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 ) , ( f i ) i ∈ I , ( ρ i ) 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 ( ( I N , d Λ ) , ( τ i ) i ∈ I , ( ρ i ) i ∈ I ) \left(\left({I}^{{\mathbb{N}}},{d}_{\Lambda }),{\left({\tau }_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I}) , where d Λ {d}_{\Lambda } is Baire’s metric on the code space I N {I}^{{\mathbb{N}}} and τ i : I N → I N {\tau }_{i}:{I}^{{\mathbb{N}}}\to {I}^{{\mathbb{N}}} is given by τ i ( ( ω 1 , ω 2 , … ) ) ≔ ( i , ω 1 , ω 2 , … ) {\tau }_{i}\left(\left({\omega }_{1},{\omega }_{2},\ldots )):= \left(i,{\omega }_{1},{\omega }_{2},\ldots ) for every ( ω 1 , ω 2 , … ) ∈ I N \left({\omega }_{1},{\omega }_{2},\ldots )\in {I}^{{\mathbb{N}}} and every i ∈ I i\in I ; (b) the canonical projections of certain iterated function systems associated with the fuzzy fractal under consideration.