Sala Conferenze (Puteano, Centro De Giorgi).
The notion of non-tangential convergence is one of the basic concept of complex geoemtry in one and several variables. It is known that univalent maps from the disc into the complex plane admit non-tangential limit almost everywhere on the boundary of the disc, and Carathéodory’s prime ends theory fully describe the boundary behavior. However, starting with a simply connected domain, it is not clear in general when a given sequence in that domain converges non-tangentially to the boundary of the disc after pulling back via a Riemann map of the domain. The aim of this talk is to give a complete answer to such a question in terms of hyperbolic geometry, using Gromov’s hyperbolicity theory. Since every holomorphic self-map of the unit disc has a (essentially unique) holomorphic linearizing model (where dynamical properties of the map can be read via the geometry of the model domain), the previous characterization allows to study the type of convergence to the Denjoy-Wolff point for maps without inner fixed points. In this talk we will restrict ourselves mainly to the case of semigroups of holomorphic self-maps of the unit disc and show how to construct “pathological” examples of semigroups whose orbits converge to the Denjoy-Wolff point oscillating but non-tangentially, or partially tangential and partially non-tangentially.