On the geometry of unbounded wandering domains

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. Second, 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.