Mandelbrot Set and Julia Set in Fuzzy Iterated Function Systems

Abstract

Classical Iterated Function Systems (IFSs) are based on the repeated application of functions an integer number of times, thereby restricting the underlying dynamics to a discrete setting. Motivated by developments in fractional dynamics and fuzzy set theory, this paper proposes a generalized framework that allows iterations to be indexed by non-negative real numbers. The aim of the study is to extend the classical notion of iteration by introducing a fractional iteration scheme realized through fuzzy set representations, in which the degree of iteration is governed by a fuzzy cardinality rather than an integer count. The proposed methodology combines tools from fuzzy set theory with the theory of IFSs to define and analyze this continuous-scale iteration process. As an application, the framework is employed to study the quadratic complex map $f_c(z)=z^2+c$ , leading to generalized definitions of Mandelbrot and Julia sets under fuzzy iteration. The results show that this approach generates new families of fractal sets that retain essential features of classical dynamics while allowing greater flexibility in controlling their geometric and dynamical behavior. These findings provide a meaningful extension of discrete iteration theory and open new directions for fractal analysis within generalized IFS frameworks.