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

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

2002-07-12

arxiv.org/abs/math/0207103v1

Mathematics

Dynamical Systems

25 pages, 14 figures. Accepted for publication in the International Journal of bifurcation and Chaos

Scientific paper

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.

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