Baby Julia Sets in Dynamical and Parameter Space for a Rational Family of Maps When Both Critical Orbits are Bounded

Abstract

We study the family of complex rational functions known as Generalized McMullen maps, F_{n,a,b}(z) = z^n + a/z^n + b, for a ≠ 0 and n ≥ 3 fixed. First, we reveal and provide a combinatorial model for some new dynamical behavior. In particular, we describe a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets and infinitely many subsets homeomorphic to a set which is obtained by starting with a quadratic Julia set, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we will describe. Next, we study the boundedness locus in a one-dimensional slice of the (a, b)-parameter space by imposing a critical orbit relation. Specifically, we consider the subfamily where one of two critical orbits is set to be a super-attracting fixed point. We show that parameters for which the other critical orbit is in its (not immediate) basin of attraction lead to Julia set copies in the parameter space slice. We close by providing a rather exhaustive catalog of additional types of Julia sets of Generalized McMullen maps, generalizing the results from the beginning of the thesis.