Abstract:

We consider the family of entire transcendental maps given by $F$_λ,m(z)=λ z^m exp(z) where $m\; \ge \; 2$. All functions $F$_λ,m have a superattracting fixed point at $z=0$, and a critical point at $z=-m$. In the dynamical plane we study the topology of the basin of attraction of $z=0$. In the parameter plane we focus on the capture behaviour, i.e., $\lambda$ values such that the critical point belongs to the basin of attraction of $z=0$. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then non-locally connected. However, there are parameter values for which the boundary of the immediate basin of $z=0$ is a quasicircle.