The Boundedness Locus and baby Mandelbrot sets for some generalized McMullen maps

TL;DR

This paper investigates the parameter spaces of certain rational functions, revealing embedded Mandelbrot set copies and employing polynomial-like map techniques to analyze their boundedness loci.

Contribution

It extends the understanding of Mandelbrot set embeddings to generalized McMullen maps with fixed degree, using polynomial-like map methods.

Findings

Identified homeomorphic copies of Mandelbrot sets in parameter planes.

Applied Douady-Hubbard techniques to rational functions.

Analyzed boundedness loci for specific parameter ranges.

Abstract

In this paper we study rational functions of the form $ R_{n,a,c}(z) = z^n + \dfrac{a}{z^n} + c, $ with $n$ fixed and at least $3$, and hold either $a$ or $c$ fixed while the other varies. We locate some homeomorphic copies of the Mandelbrot set in the $c$-parameter plane for certain ranges of $a$, as well as in the $a$-plane for some $c$-ranges. 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.