Attractor and self-similar group of generalized fuzzy contraction mapping in fuzzy metric space

In this paper, we construct a deterministic fractal in fuzzy metric space using generalized fuzzy contraction mapping and its fixed-point theorem in hyperspace of non-empty compact sets. Moreover, we present the self-similar group of -contraction in fuzzy metric space and prove some familiar results of self-similar group for fuzzy metric space.


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The Euclidean geometry handles regular objects with integer dimension, while the fractal geometry directs irregular objects with noninteger dimension. According to self-similar property, fractal can be characterized into two types, they are, an object having approximate self-similarity is called random fractal and another one is an object having exact self-similarity is called deterministic fractal. Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snowflakes. However, not all self-similar objects are fractals for example, the real line is formally self-similar but fails to have other fractal characteristics. Fractals and Fuzzy spaces play a significant role in the Nonlinear Analysis. Hence, the above studies motivate our direction to investigate the fractal concepts in particular self-similar property in fuzzy setting. Hutchinson (1981) introduced the formal definition of iterated function systems (IFS) and Barnsley (1993) developed the theory of IFS called the Hutchinson-Barnsley theory (HB Theory) in order to define and construct the fractal as a compact invariant subset of a complete metric space generated by the IFS of Banach contractions. That is, Hutchinson introduced an operator on hyperspace of nonempty compact sets called as Hutchinson-Barnsley operator (HB operator) to define a fractal set as a unique fixed point using the Banach contraction theorem in the complete metric space, in order to generate fractal as a unique fixed point using Banach fixed-point theorem having the aforesaid exact self-similar property. Moreover, these fractal sets have Hausdorff dimension greater than its topological dimension, in such a way that self-similarity is the most fundamental property of the fractals. In order to analyze self-similar sets in depth, we must realize their group structure. In this study, we present the self-similar group in fuzzy setting. Self-similar group is defined through Banach contraction and topological group in the classical metric space, while the fuzzy self-similar group is defined by fuzzy H-contraction and fuzzy topological group in the fuzzy metric space.
Fuzzy set theory was introduced by Zadeh (1965). Kramosil and Michalek (1975) introduced the notion of fuzzy metric space. Many authors have introduced and discussed several notions of fuzzy metric space in different ways and also proved fixed-point theorems with interesting consequent results in the fuzzy metric spaces (Farnoosh, Aghajani, & Azhdari, 2009;George & Veeramani, 1997;Grabiec, 1988;Gregori & Sapena, 2002;Mihet, 2007;Rodriguez-Lopez & Romaguera, 2004;Uthayakumar & Gowrisankar, 2014;Wardowski, 2013). George and Veeramani (1994) imposed some stronger conditions on the fuzzy metric space in order to obtain a Hausdorff topology. Rodriguez-Lopez and Romaguera defined the Hausdorff metric on fuzzy hyperspace and constructed the Hausdorff fuzzy metric space. Besides that, the necessary results of the Hausdorff fuzzy metric on fuzzy hyperspaces are proved in Rodriguez-Lopez and Romaguera (2004). Uthayakumar and Easwaramoorthy (2011), Easwaramoorthy and Uthayakumar (2011), investigated the fuzzy IFS fractals in the fuzzy metric space. On the basis of self-similar group of Banach contraction in classical metric space given by Saltan and Demir (2013), in this paper, we introduce the definition and property of self-similar group and strong self-similar group of -contraction. If G is a self-similar group (strong self-similar group) of -contraction, then G is also described as the attractor of a -IFS and one of the -contractions of -IFS is a group homomorphism (isomorphism). The image of G under this  -contraction map is its proper subgroup H being homomorphic (isomorphic) to G. Fractal set can be defined as a self-similar and strong self-similar group in the sense of -IFS of compact topological space.
The paper is organized into two directions, first one is to construct the fractals in fuzzy metric space using generalized fuzzy contraction mapping. Second direction is that we investigate a fuzzy metric group on self-similar property of fractal set in order to define the topological group with generalized fuzzy contraction. In this paper, we will start with short introduction of deterministic fractals in fuzzy metric space in Section 2 and some of its properties which will be used frequently in the sequel. In Section 3, we present generalization of the fuzzy contraction mappings together with their fixed-point properties. Further, in Section 4, we define the self-similar groups in fuzzy metric space and investigate the properties of these groups. At the end of the paper, two substantial examples are given, which shows the existence of fuzzy self-similar groups.

