Talk in the Oberseminar Funktionentheorie - Prof. Dr. Filippo Bracci

Abstract:

Given a semigroup of holomorphic self-maps of the unit disc without fixed points in the unit disc, by the Denjoy-Wolff theorem, every orbit of the semigroup tends to the same point on the boundary. It is an interesting and difficult question to understand when the convergence is non-tangential or tangential, or every orbit lands with a precise slope or oscillating. In this talk I will explain how to solve this problem in terms of the geometry of the associated Koenigs function. Indeed, there exists a (essentially unique) univalent map from the unit disc to C so that its image is starlike at infinity and conjugates the semigroup to the simple translation z—>z+it, such a function is called the Koenigs function of the semigroup. I will show that the orbits converge non-tangentially if and only if the image of the Koenigs function is “almost symmetric” with respect to one—and hence any—orbit. This result is based on the Gromov’s shadowing lemma and precise new estimates on the hyperbolic distance. In fact, one can prove that non-tangential convergence of the orbits is equivalent for the orbits to be quasi-geodesic in the sense of Gromov.