Combinatorial Models for Some Generalized McMullen Maps in the Case of Two Bounded Critical Orbits

Kelsey Brouwer
PhD Student
University of Wisconsin – Milwaukee

The family of generalized McMullen maps R(z)= z^n + b + a/z^n has two independent critical orbits. We consider the case in which one critical value lies in the immediate basin of an attracting cycle and the other critical value eventually lands in that immediate basin. Computer-generated images of the dynamical plane suggest the presence of both baby quadratic Julia sets and some sets which appear to be modifications of those. We present combinatorial models of the dynamics which help to explain this phenomena.