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

TL;DR

This paper investigates the appearance of baby Julia sets in parameter space slices of Generalized McMullen maps, revealing how critical orbit relations influence these fractal structures and proposing conjectures for future research.

Contribution

It introduces a specific subfamily of Generalized McMullen maps with critical orbit relations and describes the mechanisms behind baby Julia set formation in parameter space.

Findings

Baby Julia sets appear in parameter slices with critical orbit relations.

Parameters with a critical orbit in the basin of a super-attracting fixed point cause Julia set copies.

Several conjectures are proposed to explain the observed phenomena.

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.