Wandering domains with nearly bounded orbits

A major open question in transcendental dynamics asks if it is possible for points in a wandering domain to have bounded orbits, and more strongly, for a wandering domain to iterate only in a bounded domain. In this paper we give a partial answer to this question, by constructing a bounded wandering domain that spends, in a precise sense, nearly all of its time iterating in a bounded domain. This is in strong contrast to all previously known examples of wandering domains.