Fuzzy iterated function system
In this section, we recall some pertinent concepts on fuzzy metric spaces in the sense of George and Veeramani. Hausdorff fuzzy metric for a given fuzzy metric space on the set of its non-empty compact subsets as well as Fuzzy IFS Fractals in the fuzzy metric space.
George and Veeramani modified the Kramosil and Michalek (1975) fuzzy metric space as follows.
Definition 2.2 (George & Veeramani, 1997, 1994 The 3-tuple (X, M, * ) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on X × X × (0, ∞) satisfying the following conditions: for all x, y ∈ X and t > 0. Here, k is called the fuzzy contractivity ratio of f . Definition 2.5  Let (X, M, * ) be a fuzzy metric space and f n :X ⟶ X, n = 1, 2, 3, … , N (N ∈ ℕ) be N -fuzzy contractive mappings with the corresponding contractivity ratios k n , n = 1, 2, 3, … , N. Then the system X;f n , n = 1, 2, 3, … , N is called a Fuzzy Iterated Function System (FIFS) of fuzzy contractions in the fuzzy metric space (X, M, * ). (X, M, * ) be a complete fuzzy metric space. Let X;f n , n = 1, 2, 3, … , N;N ∈ ℕ be a FIFS of fuzzy contractions and F be the FHB operator of the FIFS of fuzzy contractions. We say that the set A ∞ ∈ (X) is Fuzzy Attractor (Fuzzy Fractal) of the given FIFS of fuzzy contractions, if A ∞ is a unique fixed point of the FHB operator F of fuzzy contractions. Usually, such A ∞ ∈ (X) is also called as Fractal generated by the FIFS of fuzzy contractions.

Attractor of generalized fuzzy contraction
In this section, we generate a fractal in fuzzy metric space, which is a generalization of a fractal initiated in the article . Moreover, we develop the -IFS theory in order to define and construct the fractal as a compact invariant subset of M-complete fuzzy metric space generated by the fixed-point theorem.
 denotes a collection of mappings : Then the condition (1) reduces to for all x, y ∈ X and t > 0.
Then f has a unique fixed point A * and for each A 0 , the sequence < f n (A 0 ) > n∈ℕ converges to A * .
for all n ≥ 1 and t > 0. For m, n ∈ ℕ, m < n, t < 0 and let < a i > i∈ℕ be a strictly decreasing sequence of positive number such that ∑ ∞ i=1 a i = 1.
Easy to verify that a sequence < (H M (A 0 , A 1 , a i t)) < i∈ℕ is non-decreasing and, by (c), bounded, hence we have a convergence of the series ) < for all m, n ≥ N, m < n. Thus, by Proposition 3.1, < A n < i∈ℕ is an M-Cauchy sequence. By the M-completeness of X, there exists A * ∈ X such that lim n⟶∞ A n = A * . Due to Proposition 3.2, lim n⟶∞ (H M (A n , A * , t)) = 0 for each t > 0. Hence for all t > 0, we obtain (H M (f (A * ), A n+1 , t)) ≤ k (H M (A * , A n , t)) ⟶ 0 as n ⟶ ∞. Finally, from the Proposition 3.2, we have A * = lim n⟶∞ A n+1 = f (A * ).

Suppose that there exists
Definition 3.2 Let (X, M, * ) be a fuzzy metric space and f n :X ⟶ X, n = 1, 2, 3, … , N(N ∈ ℕ) be N - contractive mappings. Then the system X;f n , n = 1, 2, 3, … , N is called a -Iterated Function System (-IFS) of -contractions in (X, M, * ). The Hutchinson-Barnsley operator (HB operator) of the -IFS is a function F: (X) ⟶ (X) defined by Definition 3.3 Let (X, M, * ) be a complete fuzzy metric space. Let X;f n , n = 1, 2, 3, … , N;N ∈ ℕ be a -IFS and F be the HB operator of the -IFS. If F has a unique fixed point A * in (X, M, * ), then the set A * ∈ (X) is called the Attractor (or Fractal) generated by the -IFS of -contractions. Proof Suppose that there exist a connected subset B of A contains more than two points. A is an attractor of given -IFS, Clearly, it gives the contradiction to f 0 , f 2 , … , f n are one-to-one on A. Therefore, only connected subset of A is single point set, there are no other connected subsets in A. Hence, A is totally disconnected. Romaguera and Sanchis (2001) extended the classical theorems on metric groups to the fuzzy setting. According to the definition of self-similar group of Banach contraction in classical metric space given by Saltan and Demir (2013), in this section, we introduce the definition and property of fuzzy self-similar group and strong fuzzy self-similar group of fuzzy contraction in fuzzy metric space. Then, we investigate some properties of strong fuzzy self-similar and fuzzy profinite groups.

