Visualization of Mandelbrot and Julia Sets of Möbius Transformations

Leah K Mork,Darin J Ulness

doi:10.3390/fractalfract5030073

Leah K Mork, Darin J Ulness

Open Access

https://doi.org/10.3390/fractalfract5030073

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Abstract

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

Highlights

The iteration procedure is briefly discussed. This is followed by a brief statement about the effect of affine transformations on the Mandelbrot set and its corresponding filled-in Julia sets
This work will focus on the fractal character of the filled-in Julia sets and their corresponding Mandelbrot sets [1,2,3]
Fractals from Möbius transformed functions are created in a similar fashion except that a fixed Möbius transformation is applied at each step in the iteration

Summary

Introduction

The current work will address the nature of these sets with a straightforward generalization. Abbas, Iqbal, and De la Sen have recently discussed the generation of Julia and Mandelbrot sets via fixed points [14]. 2021, 5, 73 using generalized Fibonacci numbers [18] In their work, they use the fact that this can be thought of in terms of consecutive values of an orbit in a quadratic dynamics related to the Mandelbrot set. The current authors have discussed the nature of filled-in Julia sets and Mandelbrot sets for centered polygonal lacunary functions [21,22]. This is followed by a brief statement about the effect of affine transformations (phase rotation, scaling, and translation) on the Mandelbrot set and its corresponding filled-in Julia sets.

Iteration Procedure

Affine Transformations

Inversion

Conclusions