Invariance And Inner Fractals In Polynomial And Transcendental Fractals

TL;DR

This paper explores invariance and embedded Multibrot fractals within polynomial and transcendental fractals, revealing shape-preserving transformations and the presence of smaller fractals inside larger ones, supported by theoretical methods and visual examples.

Contribution

It introduces a method to predict embedded fractals in polynomial and transcendental maps and demonstrates shape-preserving transformations and embedded Multibrot fractals within these complex structures.

Findings

Existence of shape-preserving transformations in polynomial fractals

Embedded Multibrot fractals within polynomial and transcendental maps

Method to predict embedded fractals

Abstract

A lot of formal and informal recreational study took place in the fields of Meromorphic Maps, since Mandelbrot popularized the map z <- z^2 + c. An immediate generalization of the Mandelbrot z <-z^n + c also known as the Multibrot family were also studied. In the current paper, general truncated polynomial maps of the form z <- \sum_{p>=2} a_px^p +c are studied. Two fundamental properties of these polynomial maps are hereby presented. One of them is the existence of shape preserving transformations on fractal images, and another one is the existence of embedded Multibrot fractals inside a polynomial fractal. Any transform expression with transcendental terms also shows embedded Multibrot fractals, due to Taylor series expansion possible on the transcendental functions. We present a method by which existence of embedded fractals can be predicted. A gallery of images is presented alongside to showcase the findings.