# Semigroup Actions on Intuitionistic Fuzzy Metric Spaces

Authors: Yaoyao Lan; Qingguo Li
Journal: Advances in Fuzzy Systems
 (2009)

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## Abstract

This paper investigates the dynamical systems in the context of topological semigroup actions on intuitionistic fuzzy metric spaces. We give some concepts such
as topological transitivity, point transitivity, and densely point transitivity for such
dynamical systems. Particularly, we consider the implications of nonsensitivity and
its relation to dynamical properties such as transitivity and equicontinuity.


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## Body

## 1. Introduction

As a generalization of fuzzy sets [1], the concepts of intuitionistic fuzzy sets was introduced by Atanassov [2], and later there has been much progress in the study of intuitionistic fuzzy sets by many authors [3–6]. Following George and Veeramani [7], Park [8] has defined intuitionistic fuzzy metric spaces and obtained several classical theorems on this new structure. Furthermore, Alaca et al. [9] studied new properties of intuitionistic fuzzy metric spaces. This concept has very important applications in quantum particle physics particularly in connection with both string and ϵ∞ theory which were given and studied by El Naschie [10, 11] and by Sigalotti and Mejias [12].On the other hand, the traditional theory of dynamical systems is founded on topology and gives topological descriptions of orbits, recurrence and stability. However, even the probabilistic approach cannot represent uncertainties attached to systems where some deterministic dynamical characteristics are unknown or deliberately ignored as well as uncertainties attached to their mathematical model. This observation led Chang and Zadeh to the concepts of fuzzy systems [13]. Many efforts have been made in this direction [14–16].Recently, an enormous amout of work has been conducted on chaotic dynamical systems. Most of this work has been concerned with the iteration of single maps, in other words, with group (or semigroup) actions of the additive group [17–20]. However, the abovementioned work has been studied mainly in the context of classical metric spaces.In this paper, we consider the dynamical systems in the context of topological semigroup actions on intuitionistic fuzzy metric spaces. We discuss topological transitivity, point transitivity, densely point transitivity and equicontinuity for such dynamical systems. We also investigate the relations between nonsensitivity and some dynamical properties such as transitivity and equicontinuity.


## 2. Preliminaries

Definition 1.
A binary operation*:[0,1]×[0,1]→[0,1] is continuous t-norm if * satisfies the following conditions:(a)
* is commutative and associative,(b)
* is continuous,(c)
a*1=a for all a∈[0,1],(d)
a*b≤c*d whenever a≤c and b≤d, for each a,b,c,d∈[0,1].Definition 2.
A binary  operation◊:[0,1]×[0,1]→[0,1] is continuous t-conorm if ◊ satisfies the following conditions:(a)
◊ is commutative and associative,(b)
◊ is continuous,(c)
a◊0=a for all a∈[0,1],(d)
a◊b≤c◊d whenever a≤c and b≤d, for each a,b,c,d∈[0,1].Definition 3.
A 5-tuple(X,M,N,*,◊) is called intuitionistic fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, ◊ is a continuous t-conorm and M, N are fuzzy sets on X2×(0,∞) satisfying the following conditions: for all x,y,z∈X,s,t>0,–
(IFm-1):M(x,y,t)+N(x,y,t)≤1,–
(IFm-2):M(x,y,t)>0,–
(IFm-3):M(x,y,t)=1 if and only if x=y,–
(IFm-4):M(x,y,t)=M(y,x,t),–
(IFm-5):M(x,y,t)*M(y,z,s)≤M(x,z,t+s),–
(IFm-6):M(x,y,t):(0,∞)→(0,1] is continuous,–
(IFm-7):N(x,y,t)<1,–
(IFm-8):N(x,y,t)=0 if and only if x=y,–
(IFm-9):N(x,y,t)=N(y,x,t),–
(IFm-10):N(x,y,t)◊N(y,z,s)≥N(x,z,t+s),–
(IFm-11):N(x,y,t):(0,∞)→(0,1] is continuous.Then(M,N) is called an intuitionistic fuzzy metric on X. The function M(x,y,t) and N(x,y,t) denote the degree of nearness and the degree of nonnearness between x and y respect to t, respectively.Remark 1.
In intuitionistic fuzzy metric spacesX, M(x,y,·) is nondecreasing and N(x,y,·) is nonincreasing for all x,y∈X.


