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

: In this article, we present a structure result concerning fuzzy fractals generated by an orbital fuzzy iterated function system X d f ρ , , , i i I i i I (( ) ( ) ( ) ) ∈ ∈ . Our result involves the following two main ingredients: ( a ) the fuzzy fractal associated with the canonical iterated fuzzy function system I d τ ρ , , , i i I i i I Λ (cid:2) (( ) ( ) ( ) ) ∈ ∈ , where d Λ is Baire ’ s metric on the code space I (cid:2) and τ I I : i (cid:2) (cid:2) → is given by τ ω ω i ω ω , , , , , i


(( ) ( ) ( ) )
∈ ∈ .Our result involves the following two main ingredients: (a) the fuzzy fractal associated with the canonical iterated fuzzy function system I d τ ρ , , , , where d Λ is Baire's metric on the code space I and τ I I : i → is given by τ ω ω i ω ω , , , , ,

.1 Generalities concerning iterated function systems, iterated fuzzy set systems and orbital fuzzy iterated function systems
Iterated function systems (which were initiated by Hutchinson in [1]) provide a standard framework for selfsimilarity.Some nice surveys on this theory are presented in [2][3][4][5].Because of their attribute to squeeze large amount of data into a few number of parameters, they have nice applications in image processing via the so-called inverse problem, which asks to find an iterated function system whose attractor approximates a target set with a prescribed precision.In the case of image with gray levels the aforementioned theory involves a class of measures generated by adding a probability system to the iterated function system.An alternative approach, based on considering images as functions rather that measures, was introduced by Cabrelli and Molter [6] and Cabrelli et al. [7].More precisely, they combined the idea of representation of a image as a fuzzy set with the Hutchinson's theory and introduced the concept of iterated fuzzy set system (IFZS).In this way, they succeeded in generating images with gray levels as attractors of IFZSs, i.e., as the fixed point of corresponding fuzzy operators.In addition, they proved that one can find an IFZS whose attractor approximates a target set with a imposed precision.For some other results along this line of research, see [8][9][10].
The articles [11][12][13] deal with the concept of orbital iterated function system, which is an iterated function system consisting of continuous functions satisfying Banach's orbital condition.It is a real generalization of the idea of the iterated function system since its associated fractal operator is weak Picard, but it is not necessarily Picard.The fuzzification idea applied to the framework of orbital iterated function systems leads to the study of the orbital fuzzy iterated function system in [14], whose corresponding fuzzy operator is weak Picard.Its fixed points are called fuzzy fractals.In [15], we presented a structure result concerning fuzzy fractals associated with an orbital fuzzy iterated function system.We proved that such an object is perfectly determined by the action of the initial term of the Picard iteration sequence on the closure of the orbits of certain elements.

The results of this article
Given a fuzzy iterated function system X d f ρ , , , let us denote by u its fuzzy fractal and by u Λ the fuzzy fractal of the canonical iterated fuzzy function system , where d Λ is Baire's metric on the code space I Λ( ) and τ and every i I ∈ .First, we prove (Theorem 3.4) that π u u , where π is the canonical projection associated with the iterated function system X d f , , i i I (( ) ( ) ) ∈ .In addition (Theorem 3.2), we prove that u Λ is perfectly determined by the admissible system of gray level maps ρ i i I ( ) ∈ .Finally (Theorem 3.5), on the basis of the main result from [15] and the aforementioned results, we are able to provide a structure result concerning the fuzzy fractals generated by an orbital fuzzy iterated function system Z .It involves u Λ and the canonical projections of certain iterated function systems associated with the fuzzy fractal under consideration.
and is bounded and closed For a family of functions f i i I ( ) ∈ , where f X : i → , we shall use the following notation: ( ) ∈ ∈ of real numbers, where I is finite and J is infinite, we have for every i I ∈ .
Proof.For j J ∈ , we shall consider i I j ∈ such that a a max .
On the one hand, we have a a max max max max .
for all i I ∈ and j J ∈ .Hence, for all i I ∈ and the justification of (1) is completed.On the other hand, we have a a max max max max . Indeed, for all i I ∈ and j J ∈ .Therefore, i.e., a a max max , for every j J ∈ .Hence, a a a a sup max max max max max and the justification of (2) is finished.By taking into account (1) and (2), the proof is completed.□

Fuzzy sets
Let us consider the sets X and Y .The elements of u X : 0,1 is called a gray level map.For a gray level map ρ and u X ∈ , one can consider ρ u ρ u .
, we shall use the following notations: , which is described in the following way: ) and u X ∈ , we shall use the following notations: ), the function d : 0 , , given by See [16] for more details.
Lemma 2.2.Let us consider a continuous function f X X : → and an increasing function In view of (3) and (4), the proof is completed.□

The code space
Let us consider a nonempty set I and n ∈ .
The set I will be denoted by I Λ( ).A standard element ω of I Λ( ) takes the form ω ω ω ω ω will be denoted by ) is compact provided that I is finite.For every i I ∈ , the function τ given by for every ω θ I , Λ( ) ∈ .

