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Semigroup Actions on Intuitionistic Fuzzy Metric Spaces
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.
As a generalization of fuzzy sets , the concepts of intuitionistic fuzzy sets was introduced by Atanassov , and later there has been much progress in the study of intuitionistic fuzzy sets by many authors [3–6]. Following George and Veeramani , Park  has defined intuitionistic fuzzy metric spaces and obtained several classical theorems on this new structure. Furthermore, Alaca et al.  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 theory which were given and studied by El Naschie [10, 11] and by Sigalotti and Mejias .
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 . 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.
Definition 1. A binary operation is continuous -norm if satisfies the following conditions:(a) is commutative and associative,(b) is continuous,(c) for all ,(d) whenever and , for each .
Definition 2. A binary operation ◊ is continuous -conorm if ◊ satisfies the following conditions:(a)◊ is commutative and associative,(b)◊ is continuous,(c)◊ for all ,(d)◊◊ whenever and , for each .
Definition 3. A 5-tuple ◊ is called intuitionistic fuzzy metric space if is an arbitrary set, is a continuous -norm, ◊ is a continuous -conorm and , are fuzzy sets on satisfying the following conditions: for all ,–(IFm-1): –(IFm-2): ,–(IFm-3): if and only if ,–(IFm-4): ,–(IFm-5): ,–(IFm-6): is continuous,–(IFm-7): ,–(IFm-8): if and only if ,–(IFm-9): ,–(IFm-10): ◊,–(IFm-11): is continuous.
Then is called an intuitionistic fuzzy metric on . The function and denote the degree of nearness and the degree of nonnearness between and respect to , respectively.
Remark 1. In intuitionistic fuzzy metric spaces , is nondecreasing and is nonincreasing for all .
3. Main Results
For brevity, we write the intuitionistic fuzzy metric spaces as , wherever there is no risk of confusion.
Definition 4. A dynamical system in is a triple (we note for short), where is a topological semigroup, at least is Hausdorff and is a continuous action on . Thus, holds for each in . The orbit of is the set . A subset is said to be -invariant if . If is a closed -invariant subset of , then we call the restricted action, is a subsystem of . For and denote .
Note that if and is continuous, then is a classical dynamical system in . Notation: .
Definition 5. Let be a topological semigroup, we say that is an(1)-semigroup if for every the subset is finite,(2)-semigroup if for every , the closure of the subset is compact in (i.e., is relatively compact).
Obviously, every topological group is of course an -semigroup and also a -semigroup.
Definition 6. Let be a dynamical system.(1)A subset of acts equicontinuously at if for every and , there exists such that and imply and for every .(2)A point is a point of equicontinuity (notation: ) if acts equicontinuously at . is equicontinuous if .(3) is almost equicontinuous if is dense in .
Observe that every equicontinuous system is almost equicontinuous. Theorem 3 show that the relation between almost equicontinuity and nonsensitivity.
Lemma 1. Let be a dynamical system. If is a related compact subset of , then acts equicontinuously on .
Proof. Let and . Suppose that is in the neighborhood of defined by and for some and . Given . Note that , by the continuity of , for each there is an open neighborhood of and a such that if and , then Therefore, Taking and , thus we have and . From the compactness of it easily follows that there is a such that if , and then Hence .
Definition 7. The dynamical system is called as follows.(1)TT: topological transitive if for every pair of nonempty subsets and in , there exists such that .(2)PT: point transitivity if has a dense orbit, that is, there is a point whose orbit is dense in . Such a point is called transitive point. Notation: .(3)DPT: densely point transitive if there is a dense set of transitive point, that is, is dense in .
Remark 2. (1) One can easily verify that , it is say that .
(2) Clearly, DPT implies PT. Generally, TT and PT are independent properties.
Proposition 1. Let be a dynamical system.(1)Let be perfect (has no isolated point) and an -semigroup. Then PT implies TT . Furthermore, if is separable and second category, then TT implies DPT and hence also PT.(2)Every DPT system is TT.
