Exponential elliptics give dimension two

Abstract

Using the theory of infinite iterated function systems, we show that the Julia set of any function of the type $G=\lambda\exp \circ F$, $\lambda \in \mathbb{C} \setminus \{0\}$, with $F:\mathbb{C} \to \hat{\mathbb{C}}$ a non-constant elliptic function, has Hausdorff dimension two. However, there exist elliptic functions $F$ such that the Julia sets of the maps $G=\exp \circ F$ are nowhere dense in $\mathbb{C}$.