（Bn）, then e-t f（., t） is an extreme point of So（Bn） for t≥0 （respectively a support point of So（Bn） for t C [O, t0] and some to〉 0）. This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to So（Bn） and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.展开更多
基金supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. A9221)Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science, 2011 (Grant No. 22540213)the Romanian Ministry of Education and Research, UEFISCSU-CNCSIS(Grants Nos. PN-II-ID 524/2007, 525/2007)
文摘In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class So（Bn） of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f（-, O） of a Loewner chain f（x,t） = etz ＋ ... such that {e-tf（.,t）}t≥o is a normal family on Bn. We show that if f（.,O） is an extreme point （respectively a support point） of So（Bn）, then e-t f（., t） is an extreme point of So（Bn） for t≥0 （respectively a support point of So（Bn） for t C [O, t0] and some to〉 0）. This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to So（Bn） and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.