Fuzzy self-similar group
Topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Now we recall the definition of self-similar group in compact topological space and profinite group.    (Dixon, Du Sautoy, Mann, & Segal, 1999;Saltan & Demir, 2013) A topological group G is profinite, if it is topologically isomorphic to an inverse limit of finite discrete topological groups.
Equivalently, a profinite group is a compact, Hausdorff, and totally disconnected topological group.
there exists a proper subgroup H of finite index and a surjective homomorphism :G ⟶ H, which is a -contraction with respect to ∈  on fuzzy metric (M, * ).
Definition 4.5 Let (G, ., M, * ) be a compact fuzzy topological group with a translation-invariant fuzzy metric (M, * ). Then G is called a strong self-similar group of -contraction, if there exists a proper subgroup H of finite index and a group isomorphism :G ⟶ H, which is a -contraction with respect to ∈  on fuzzy metric (M, * ).
Definition 4.6 A fuzzy topological group G is fuzzy profinite, if it is topologically isomorphic to an inverse limit of finite discrete topological fuzzy groups. Proof Assume that G is profinite group, then G is Hausdorff, compact, and totally disconnected.
Since every topological group is Hausdorff, and finite discrete groups are compact and totally disconnected.
Conversely, Let G be Hausdorff, compact, and totally disconnected. Since all components, i.e. all points of G, are closed and e = comp {e} is the intersection of all open-closed neighborhoods of e. It is easy to show that every open-closed neighborhood of e contains an open normal subgroup, which implies G has a topological isomorphic to an inverse limit of finite discrete topological group.

Proposition 4.1 A strong self-similar group of -contraction is the attractor of -IFS.
Proof Let (X, ., M, * ) be a strong self-similar group of -contraction. Hence, there is a proper subgroup H of X with [X:H] = n such that the mapping o :X ⟶ H is a group isomorphism and is a -contraction with respect to ∈ . Let x o = e be the identity element of X. For all i, j ∈ 0, 1, 2 … , n − 1 and i ≠ j, there are cosets of H in X such that (H.x i ) ∩ (H.x j ) = and X = H ∪ (H.x 1 ) ∪ (H.x 2 ) ∪ ⋯ ∪ (H.x n − 1). Define i :X ⟶ X by i (g) = o (g).x i , i = 1, 2, 3, … , n − 1. Clearly, because of o is surjective. Since o is a -contraction mapping with respect to and (M, * ) is a translation invariant fuzzy metric, we obtain that for all g, h ∈ X. Therefore, i is a -contraction mapping with respect to for i = 1, 2, 3, … , n − 1 and Thus, X is the attractor of the -IFS X; o , 1 , … , n−1 .
It gives that is group homomorphism. 1 , 2 , … , n are bijective implies is bijective. Take -contraction ratio b i of -contraction mappings i for i = 1, 2, … , n and choose b = max b 1 , b 2 , … , b n . Then, Hence, is -contraction mapping with respect to ∈ . It gives G 1 × G 2 × ⋯ × G n is a strong selfsimilar group of -contraction.
The above Theorem 4.3 shows that, finite product of strong self-similar groups of -contraction is also a strong self-similar group of -contraction. Proposition 4.2 A self-similar group of -contraction is a disconnected set. Proof Let G be a self-similar group of -contraction. Then G is a topological fuzzy group. Proposition 4.1 shows that G is the attractor of the -IFS 0 , … , n−1 . Hence, for all i, j ∈ 0, 1, 2, … , n − 1 and i ≠ j. For every i = 1, 2, … , n − 1, the mappings i : G ⟶ i (G) are -contraction with respect to ∈ . -contraction mapping is t-uniform continuous. Further, t -uniformly continuous function maps compact set into compact set. Hence, i (G) is compact subspace in (G, M, * ). i (G) is a closed set in (G, M, * ) for all i ∈ 0, 1, 2, … , n − 1 , since (G, M, * ) is Hausdroff space. Due to the fact that hence G can be written as disjoint union of non-empty closed sets 0 (G), [ 1 (G) ∪ ⋯ ∪ n−1 (G)]. That is, G is a disconnected set. Proposition 4.3 A strong self-similar group of -contraction is a totally disconnected set.