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Wandering domains in complex dynamics

Complex dynamics concerns the iteration of analytic functions of the complex plane. For each function, the plane is split into two sets: the Fatou set (where the behaviour of the iterates is stable under local variation) and the Julia set (where the behaviour is chaotic). One of the most dramatic breakthroughs was given by Sullivan in the 1980s when he proved that, for rational functions, all components of the Fatou set are eventually periodic and there are no so-called wandering domains. For transcendental functions, however, wandering domains can exist and the rich variety of possible behaviours that can occur is only just becoming apparent. We discuss a recent classification of the dynamical behaviour of the iterates inside wandering domains and the relationship between the behaviour of orbits of points in the interior, and orbits of points on the boundary. These new results hold in a very general context including non-autonomous dynamical systems of holomorphic maps on simply connected domains.