The structure of fuzzy fractals generated by an orbital fuzzy iterated function system

Abstract

In this article, we present a structure result concerning fuzzy fractals generated by an orbital fuzzy iterated function system ((X,d),(fi)i∈I,(ρi)i∈I)\left(\left(X,d),{({f}_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I}). Our result involves the following two main ingredients: (a) the fuzzy fractal associated with the canonical iterated fuzzy function system ((IN,dΛ),(τi)i∈I,(ρi)i∈I)\left(\left({I}^{{\mathbb{N}}},{d}_{\Lambda }),{\left({\tau }_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I}), where dΛ{d}_{\Lambda } is Baire’s metric on the code space IN{I}^{{\mathbb{N}}} and τi:IN→IN{\tau }_{i}:{I}^{{\mathbb{N}}}\to {I}^{{\mathbb{N}}} is given by τi((ω1,ω2,…))≔(i,ω1,ω2,…){\tau }_{i}\left(\left({\omega }_{1},{\omega }_{2},\ldots )):= \left(i,{\omega }_{1},{\omega }_{2},\ldots ) for every (ω1,ω2,…)∈IN\left({\omega }_{1},{\omega }_{2},\ldots )\in {I}^{{\mathbb{N}}} and every i∈Ii\in I; (b) the canonical projections of certain iterated function systems associated with the fuzzy fractal under consideration.

Keywords

• orbital fuzzy iterated function system
• fuzzy hutchinson-barnsley operator
• fuzzy fractal
• canonical projection
• 28a80
• 37c70
• 54h20