Exponential Elliptics Give Dimension Two

Using the theory of infinite iterated function systems, we show that the Julia set of any function of the type G = λ exp ◦F , λ ∈ C \ {0} and with F : C → Ĉ 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 ◦F are nowhere dense in C.