Fuzzy Attractors Appearing from GIFZS

TL;DR

This paper extends fuzzy iterated function systems to mappings on finite Cartesian products, introducing generalized fuzzy fractals and proving their uniqueness and basic properties.

Contribution

It introduces the concept of generalized iterated fuzzy function systems (GIFZS) and demonstrates their ability to generate unique fuzzy fractals, expanding the theory to new settings.

Findings

GIFZS generate unique fuzzy fractal sets.

Basic properties of GIFZS and their fractals are established.

Potential for new fuzzy fractal sets is explored.

Abstract

Cabrelli, Forte, Molter and Vrscay in 1992 considered a {fuzzy} version of the theory of iterated function systems (IFSs in short) and their fractals%The idea was to extend the classical Hutchinson-Barnsley operator to selfmaps of a metric space to appropriate selfmaps of space of fuzzy , which now is quite rich and important part of the fractals theory. On the other hand, Miculescu and Mihail in 2008 introduced another generalization of the IFSs' theory - instead of selfmaps of a metric space $X$, they considered mappings defined on the finite Cartesian product $X^m$. %It turns out that many parts of the classical Hutchinson-Barnsley fractals theory have natural counterparts in this generalized setting. In particular, if $X$ is complete, then appropriately contractive systems of such maps generate unique fractal sets. In this paper we show that the \emph{fuzzyfication} ideas of Cabrelli et al. can be naturally adjusted to the case of mappings defined on finite Cartesian product. In particular, we define the notion of a generalized iterated fuzzy function system (GIFZS in short) and prove that it generates a unique fuzzy fractal set. We also study some basic properties of GIFZSs and their fractals, and consider the question whether our setting gives us some new fuzzy fractal sets.