Baby Mandelbrot Sets and Spines in Some One-Dimensional Subspaces of the Parameter Space for Generalized McMullen Maps

Document Type

Article

Publication Date

8-1-2025

Publication Source

Qualitative Theory of Dynamical Systems

Volume Number

24

Issue Number

4

ISSN

15755460

E-ISSN

16623592

Article Number

157

Abstract

For the family of complex rational functions of the form Rn,a,c(z)=zn+azn+c, known as “Generalized McMullen maps”, for a≠0 and n≥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|≥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 ([9]) and applied for the subfamily Rn,a,0 by Devaney ([4]). 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.

Keywords

Complex Dynamical Systems, Douady, Hubbard, Iteration, Mandelbrot, Polynomial-Like

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