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 holomorphicself-map of D. In this paper we introduce and study the dilatation(f) of f defined, if f has no fixed points in D, as the usualboundary dilatation coefficient of f at its Wolff point, or,if f has some fixed points in D, as the ratio of shrinking ofthe Kobayashi balls around a fixed point of f. In particular,we show that the map , defined as : f (f) [0,1], is lowersemicontinuous. Among other things, this allows us to studythe limits of a family of holomorphic self-maps of D. In thecase of an inner fixed point, the dilatation is an intrinsicmeasure of the order of contact of f(D) to D. Finally, using complex geodesics, we define and study a directionaldilatation, which is a measure of the shrinking of the domainalong a given direction. Again, results of semicontinuity aregiven and applied to a family of holomorphic self-maps. 2000Mathematical Subject Classification: primary 32H99; secondary30F99, 32H15.