Fuzzy Topological Dynamical Systems

In this paper by considering a fuzzy continuous action of a fuzzy topological group on a fuzzy topological space we have fuzziﬁifed the notion of topological dynamical system. Some properties of this fuzzy structure are investigated. Then we have constructed mixed fuzzy topological dynamical system from two given fuzzy topological dynamical systems.


Introduction
In general, the theory of dynamical systems deals with the action of groups of continuous transformations of topological spaces.A classical dynamical system(Wieslaw Szlenk, 1984) is a structure ( π, G, X,) where G is a topological group, X is a topological space and π is a continuous function from G × X → X satisfying π(0, x) = x and π(s, (t, x)) = π(s + t, x), where 0 is the identity of G.In this paper we fuzzify the above concept as a natural transition from the corresponding crisp structure.For this fuzzification we will consider a fuzzy topological group (Wieslaw Szlenk, 1984), a fuzzy topological space and a fuzzy continuous map from G × X → X satisfying the above stated conditions.In (N.R. Das, 1995, p77-784) Das and Baishya introduced the notion of fuzzy mixed topology and in (N.R. Das, 2000, p401-408) they constructed fuzzy mixed topological group.In this paper we will construct fuzzy mixed topological dynamical system.Throughout our discussion the fuzzy topology on any set will contain all the constant fuzzy subsets.In other words we will use Lowen(R.Lowen, 1976, p621-633) definition of fuzzy topology.

Preliminaries
In this section we recall some preliminary definitions and results to be used in the sequel.
Let X be a non-empty set.A fuzzy set in X is an element of the set I X of all functions from X into the unit interval I.A fuzzy point of a set X is a fuzzy subset which takes non-zero value at a single point and zero at every other point.The fuzzy point which takes value α 0 at x ∈ X, and zero elsewhere is denoted by x α .Let λ be a fuzzy subset of X. Suppose λ(x) = α for x ∈ X.Then λ can be expressed as union of all its fuzzy points, i.e, λ = ∨ x∈X x α .Here ∨ denote union.We will use the same notation ∨ to denote supremum of a set of numbers.Similarly ∧ will be used to denote intersection of fuzzy sets as well as infimum of a set of real numbers.
Let λ and μ be fuzzy subsets of X, then we write λ ⊆ μ whenever λ(x) ≤ μ(x).Let λ be a fuzzy subset of a group (G, +).Then we define a fuzzy subset For the definition of a fuzzy topology, we will use the one given by Lowen (R. Lowen, 1976, p621-633) since his definition is more appropriate in our case.So, throughout this paper, by a fuzzy topology on a set X we will mean a sub-collection τ of I X satisfying the following conditions: (i) τ contains every constant fuzzy subset in X ; A fuzzy topological space is a set X on which there is given a fuzzy topology τ.Given a crisp topological space (X, T ), the collection (T ), of all fuzzy sets in X which are lower semicontinuous, as functions from X to the unit interval I = [0, 1] equipped with the usual topology, is a fuzzy topology on X ((R.Lowen, 1976, p621-633)).We will refer to the fuzzy topology (T ) as the fuzzy topology generated by the usual topology T. If (X, T j ) j∈J is a family of crisp topological spaces and T the product topology on X = Π j∈J X j , then (T ) is the product of the fuzzy topologies (T j ), j ∈ J, (R. Lowen, 1997, p11-21).
Result 2.2 Let ( π, G, X) be a classical dynamical system.If we equip G and X with the induced fuzzy topologies and G × X, with the corresponding product fuzzy topology, then the mapping π : G × X → X is fuzzy continuous.
Proof.It follows from the previous result.
Result 2.5 (Liu Yingming, 1997) Let (X, δ), (Y, τ) and (Z, k) be fuzzy topological spaces and f : X → Y and g : Y → Z be any mappings.Then f, g are fuzzy continuous ⇒ gof is fuzzy continuous.

Fuzzy topological Dynamical systems
In this section we will introduce the concept of fuzzy topological dynamical systems.Definition 3.1 Let X be a fuzzy topological space, G be a fuzzy topological group.
) is called a fuzzy topological dynamical system Throughout this section X will stand for a fuzzy topological dynamical system.Definition 3.2 Let t ∈ G, then the t-transition of ( π, G, X) denoted by π t is the mapping : π t : X → X such that π t (x) = π(t, x).Result 3.3 (i) π 0 is the identity mapping of X.
(ii) π s π t = π s+t for s, t ∈ G (iii) π t is one-to-one mapping of X onto X and −(π t ) = π −t (iv) For t ∈ G, π t is a fuzzy homeomorphism of X onto X. Proof.Straightforward.
Definition 3.4 The transition group of ( π, G, X) is the set G t = {π t : t ∈ G}.The transition projection of ( π, G, X) is the mapping θ : G → G t defined as θ(t) = π t .Definition 3.5 ( π, G, X) is said to be effective if t ∈ G with t 0 ⇒ π t (x) x for some x.
Result 3.8 π x is a fuzzy continuous mapping of G into X.

