DILATATION AND ORDER OF CONTACT FOR HOLOMORPHIC SELF-MAPS OF STRONGLY CONVEX DOMAINS

Abstract

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 $\alpha (f)$ 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 $\alpha$, defined as $\alpha : f \mapsto \alpha (f) \in [0,1]$, 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 $\partial D$.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.