Proof. (1) Let be a transitive point with dense orbit and a pair of nonempty open subset of , then there exists such that . Since is an -semigroup, the subset is finite. On the other hand, is dense in . Indeed, removing the finite subset from the dense orbit we obtain again a dense subset because has no isolated point. Then there exists such that , that is, . Thus, and hence .
For the second half, assume that has no dense orbit and is a countable base. For each there is some such that . But is open and meets every open set in since is TT. Let , then and is closed and has empty interior. However, is a countable union, contradicting the fact that is of second category.
(2) Let and be nonempty open subsets of . Since is DPT, we can choose a transitive point . On the other hand, there exists such that since the orbit of is dense in . Therefore, .
Proposition 2. If the dynamical system is TT , then .
Proof. Let and . For the orbit of and the -neighborhood of , we show that for and . Since , there exists a neighborhood of such that Therefore, Taking and , we then have and . This completes the proof.
Proposition 3. Let be a -semigroup. If is PT and , then , that is, every transitive point is an equicontinuous point.
Proof. Let and . We claim that . For given there is a neighborhood of such that for all . Since so there exists such that . Thus is a neighborhood of . Therefore, for each , Since is a -semigroup the subset is compact. By Lemma 1, acts equicontinuously on . Hence we can choose a neighborhood of such that for all , Let , obviously is a neighborhood of . Since , we have for each , This proves that and hence .
Definition 8. A (not necessarily compact) dynamical system is called minimal if every point of is transitive.
Theorem 1. Let be a -semigroup. If is minimal and , then is equicontinuous.
Proof. By the definition of minimal, we have . Using Proposition 3, if then every transitive point is an equicontinuity point. Thus, .
Theorem 2. Let be a -semigroup. Assume that the dynamical system is separable and second category. Then is almost equicontinuous if and only if .
Proof. Clearly, if is almost equicontinuous then . On the other hand, using Proposition 1(1), we obtain that is DPT, that is, is dense in . If , then by Proposition 3, it follows that is also dense in and hence is almost equicontinuous.
Definition 9 (sensitive dependence on initial conditions). A dynamical system has sensitive dependence on initial conditions or more briefly, is sensitive, if there exists and such that for every and every neighborhood of , there exists with When is not sensitive, we say that 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 when .
(2) Spelling the property nonsensitive out we have: for every and , there exists and a neighborhood of , such that for each , We observe that trivially is nonsensitive whenever has an isolated point.
(3) Without loss of generality, we sometimes use the open ball instead of the neighborhood of in Definition 9.
Proposition 4. For a PT dynamical system with no isolated points, being nonsensitive is equivalent to the following property: for every and , there exists a transitive point and a neighborhood of such that for every and every ,
Proof. Let be given and let and be as in the definition of nonsensitive. Since is PT, there is a point in whose orbit is dense, that is, there exists a such that . Denote , then
On the other hand, there exists and such that . Let , then for every and ,
for some , and
for some .
Taking and , then and . Since has no isolated points, the point is also transitive and the proof is complete.
Theorem 3. Let be a -semigroup. Assumed that a TT dynamical system is separable and second category. Then is almost equicontinuous if and only if it is nonsensitive.
Proof. Suppose that is a countable base. Since is nonsensitive, for any , there exists an and a neighborhood of such that for all and ,
Since is a base, there exists such that . Without loss of generality, we can assume that , for all . Let
Clearly, every is open and meets every open subset of since is TT. This means that each is dense in . Since is second category, by Baire theorem, is also dense. Now it remains to show that . Given , then there exists such that . Let . Hence for and each we have
for some and
for some .
Taking then and . But is compact because is an -semigroup. Then by Lemma 1, the set acts equicontinuously on . This means that if is an open neighborhood of , then for all and for each , holds. Let be an open neighborhood of . for all and all . Hence . This completes the proof.
This research was partially supported by the NSF of China (Grant 10771056) and the Special Research Found for the Doctoral Program of Higher Education.
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Copyright © 2009 Yaoyao Lan and Qingguo Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.