Speaker: Vasiliki Evdoridou (The Open University and MSRI, Berkeley)
Abstract: In the iteration of holomorphic self-maps of the unit disc the dynamical behaviour of points in the disc is determined by the well-known Denjoy-Wolff theorem. Specifically for inner functions, there is a remarkable dichotomy in the behaviour of boundary points. Motivated by questions on the boundary behaviour of wandering domains, i.e. Fatou components that are not eventually periodic, we extend the notion of the Denjoy-Wolff point to sequences of holomorphic functions between simply connected domains and define the Denjoy-Wolff set; those points on the boundary whose images have the same limiting behaviour as the images of all interior points. Moreover, we study the aforementioned dichotomy in this more general setting. We will focus on the special case of simply connected wandering domains of transcendental entire functions and see how these results help us relate the internal behaviour with the behaviour of boundary points. This is joint work with A.M. Benini, N. Fagella, P. Rippon and G. Stallard.