The Parameter Planes of $\lambda \mathrm{z}^{m} \exp (\mathrm{z})$ for $m \geq 2^{*}$
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...
| Autores: | , |
|---|---|
| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2007 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/164138 |
| Acesso em linha: | https://hdl.handle.net/2445/164138 |
| Access Level: | acceso abierto |
| Palavra-chave: | Funcions enteres Sistemes dinàmics complexos Entire functions Complex dynamical systems |
| Resumo: | 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 $F_{\lambda, m}$. |
|---|