Computer Visualization of Julia Sets for Maps beyond Complex Analyticity

Alexey Toporensky,Ivan Stepanyan

doi:10.1051/epjconf/202124801013

Abstract

Using the computer program creating Julia sets for two-dimensional maps we have made computer animation showing how Julia sets for the Peckham map alters when the parameter of the map is changing. The Peckham map is a one-parameter map which includes the complex map z=z^2+c, and is nonanalytical for other values of the parameter. Computer animation of Julia fractal sets allows seeing how patterns typical for complex maps gradually destroy.

Highlights

Iterations o f co mplex a nalytic maps ap pears to be a wonderful s ource o f fractal s tructures, i nteresting from both mathematical and aesthetic viewpoints
T here i s a well d eveloped m athematical t heory for s uch kind o f maps. Julia sets for such maps has roughly the same structure, though can get additional symmetries in t he p olynomial cas e
The letter follows directly from t he f undamental t heorem o f algebra, t he former follows from the fact that a complex multiplication is a combination of stretching and rotation, while a map in a saddle point stretches in one dimension and contracts in another one

Summary

Introduction

Iterations o f co mplex a nalytic maps ap pears to be a wonderful s ource o f fractal s tructures, i nteresting from both mathematical and aesthetic viewpoints. One way i s t o retain th e c omplex a nalyticity and consider p olynomial o r t ranscendent maps. T here i s a well d eveloped m athematical t heory for s uch kind o f maps ( see, for ex ample, Milnor’s l ectures [ 2] an d references therein). Julia sets for such maps has roughly the same structure, though can get additional symmetries in t he p olynomial cas e. P articular cas e o f Z hukovski map have been considered in [3], for more general forms of maps within complex analyticity see, for example, [4]. Among r ecent r esults we c an i ndicate t he p aper [ 5] where it was shown that the Julia set of a transcendental meromorphic map with at most finitely many pol es cannot be contained in any finite set of straight lines and the pa per [ 6] which de scribes t he c onditions of existence of Sierpiński carpet Julia sets and estimation of it’s Hausdorff dimension

Objectives

Results

Conclusion