The Parameter Planes of $\lambda \mathrm{z}^{m} \exp (\mathrm{z})$ for $m \geq 2^{*}$

Abstract

We consider the families of entire transcendental maps given by $F_{\lambda, m}(\mathrm{z})=\lambda \mathrm{z}^{m} \exp (\mathrm{z}),$ where $m \geq 2 .$ All functions $F_{\lambda, m}$ have a superattracting fixed point at $z=0,$ and a critical point at z $=-m .$ In the parameter planes we focus on the capture zones, i.e., $\lambda$ values for which the critical point belongs to the basin of attraction of $\mathrm{z}=0,$ denoted by $A(\mathrm{o}) .$ In particular, we study the main capture zone (parameter values for which the critical point lies in the immediate basin, $A^{*}(\mathrm{o})$ ) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter $\lambda$ in the main capture zone, $A(o)$ consists of a single connected component with non-locally connected boundary. For all remaining values of $\lambda, A^{*}$ (o) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of $\mathbb{C}^{*}$ which serve as a model for $F_{\lambda, m},$ in the sense that they are related by means of quasiconformal surgery to $F_{\lambda, m}$.