Existence of the Mandelbrot set in parameter planes for some generalized McMullen maps

Suzanne Boyd, Alexander Mitchell

Published 2022-06-21Version 1

In this paper we study rational functions of the form $ R_{n,a,c}(z) = z^n + \dfrac{a}{z^n} + c, $ and hold either $a$ or $c$ fixed while the other varies. We show that for certain ranges of $a$, the $c$-parameter plane contains homeomorphic copies of the Mandelbrot set, as well as the $a$-plane for certain ranges of $c$. We use techniques first introduced by Douady and Hubbard, that were applied for the subfamily $R_{n,a,0}$ by Robert Devaney. These techniques involve polynomial-like maps of degree two.