On the geometry at the boundary of holomorphic self-maps of the Unit Ball of C n

Abstract

We introduce two new tools to study a holomorphic self-map f of B n (the unit ball of C n, n>1): the inner space A (f) and the generalized inner space AG(f) After having defined the differential at the boundary for f, k-dfτ, in its Wolff pont τ ∈ ∂B n, we prove that the boundary dilatation coefficient α(f) is an eigenvalue for k-dfτ and we define AG(f) to be the generalized eigenspace associated to α(f); the inner space A(f) will be the span of the eigenvectors not belonging to the complex tangent space of ∂ B n at the Wolff point τ and contained in AG(f). Among other things it turns out that A(f) is the space of all the direction of complex geodesics that are mapped into themselves by f, and that the generalized inner space AG(f) is a direct addend of a boundary Cartan-type decomposition for C n. Using A(f) and AG(f) we obtain several new results on the geometry of holomorphic self-maps of B n, including some necessary conditions for commutation under composition.