Filippo Bracci

The classical Julia-Wolff-Carathéodory theorem states that a holomorphic self-map of the unit disc has positive angular derivative at any boundary regular fixed point, where, a  point on the boundary of the disc is a boundary regular fixed point for the map if it  is fixed for the map in the sense of non-tangential limits and  the speed of approaching of the map at the boundary along a sequence converging to such a point is comparable to the speed of approaching of the sequence itself to the boundary. As a consequence, the map is isogonal and every angle contained in the unit disc with vertex at a boundary regular fixed point is eventually contained in the local image of the map.

The Julia-Wolff-Carathéodory theorem, in its qualitative form, has been generalized to the unit ball by Rudin and to strongly pseudoconvex domains by Abate.

In this talk we will discuss the quantitative version of such results. Given a holomorphic map between two bounded strongly pseudoconvex domains A and B with smooth boundaries we say that a point p on the boundary of A is a super-regular contact point if f(p) (in the sense of non-tangential limits) belongs to the boundary of B, if p is regular as in the one-dimensional case and if  the Jacobian of the map is bounded away from zero when moving toward p along non-tangential, normal directions. In such a case, every cone contained in B (or more general, every region which is Kobayashi asymptotic to a cone) with vertex f(p) is eventually contained in the local image of f at p.

This is a joint work with J. E. Fornaess