## 3. Main Results

For brevity, we write the intuitionistic fuzzy metric spaces asX, wherever there is no risk of confusion.Definition 4.
A dynamical system inX is a triple (S,X,π) (we note (S,X) for short), where S is a topological semigroup, X at least is Hausdorff and
(1)π:S×X→X(s,x)↦sx
is a continuous action on X. Thus, s1(s2x)=(s1s2)x holds for each (s1,s2,x) in S×S×X. The orbit of x is the set Sx={sx:s∈S}. A subset Y⊂X is said to be S-invariant if f(Y)⊂Y. If Y is a closed S-invariant subset of X, then we call (S,Y) the restricted action, is a subsystem of (S,X). For U⊂X and s∈S denote s-1U={x∈X:sx∈U}.Note that ifS={fn:n=1,2,…} and f:X→X is continuous, then (S,X) is a classical dynamical system in X. Notation: (X,f).Definition 5.
LetS be a topological semigroup, we say that S is an(1)
F-semigroup if for every s0∈S the subset S∖Ss0 is finite,(2)
C-semigroup if for every s0∈S, the closure of the subset S∖Ss0 is compact in S (i.e., S∖Ss0 is relatively compact).Obviously, every topological group is of course anF-semigroup and also a C-semigroup.Definition 6.
Let(S,X) be a dynamical system.(1)
A subsetA of S acts equicontinuously at x0∈X if for every ε∈(0,1) and t>0, there exists δ∈(0,1) such that M(x0,x,t)>δ and N(x0,x,t)<1-δ imply M(ax0,ax,t)>ε and N(x0,x,t)<1-ε for every a∈A.(2)
A pointx0∈X is a point of equicontinuity (notation: x0∈Eq(X)) if S acts equicontinuously at x0. (S,X) is equicontinuous if Eq(X)=X.(3)
(S,X) is almost equicontinuous if Eq(X) is dense in X.Observe that every equicontinuous system is almost equicontinuous. Theorem3 show that the relation between almost equicontinuity and nonsensitivity.Lemma 1.
Let(S,X) be a dynamical system. If A is a related compact subset of S, then A acts equicontinuously on X.Proof.
Letx∈X and s1∈S. Suppose that s2∈S is in the neighborhood of s1 defined by M(s1x,s2x,t)>r and N(s1x,s2x,t)<1-r for some r∈(0,1) and t>0. Given ε>0. Note that π(s,x)=sx, by the continuity of π, for each s∈S there is an open neighborhood Us of s and a δs>0 such that if s′∈Us and y∈B(x,δs,t), then
(2)M(s′y,sx,t)>ε,N(s′y,sx,t)<1-ε,M(sx,s′x,t)>ε,N(sx,s′x,t)<1-ε
Therefore,
(3)M(s′x,s′y,2t)≥M(s′y,sx,t)*M(sx,s′x,t)>ε*ε>ε1forsome0<ε1<1,N(s′x,s′y,2t)≤N(s′y,sx,t)⋄N(sx,s′x,t)<(1-ε)⋄(1-ε)<1-ε2forsome0<ε2<1.
Taking ε′=max{ε1,ε2} and t′=2t, thus we have M(s′x,s′y,t′)>ε′ and N(s′x,s′y,t′)<1-ε′. From the compactness of A¯ it easily follows that there is a δ>0 such that if s′∈S, M(x,y,t)>δ and N(x,y,t)<1-δ then
(4)M(s′x,s′y,t′)>ε′,N(s′x,s′y,t′)<1-ε′.
Hence x∈Eq(X).Definition 7.
The dynamical system(S,X) is called as follows.(1)
TT: topological transitive if for every pair of nonempty subsets U and V in X, there exists s∈S such that sU∩V≠∅.(2)
PT: point transitivity if S has a dense orbit, that is, there is a point x0∈X whose orbit is dense in X. Such a point is called transitive point. Notation: x0∈Trans(X).(3)
DPT: densely point transitive if there is a dense set of transitive point, that is, Trans(X) is dense in X.Remark 2.
(1)   One can easily verify thats(s-1V∩U)=V∩sU≠∅, it is say that U∩s-1V≠∅.
(2)  Clearly,DPT implies PT. Generally, TT and PT are independent properties.Proposition 1.
Let(S,X) be a dynamical system.(1)
LetX be perfect (has no isolated point) and S an F-semigroup. Then PT  implies TT . Furthermore, if X is separable and second category, then TT  implies DPT  and hence also PT.(2)
Every DPT  system(S,X) is TT.Proof.
