The Parameter Planes of $ λz^{m} $ exp(z) for m ≥ 2*

Abstract We consider the families of entire transcendental maps given by Fλ,m (z) = $ λz^{m} $ exp(z), where m ≥ 2. All functions Fλ,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 crit...
Ausführliche Beschreibung

Abstract We consider the families of entire transcendental maps given by Fλ,m (z) = $ λz^{m} $ exp(z), where m ≥ 2. All functions Fλ,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 Fλ,m, in the sense that they are related by means of quasiconformal surgery to Fλ,m. Ausführliche Beschreibung