The Parameter Planes of λz m exp(z) for m ≥ 2*

Abstract

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