Proof. Straightforward.
Notation: We will denote π(λ × μ) by λμ Result 3.9 (Tazid Ali and Sampa Das, 2009) (ii) Let G and X be product related, then for a fuzzy subset λ of G and a fuzzy subset μ of X, π(clλ  Corollary 3.13 Let t ∈ G and μ ∈ I X be a fuzzy nbd. of x, for some x ∈ X.Then π(t α × μ) is a fuzzy nbd. of π(t α × x) Result 3.14 Let μ be a fuzzy closed subset of X. then for any fuzzy point t α of G, π(t α × μ) is fuzzy closed.
Corollary 3.15 Let λ be any fuzzy subset of G and μ ∈ I X be fuzzy closed.If suppλ is finite, then π(λ × μ) is fuzzy closed.
Proof.Without loss of generality, we may assume that μ is open.Since the map π : G × X → X is continuous, the fuzzy set there exist open fuzzy sets μ 1 , μ 2 in G and X respectively with Remark I. Let (π, G, X) be a fuzzy topological dynamical system.Then for any μ ∈ I X , π(s, π(t, μ) = π(s + t, μ).
We have for s, t ∈ G, In the following results we will consider the topological group (K, +)(R or C) equipped with usual fuzzy topology.
Result 3.17.Let X be a fuzzy topological dynamical system, a ∈ X and μ a neighborhood of a.Then, for each 0 < θ < μ(a) there exists an open neighborhood of zero in K such that α(0) > θ and α(t) ≤ μπ(t, a) for all t ∈ K.
Proof.By the preceding result, there exists an open neighborhood α of zero in K such that α(0) > θ and α(t) ≤ μ(tx) for all t ∈ K. Since the set is open and contains 0, there exists ε > 0 such that t ∈ V whenever |t| ≤ ε.

Clearly
Result 3.19 Let X be a fuzzy topological dynamical system and μ a neighborhood of a.Then, for each real number θ with 0 < θ < μ(a) there exist an open neighborhood ρ of a ∈ X, with ρ ≤ μ and ρ(a) > θ, and a positive real number such that π(t, ρ) ≤ μ for each t ∈ K with |t| ≤ ε.

We have tμ
We propose the following definition Definition 3.22 If in definition 3.1, the fuzzy topological group is replaced by a fuzzy topological semi-group, then the system will be called a fuzzy topological semi-dynamical system.
(R, .)with usual fuzzy topology is a fuzzy topological semi group.
Result 3.23 Let (π, R, X) be a fuzzy topological semi-dynamical system and μ a neighborhood of a.Then, for each real number θ with 0 < θ < μ(a) there exist an open neighborhood ρ of a ∈ X, with ρ ≤ μ and ρ(a) > θ, and a positive real number ε such that π(t, ρ) ≤ μ for each t ∈ K with |t| ≤ ε.
¢ www.ccsenet.org/jmr Definition 2.7 (N. Palaniappan, 2005) Let (X, δ) and (Y, τ) be two fuzzy topological spaces.Then f : X → Y is fuzzy open (closed) if the image of every fuzzy open(closed) subset of X is fuzzy open(closed) in Y.
is a fuzzy open (closed), then tμ is fuzzy open(closed).Result 3.10 Let α be a constant fuzzy subset of G and μ ∈ I X be fuzzy open.Then π(α × μ) is fuzzy open.
open and μ is open so π t μ is open.Also by definition of fuzzy topology α is open.Consequently α ∧ {∨(π t μ)} is open.Hence π(α × μ) is open.Corollary 3.11 Let μ be a fuzzy open subset of X. then for any fuzzy point t α of G, π(t α × μ) is fuzzy open.Corollary 3.12 Let λ be any fuzzy subset of G and μ ∈ I X be fuzzy open, then π(λ × μ) is fuzzy open.
The elements of τ are the open fuzzy sets in X. Complement of an open fuzzy set is called a closed fuzzy set.Interior of a fuzzy set λ is the union of all the open fuzzy sets contained in λ and the closure of λ is the intersection of all fuzzy sets containing λ.The interior and closure of λ will be denoted by λ o and clλ respectively.A map f from a fuzzy topological space X to a fuzzy topological space Y, is called continuous if f −1 (μ) is open in X for each open fuzzy set μ in Y. Let X be a fuzzy topological space and x ∈ X.A fuzzy set μ in X is called a neighborhood of x if there exists an open fuzzy set ρ with ρ ⊆ μ and ρ(x) = μ(x) > 0.