(1)  Letx be a transitive point with dense orbit Sx and (U,V) a pair of nonempty open subset of X, then there exists s1∈S such that s1x∈U. Since S is an F-semigroup, the subset S∖Ss1 is finite. On the other hand, Ss1x is dense in X. Indeed, removing the finite subset (S∖Ss1)x from the dense orbit Sx we obtain again a dense subset because X has no isolated point. Then there exists s2∈S such that s2s1x∈V, that is, s1x∈s2-1V. Thus, s1x∈U∩s2-1V and hence U∩s2-1V≠∅.
For the second half, assume thatS has no dense orbit and {Un}n=1∞ is a countable base. For each x∈X there is some Un such that sx∉Un. But ⋃s∈SsU is open and meets every open set in X since (S,X) is TT. Let Vn=X∖⋃s∈SsU, then x∈Vn and Vn is closed and has empty interior. However, X=⋃Vn is a countable union, contradicting the fact that X is of second category.
(2)  LetU and V be nonempty open subsets of X. Since (S,X) is DPT, we can choose a transitive point x0∈U. On the other hand, there exists s∈S such that sx0∈V since the orbit of x0 is dense in X. Therefore, sx0∈sU∩V.Proposition 2.
If the dynamical system(S,X) is TT , then Eq(X)⊂Trans(X).Proof.
Letx0∈Eq(X) and y∈X. For the orbit Sx0 of x0 and the r-neighborhood B(y,r,t)={z∈X:M(y,z,t)>r,N(y,z,t)<1-r} of y, we show that sx0∩B(y,r,t)≠∅ for r∈(0,1) and t>0. Since x0∈Eq(X), there exists a neighborhood U of x0 such that
(5)M(sx0,sx,t)>r,N(sx0,sx,t)<1-rforsomex∈U.
Therefore,
(6)M(s0x0,y,2t)≥M(s0x0,sx,t)*M(s0x,y,t)>r*r>r1forsome0<r1<1,N(s0x0,y,2t)≤N(s0x0,sx,t)⋄N(s0x,y,t)<(1-r)⋄(1-r)<r2forsome0<r2<1.
Taking r′=max{r1,r2} and t′=2t, we then have M(s0x0,y,t′)≥r′ and N(s0x0,y,t′)≤1-r′. This completes the proof.Proposition 3.
LetS be a C-semigroup. If (S,X) is PT  and Eq(X)≠∅, then Trans(X)⊂Eq(X), that is, every transitive point is an equicontinuous point.Proof.
Lety∈Trans(X) and x∈Eq(X). We claim that y∈Eq(X). For given r>0 there is a neighborhood O(x) of x such that
(7)M(sx”,sx′,t)>ε,N(sx”,sx′,t)<1-ε
for all (s,x′,x”)∈S×O(x)×O(x). Since y∈Trans(X) so there exists s0∈S such that s0y∈O(x). Thus O(y)=s0-1O(x) is a neighborhood of y. Therefore, for each (s,y′,y”)∈S×O(y)×O(y),
(8)M(ss0y′,ss0y′′,t)>ε,N(ss0y′,ss0y′′,t)<1-ε.
Since S is a C-semigroup the subset M=S∖Ss0¯ is compact. By Lemma 1, S acts equicontinuously on (S,X). Hence we can choose a neighborhood Uy of y such that for all (m,y′,y”)∈M×U(y)×U(y),
(9)M(my′,my′′,t)>ε,N(my′,my′′,t)<1-ε.
Let V=O(y)∩U(y), obviously V is a neighborhood of y. Since S=Ss0⋃M, we have for each (s,y′,y′′)∈S×V×V,
(10)M(sy′,sy′′,t)>ε,N(sy′,sy′′,t)<1-ε.
This proves that y∈Eq(X) and hence Trans(X)⊂Eq(X).Definition 8.
A (not necessarily compact) dynamical system(S,X) is called minimal if every point of X is transitive.Theorem 1.
LetS be a C-semigroup. If (S,X) is minimal and Eq(X)≠∅, then X is equicontinuous.Proof.
By the definition of minimal, we haveTrans(X)=X. Using Proposition 3, if Eq(X)≠∅ then every transitive point is an equicontinuity point. Thus, Eq(X)=X.Theorem 2.
LetS be a C-semigroup. Assume that the dynamical system (S,X) is separable and second category. Then (S,X) is almost equicontinuous if and only if Eq(X)≠∅.Proof.
Clearly, if(S,X) is almost equicontinuous then Eq(X)≠∅. On the other hand, using Proposition  1(1), we obtain that (S,X) is DPT, that is, Trans(X) is dense in X. If Eq(X)≠∅, then Trans(X)⊂Eq(X) by Proposition 3, it follows that Eq(X) is also dense in X and hence (S,X) is almost equicontinuous.Definition 9 (sensitive dependence on initial conditions).
