On the geometry of unbounded wandering domains

Abstract

We study the geometry of unbounded wandering domains of entire functions using Arakelian approximation. First, we show that, given a uniformly accessible closed set contained in a strip, the connected components of its interior can be realized as escaping or oscillating wandering domains of some entire function. The iterates of the function are univalent on these wandering domains, and any unbounded wandering domain remains unbounded under iteration. Next, we construct escaping or oscillating wandering domains by lifting bounded wandering domains with prescribed geometry via the exponential map. In particular, there exists an entire function that has a half-plane as a wandering domain. Finally, we show that, in some precise sense, any simply connected open set can be approximated by escaping or oscillating wandering domains. As a direct consequence, we obtain wandering domains whose complements have arbitrarily small areas.

arXiv comment33 pages, 12 figures. New abstract. A new section has been added dedicated to the construction of wandering domains by means of lifting via the exponential map. This answers one of the open questions from the first version of the paper. Open questions stated in the paper have been reformulated. Some minor typographical mistakes have also been addressed