A Study on Nonconventional Iterated Function Systems for the Hutchinson Barnsley Fractal

Abstract

Fractals were largely alien to mathematics until Mandelbrot brought them into focus. However, newlinetheir existence seemed natural when Hutchinson applied the Banach contraction principle newlineto demonstrate their foundation. While Mandelbrot and Hutchinson played crucial roles in introducing newlinefractals to the mathematical community, their widespread popularity is largely attributed to newlineBarnsley. He advanced the concept of numerical interpolation, specifically fractal interpolation (FI), newlineby leveraging the Iterated Function System (IFS) and#1048576; a framework he developed that generates a variety newlineof well-known fractals, such as the Cantor set, Sierpi´nski gasket, and Koch curve. The success newlineof IFSs established the term fractal as synonymous with the fixed points of collage mappings, as newlineformalized by Hutchinson. newlineThe generation of fractals through Iterated Function Systems (IFS) has become a significant newlinetopic for fixed point theorists in the 21st century. Over the past decade, numerous IFSs have been newlinedeveloped by incorporating collections of fixed point mappings, though primarily of the conventional newlinetype. Within the Hutchinson-Barnsley framework, nonconventional IFSs remain an active newlinearea of study. newlineThis thesis, for the first time in the literature, explores three types of nonconventional IFSs: newlineadditive-based, product-based, and maximum-based. As a result, it introduces a GP operator, constructs newlinethe Cantor set using a pair of discontinuous functions, and presents fractal curves of highly newlineirregular nature. Notably, these irregular fractal curves arise from a revised classical FI model newlineutilizing additive-based IFSs newline