Iteration of certain exponential-like meromorphic functions

Abstract

The dynamics of functions $f_\lambda (z)= \lambda \frac{\mathrm{e}^{z}}{z+1}\ \text{ for }\ z\in \mathbb {C}, \lambda >0$ is studied showing that there exists $\lambda ^* > 0$ such that the Julia set of $f_\lambda$ is disconnected for $0< \lambda < \lambda ^*$ whereas it is the whole Riemann sphere for $\lambda > \lambda ^*$. Further, for $0< \lambda < \lambda ^*$, the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of $\infty$ is shown to be disconnected for $0<\lambda < \lambda ^*$ whereas it is connected for $\lambda > \lambda ^*$. For complex $\lambda$, it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions $E_n(z) =\mathrm{e}^{z}(1+z+\frac{z^2}{2!}+\cdots +\frac{z^n}{n!})^{-1}$, which we call exponential-like, are studied as a generalization of $f(z)=\frac{\mathrm{e}^{z}}{z+1}$ which is nothing but $E_1(z)$. This name is justified by showing that $E_n$ has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only one singularity over 0 as well as over $\infty$ and both are direct. Non-existence of Herman rings are proved for $\lambda E_n$.