Invariance And Inner Fractals In Polynomial And Transcendental 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.