Baby Mandelbrot sets and Spines in some one-dimensional subspaces of the parameter space for generalized McMullen Maps

TL;DR

This paper investigates the structure of parameter spaces for generalized McMullen maps, revealing the presence of baby Mandelbrot sets in certain slices and using polynomial-like maps to analyze their locations.

Contribution

It identifies and locates baby Mandelbrot sets in specific one-dimensional slices of the parameter space for generalized McMullen maps, extending the understanding of their complex dynamics.

Findings

Existence of baby Mandelbrot sets in the parameter space for fixed $c$ with $|c| extgreater 6$

Identification of baby Mandelbrot sets in slices where $c=ta$

Use of polynomial-like maps to determine the location of these sets

Abstract

For the family of complex rational functions of the form $R_{n,c,a}(z) = z^n + \dfrac{a}{z^n}+c$, known as ``Generalized McMullen maps'', for $a\neq 0$ and $n \geq 3$ fixed, we study the boundedness locus in some one-dimensional slices of the $(a,c)$-parameter space, by fixing a parameter or imposing a relation. First, if we fix $c$ with $|c|\geq 6$ while allowing $a$ to vary, assuming a modest lower bound on $n$ in terms of $|c|$, we establish the location in the $a$-plane of $n$ ``baby" Mandelbrot sets, that is, homeomorphic copies of the original Mandelbrot set. We use polynomial-like maps, introduced by Douady and Hubbard and applied for the subfamily $R_{n,a,0}$ by Devaney. Second, for slices in which $c=ta$, we again observe what look like baby Mandelbrot sets within these slices, and begin the study of this subfamily by establishing a neighborhood containing the boundedness locus.