A dynamical system(S,X) has sensitive dependence on initial conditions or more briefly, is sensitive, if there exists ε∈(0,1) and t>0 such that for every x∈X and every neighborhood U of x, there exists (s,y)∈S×U with
(11)M(sx,sy,t)<ε,N(sx,sy,t)>1-ε.
When (S,X) is not sensitive, we say that (S,X) is nonsensitive.Remark 3.
(1)   The definition of sensitive dependence on initial conditions plays an important role in classical chaotic systems. Note that the above form is just a generalization of existing definition for(X,f) when S={fn:n=1,2.…}.
(2)  Spelling the property nonsensitive out we have: for everyε∈(0,1) and t>0, there exists x∈X and a neighborhood U of x, such that for each (s,y)∈S×U,
(12)M(sx,sy,t)≥ε,N(sx,sy,t)≤1-ε.
We observe that trivially (S,X) is nonsensitive whenever X has an isolated point.
(3)   Without loss of generality, we sometimes use the open ballB(x,r,t) instead of the neighborhood U of x in Definition 9.Proposition 4.
For a PT  dynamical system(S,X) with no isolated points, being nonsensitive is equivalent to the following property: for every ε∈(0,1) and t>0, there exists a transitive point x0∈X and a neighborhood U of x0 such that for every y∈U and every s∈S,
(13)M(sx0,sy,t)≥ε,N(sx0,sy,t)≤1-ε.Proof.
Letε′ be given and let x and U be as in the definition of nonsensitive. Since (S,X) is PT, there is a point x0 in X whose orbit is dense, that is, there exists a s0∈S such that s0x0∈U. Denote x1=s0x0, then
(14)M(sx,sx1,t)≥ε′,N(sx,sx1,t)≤1-ε′.
On the other hand, there exists r∈(0,1) and t>0 such that B(x,r,t)⊂U. Let V=B(x,r,t), then for every y∈V⊂U and s∈S,
(15)M(sx,sy,t)≥ε′,N(sx,sy,t)≤1-ε′.
Therefore,
(16)M(sx1,sy,2t)≥M(sx,sx1,t)*M(sx,sy,t)≥ε′*ε′≥ε1
for some ε1∈(0,1), and
(17)N(sx1,sy,2t)≤N(sx,sx1,t)⋄N(sx,sy,t)≤(1-ε′)⋄(1-ε′)≤1-ε2
for some ε2∈(0,1).
Takingε=max{ε1,ε2} and t′=2t, then M(sx1,sy,t′)≥ε and N(sx1,sy,t′)≤1-ε. Since X has no isolated points, the point x1 is also transitive and the proof is complete.Theorem 3.
LetS be a C-semigroup. Assumed that a TT  dynamical system (S,X) is separable and second category. Then (S,X) is almost equicontinuous if and only if it is nonsensitive.Proof.
Suppose that{Un}n=1∞ is a countable base. Since (S,X) is nonsensitive, for any ε∈(0,1), there exists an x∈X and a neighborhood U of x such that for all (s,y)∈S×U and t>0,
(18)M(sx,sy,t)≥ε,N(sx,sy,t)≤1-ε.
Since {Un}n=1∞ is a base, there exists n0∈ℕ such that y∈Un0⊂U. Without loss of generality, we can assume that sUn⊂B(sx,1/n,t), for all (s,n)∈S×ℕ. Let
(19)Vn=S-1Un,M=⋂n∈ℕVn.
Clearly, every Vn is open and meets every open subset of X since (S,X) is TT. This means that each Vn is dense in X. Since X is second category, by Baire theorem, M=⋂n∈ℕVn is also dense. Now it remains to show that M⊂Eq(X). Given x∈M, then there exists s0∈S such that s0x∈Un. Let V=s0-1Un. Hence for y∈V and each s=s′s0∈Ss0 we have
(20)M(sx,sy,t)=M(s′s0x,s′s0y,t)≥1n≥r1
for some r1∈(0,1) and
(21)N(sx,sy,t)=N(s′s0x,s′s0y,t)≤1-1n≤r2
for some r2∈(0,1).
Takingr=max{r1,r2} then M(sx,sy,t)≥r and N(sx,sy,t)≤1-r. But S∖Ss0¯ is compact because S is an C-semigroup. Then by Lemma 1, the set S∖Ss0¯ acts equicontinuously on (S,X). This means that if W is an open neighborhood of x, then for all y∈W and for each s∈S∖Ss0¯,
(22)M(sx,sy,t)≥r,N(sx,sy,t)≤1-r
holds. Let O=W∩V be an open neighborhood of x.
(23)M(sx,sy,t)≥r,N(sx,sy,t)≤1-r
for all s∈S and all y∈O. Hence x∈Eq(X). This completes the proof.



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*Source: 148193-2009-07-12.xml*