Iterated function systems, iterated fuzzy function systems, orbital iterated function systems, and orbital fuzzy iterated function systems
By an iterated function system, we mean a pair X d f , , , where: ) is a complete metric space; The fractal operator associated with is the function F P X P X : cp cp ( ) ( ) → , given by Let us recall that (see [1]): (α) F is a contraction with respect to h.As P X h , cp ( ( ) ) is a complete metric space, F is a Picard operator.Its fixed point is denoted by A and is called the attractor of ., has the following properties: -It is a continuous surjection; - It is called the canonical projection associated with .By an iterated fuzzy function system, we mean a triple , where: ∈ is an iterated function system -ρ i i I ( ) ∈ is an admissible system of gray level maps, i.e., ρ 0 0 i ( ) = , ρ i is nondecreasing and right continuous for every i I ∈ and there exists j I ∈ such that ρ 1 1 j ( ) = .
The fuzzy Hutchinson-Barnsley operator associated with Z is the function Z : X X → * * , given by ∞ is a complete metric space, Z is a Picard operator.Its unique fixed point is denoted by u and is called the fuzzy fractal generated by Z .
The canonical iterated fuzzy function system will play a central role in this article.The fuzzy Hutchinson-Barnsley operator associated with Λ will be denoted by Z Λ and u Λ will be denoted by u Λ .Hence, By an orbital iterated function system, we mean a pair X d f , , , where: ) is a complete metric space; -f i i I ( ) ∈ is a finite family of continuous functions f X X : i → having the property that there exists C 0, 1 In contrast with the case of iterated function systems, the fractal operator associated with an orbital iterated function system is a weak Picard operator.Its fixed points are called the attractors of the system.
By an orbital fuzzy iterated function system, we mean a triple X d f ρ , , , , where: ∈ is an orbital iterated function system.-ρ i i I ( ) ∈ is an admissible system of gray level maps.

Let us mention that if:
-Z designates the fuzzy Hutchinson-Barnsley operator associated with Z u x u wy X xy w u y for each there exist , such that , and 1 for every u ∈ * , is weak Picard and its fixed points are called fuzzy fractals generated by Z .
For u ∈ * and x u [ ] ∈ * (hence, there exist w y X , x x ∈ such that x y w , x x ( ) ∈ and u y 1 x ( ) = ), we shall use the following notations: where u x ∈ * is described by u y u y y w , if 0, otherwise .
x x ∈ be an orbital fuzzy iterated function system, f X X : → a continuous function and u j j J ( ) ∈ a family of elements from X * * having the property that there exists K P X cp ( ) ∈ such that u K supp j ⊆ for all j J ∈ .Then ∈ for every n ∈ and every u X ∈ * .
Proof.Let us consider a fixed, but arbitrarily chosen u X ∈ * .We are going to use the mathematical induction method to prove the thesis of our lemma.
For n 1 = , the thesis is valid in view of the definition of Z. Now supposing that it is valid for n ∈ , we have i.e., the statement is valid for n 1 + .□ Theorem 2.5.(see Theorem 3.9 from [15]).Let X d f ρ , , , be an orbital fuzzy iterated function system and u ∈ * .Then Orbital fuzzy iterated function system  7 3 The main results

A characterization of the fuzzy fractal u Λ
For a set I , let us consider the canonical iterated fuzzy function system We shall denote by 1 the element of Λ * given by ω 1 1, ( ) = for every ω I Λ( ) ∈ .
Proposition 3.1.In the aforementioned framework, we have and every n ∈ .
Proof.For ω I Λ( ) ∈ , n ∈ , and θ I Λ n ( ) ∈ , we have Consequently, we conclude that for every x X ∈ .As u Λ is upper semicontinuous, τ i is continuous and I Λ( ) is compact, we infer that and its element, denoted by a ω , belongs to A .The function π

Theorem 3 . 2 .Remark 3 . 3 .
In the aforementioned framework, we have The aforementioned result shows that u Λ is perfectly determined by the admissible system of gray level maps ρ i i I ( ) ∈ .

3. 2 ATheorem 3 . 4 .
structure result concerning fuzzy fractal of a fuzzy iterated function system In the aforementioned framework, for every fuzzy iterated function system X where π is the canonical projection associated with the iterated function system X d f , ,