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Dilatation and Order of Contact for Holomorphic Self‐Maps of Strongly Convex Domains
Author(s) -
Bracci Filippo
Publication year - 2003
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013758
Subject(s) - holomorphic function , mathematics , bounded function , geodesic , boundary (topology) , domain (mathematical analysis) , regular polygon , fixed point , order (exchange) , identity theorem , mathematical analysis , combinatorics , measure (data warehouse) , pure mathematics , geometry , finance , database , computer science , economics
Let D be a bounded strongly convex domain and let f be a holomorphic self‐map of D . In this paper we introduce and study the dilatation α( f ) of f defined, if f has no fixed points in D , as the usual boundary dilatation coefficient of f at its Wolff point, or, if f has some fixed points in D , as the ratio of shrinking of the Kobayashi balls around a fixed point of f . In particular, we show that the map α, defined as α : f ↦ α ( f ) ∈ [0,1], is lower semicontinuous. Among other things, this allows us to study the limits of a family of holomorphic self‐maps of D . In the case of an inner fixed point, the dilatation is an intrinsic measure of the order of contact of f ( D ) to ∂ D . Finally, using complex geodesics, we define and study a directional dilatation, which is a measure of the shrinking of the domain along a given direction. Again, results of semicontinuity are given and applied to a family of holomorphic self‐maps. 2000 Mathematical Subject Classification: primary 32H99; secondary 30F99, 32H15.