Julia Sets of Joukowski-Exponential Maps

Let $h$ be a transcendental entire function of finite type such that all the coefficients in its Taylor series about the origin are non-negative, $h(x)! >!0 hbox { for } x !<~0,,h(0)ge 1$ and each finite singular value of $h$ is either real or is with unit modulus. For $J(z) = z + (1/z)hbox { and } n in mathbb {N}$ , we define $f_{lambda }(z)=lambda J^n(h(z))$ . It is proved that there exists a $lambda ^{*}$ such that the Julia set of $f_lambda $ is a nowhere dense subset of $widehat{mathbb {C}}hbox { for }0< lambda le lambda ^*$ whereas it becomes equal to $widehat{mathbb {C}}$ for $lambda >lambda ^*$ . A detailed study of the Julia sets of Joukowski-exponential maps $lambda J( e^z+1)$ is undertaken when it is not equal to the whole sphere. Such a Julia set consists of a non-singleton, unbounded and forward invariant component, infinitely many non-singleton bounded components and singleton components. A bounded component of the Julia set eventually not mapped into the unbounded Julia component is singleton if and only if it is not expanding. The Julia set contains two topologically as well as dynamically distinct completely invariant subsets.