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Exploring baby Julia sets in parameter space slices for Generalized McMullen Maps

Suzanne Boyd, Kelsey Brouwer, Matthew Hoeppner

TL;DR

The study analyzes Generalized McMullen maps $R_{n,a,b}(z)=z^n+a/z^n+b$, focusing on parameter-space slices where baby Julia sets appear. By fixing $v_+$ to obtain the one-parameter subfamily $r_{n,a}$, the authors characterize the dynamical plane through a fixed Fatou component $D_+$ and a variable $D_-$, establishing polynomial-like behavior when appropriate. They locate the bifurcation spine, bound the boundedness locus within annuli near $1/8$, and identify centers $a_{2j}$ for principal components with $n-2$ associated baby-Mandelbrot-like regions, while predicting additional baby Julia set phenomena in parameter space. The work develops a framework usingDouady–Hubbard criteria and holomorphic motions to prove that certain hyperbolic components map to unit disks in parameter space, providing a rigorous topological description of disk-like regions and outlining directions for proving broader conjectures about baby Julia sets.

Abstract

For the family of complex rational functions of the form R(z)= z^n + a/z^n+b, known as "Generalized McMullen maps", for non-zero a, and integer n fixed and at least 3, we describe the apparent phenomena of baby Julia sets in parameter space appearing both in slices with independent critical orbits and a slice defined by imposing a critical orbit relation. Specifically, we introduce the subfamily where one of two critical orbits is set to be a super-attracting fixed point, provide some general results on this subfamily and describe how Julia set copies in the parameter space slice occur--due to parameters for which the other critical orbit is in the (not immediate) basin of attraction of this fixed critical point. We provide several conjectures on this intriguing phenomena to catalyze further study.

Exploring baby Julia sets in parameter space slices for Generalized McMullen Maps

TL;DR

The study analyzes Generalized McMullen maps , focusing on parameter-space slices where baby Julia sets appear. By fixing to obtain the one-parameter subfamily , the authors characterize the dynamical plane through a fixed Fatou component and a variable , establishing polynomial-like behavior when appropriate. They locate the bifurcation spine, bound the boundedness locus within annuli near , and identify centers for principal components with associated baby-Mandelbrot-like regions, while predicting additional baby Julia set phenomena in parameter space. The work develops a framework usingDouady–Hubbard criteria and holomorphic motions to prove that certain hyperbolic components map to unit disks in parameter space, providing a rigorous topological description of disk-like regions and outlining directions for proving broader conjectures about baby Julia sets.

Abstract

For the family of complex rational functions of the form R(z)= z^n + a/z^n+b, known as "Generalized McMullen maps", for non-zero a, and integer n fixed and at least 3, we describe the apparent phenomena of baby Julia sets in parameter space appearing both in slices with independent critical orbits and a slice defined by imposing a critical orbit relation. Specifically, we introduce the subfamily where one of two critical orbits is set to be a super-attracting fixed point, provide some general results on this subfamily and describe how Julia set copies in the parameter space slice occur--due to parameters for which the other critical orbit is in the (not immediate) basin of attraction of this fixed critical point. We provide several conjectures on this intriguing phenomena to catalyze further study.

Paper Structure

This paper contains 12 sections, 24 theorems, 38 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction: the big picture
  2. Preliminaries
  3. Polynomial-Like Maps
  4. Preliminaries Regarding $R_{n,a,b}$
  5. Dynamical Plane Results for $r_{n,a}$
  6. Locating the bifurcation locus of $r_{n,a}$
  7. Centers of Principal Components in the $r_{n,a}$ Parameter Space
  8. Baby Mandelbrots in the $r_{n,a}$ Parameter Space
  9. Topological Disks in the $r_{n,a}$ Parameter Space
  10. Dynamical plane
  11. Hyperbolic components
  12. Future work on the location of the topological disk components

Key Result

Theorem 1.1

The bifurcation locus in the subfamily $r_{n,a}(z)= z^n + \frac{a}{z^n}+ (a^{1/2n}-2\sqrt{a}),$ contains $n-1$ hyperbolic components, $\mathcal{H}_{2j}$ for $j=1, \ldots, n-1$, for which there are homeomorphisms $\Phi_j \colon \mathcal{H}_{2j} \to \mathbb{D}$. Each $\mathcal{H}_{2j}$ has dynamical c where at the center, $r(v_+)=v_+$ is a super-attracting fixed point, and $r(v_-)=v_+$, and such tha

Figures (12)

  • Figure 1: Parameter planes: for $P_c$ (left), and baby Mandelbrot sets in a slice of a bifurcation locus (right).
  • Figure 2: Apparent (partial) baby Julia set in a parameter slice of $R_{n,a,b}$, here the $a$-plane for $b:=6a$, and the corresponding quadratic Julia set (basillicas in the basillica)
  • Figure 3: The parameter slice of $R_{6,a,b}$ with one critical value held at a fixed point.
  • Figure 4: Parameter planes for $r_{n,a}$ for $n=4,5,6,7$.
  • Figure 5: Establishing the location of a baby Mandelbrot set requires tracking behavior in the dynamical plane as the parameter moves through the parameter plane.
  • ...and 7 more figures

Theorems & Definitions (47)

  • Conjecture 1: Location of baby Julia sets in parameter slices of Generalized McMullen Maps
  • Conjecture 2: Criteria for baby Julia sets in parameter space
  • Theorem 1.1
  • Definition 2.1: DouadyHubbard, Chapter 1
  • Theorem 2.2: DouadyHubbard, Chapter 1, Theorem 1
  • Definition 2.3: DouadyHubbard, Chapter 2
  • Definition 2.4: DouadyHubbard, Chapter 2
  • Theorem 2.5: DouadyHubbard, Chapter 2, Theorems 1 and 2
  • Proposition 2.6: DouadyHubbard, Chapter 2, Proposition 10
  • Definition 2.7
  • ...and 37 more