Capture zones of the family of functions lambda z^m exp(z)

Abstract

We consider the family of entire transcendental maps given by $F_{\lambda,m}= \lambda z^m exp(z)$ where m>=2. All functions $F_{\lambda,m}$ 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.