Several constructions in the Eremenko–Lyubich class

Abstract

We use a theorem of Bishop in [2] to construct several functions in the Eremenko–Lyubich class B. First it is verified, that in Bishop's initial construction [2] of a wandering domain in B, all wandering Fatou components must be bounded. Next we modify this construction to produce a function in B with wandering domain and uncountable singular set. Finally we construct a function in B with unbounded wandering Fatou components. It is shown that these constructions answer two questions posed in [9].