Dilatation and Order of Contact for Holomorphic Self‐Maps of Strongly Convex Domains

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.