THE PARAMETER PLANES OF $\lambda z^m \exp(z)$ for $m\geq 2$ Nuria Fagella*, Antonio Garijo** *Departament de Matematica aplicada i Analisi Universitat de Barcelona Gran Via 585 08005 Barcelona Spain e-mail: fagella@maia.ub.es *Departament d'Enginyeria informatica i Matematiques Universitat Rovira i Virgili Av. Paisos Catalans 26 13007 Tarragona, Spain agarijo@etse.urv.es ABSTRACT We consider the families of entire transcendental maps given by $F_{\lambda,m} (z ) \, = \, \lambda z^m \exp(z) $ where $ m \ge 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 $z=0$, denoted by $A(0)$. In particular, we study the main capture zone (parameter values for which the critical point lies in the inmediate basin, $A^*(0)$) 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, $A(0)$ consists of a single connected component with non-locally connected boundary. For all remaining values of $\lambda$, $A^*(0)$ is a quasidisk. On a different approach, we introduce some families of holomorphic maps of $\bc^*$ which serve as a model for $F_{\lambda,m}$, in the sense that they are related by means of quasiconformal surgery to $F_{\lambda,m}$. }