Semigroup Actions on Intuitionistic Fuzzy Metric Spaces

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.

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 spaces 𝑋, 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 as 𝑋, wherever there is no risk of confusion.

Definition 4. A dynamical system in 𝑋 is a triple (𝑆,𝑋,𝜋) (we note (𝑆,𝑋) for short), where 𝑆 is a topological semigroup, 𝑋 at least is Hausdorff and 𝜋∶𝑆×𝑋→𝑋(𝑠,𝑥)↦𝑠𝑥 (1) 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 if 𝑆={𝑓^𝑛∶𝑛=1,2,…} and 𝑓∶𝑋→𝑋 is continuous, then (𝑆,𝑋) is a classical dynamical system in 𝑋. Notation: (𝑋,𝑓).

Definition 5. Let 𝑆 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 an 𝐹-semigroup and also a 𝐶-semigroup.

Definition 6. Let (𝑆,𝑋) be a dynamical system.(1)A subset 𝐴 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 point 𝑥₀∈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. Theorem 3 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. Let 𝑥∈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 𝑈_𝑠 of 𝑠 and a 𝛿_𝑠>0 such that if 𝑠^′∈𝑈_𝑠 and 𝑦∈𝐵(𝑥,𝛿_𝑠,𝑡), then 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 (2) Therefore, 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. (3) 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 𝑀(𝑠^′𝑥,𝑠^′𝑦,𝑡^′)>𝜀^′,𝑁(𝑠^′𝑥,𝑠^′𝑦,𝑡^′)<1−𝜀^′. (4) 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 that 𝑠(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)Let 𝑋 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)  Let 𝑥 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 that 𝑆 has no dense orbit and {𝑈_𝑛}^∞_𝑛=1 is a countable base. For each 𝑥∈𝑋 there is some 𝑈_𝑛 such that 𝑠𝑥∉𝑈_𝑛. But ⋃_𝑠∈𝑆𝑠𝑈 is open and meets every open set in 𝑋 since (𝑆,𝑋) is TT. Let 𝑉_𝑛=𝑋⧵⋃_𝑠∈𝑆𝑠𝑈, then 𝑥∈𝑉_𝑛 and 𝑉_𝑛 is closed and has empty interior. However, 𝑋=⋃𝑉_𝑛 is a countable union, contradicting the fact that 𝑋 is of second category.
(2)  Let 𝑈 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. Let 𝑥₀∈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 𝑀(𝑠𝑥₀,𝑠𝑥,𝑡)>𝑟,𝑁(𝑠𝑥₀,𝑠𝑥,𝑡)<1−𝑟forsome𝑥∈𝑈. (5) Therefore, 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. (6) 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. Let 𝑆 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. Let 𝑦∈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 𝑀(𝑠𝑥”,𝑠𝑥^′,𝑡)>𝜀,𝑁(𝑠𝑥”,𝑠𝑥^′,𝑡)<1−𝜀 (7) 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), 𝑀(𝑠𝑠₀𝑦^′,𝑠𝑠₀𝑦^′′,𝑡)>𝜀,𝑁(𝑠𝑠₀𝑦^′,𝑠𝑠₀𝑦^′′,𝑡)<1−𝜀. (8) 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 𝑈_𝑦 of 𝑦 such that for all (𝑚,𝑦^′,𝑦”)∈𝑀×𝑈(𝑦)×𝑈(𝑦), 𝑀(𝑚𝑦^′,𝑚𝑦^′′,𝑡)>𝜀,𝑁(𝑚𝑦^′,𝑚𝑦^′′,𝑡)<1−𝜀. (9) 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, 𝑀(𝑠𝑦′,𝑠𝑦^′′,𝑡)>𝜀,𝑁(𝑠𝑦^′,𝑠𝑦^′′,𝑡)<1−𝜀. (10) 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. Let 𝑆 be a 𝐶-semigroup. If (𝑆,𝑋) is minimal and Eq(𝑋)≠∅, then 𝑋 is equicontinuous.

Proof. By the definition of minimal, we have Trans(𝑋)=𝑋. Using Proposition 3, if Eq(𝑋)≠∅ then every transitive point is an equicontinuity point. Thus, Eq(𝑋)=𝑋.

Theorem 2. Let 𝑆 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 𝑀(𝑠𝑥,𝑠𝑦,𝑡)<𝜀,𝑁(𝑠𝑥,𝑠𝑦,𝑡)>1−𝜀. (11) 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={𝑓^𝑛∶𝑛=1,2.…}.
(2)  Spelling the property nonsensitive out we have: for every 𝜀∈(0,1) and 𝑡>0, there exists 𝑥∈𝑋 and a neighborhood 𝑈 of 𝑥, such that for each (𝑠,𝑦)∈𝑆×𝑈, 𝑀(𝑠𝑥,𝑠𝑦,𝑡)≥𝜀,𝑁(𝑠𝑥,𝑠𝑦,𝑡)≤1−𝜀. (12) We observe that trivially (𝑆,𝑋) is nonsensitive whenever 𝑋 has an isolated point.
(3)   Without loss of generality, we sometimes use the open ball 𝐵(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, 𝑀(𝑠𝑥₀,𝑠𝑦,𝑡)≥𝜀,𝑁(𝑠𝑥₀,𝑠𝑦,𝑡)≤1−𝜀. (13)

Proof. Let 𝜀^′ 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 𝑀(𝑠𝑥,𝑠𝑥₁,𝑡)≥𝜀^′,𝑁(𝑠𝑥,𝑠𝑥₁,𝑡)≤1−𝜀^′. (14) 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, 𝑀(𝑠𝑥,𝑠𝑦,𝑡)≥𝜀^′,𝑁(𝑠𝑥,𝑠𝑦,𝑡)≤1−𝜀^′. (15) Therefore, 𝑀(𝑠𝑥₁,𝑠𝑦,2𝑡)≥𝑀(𝑠𝑥,𝑠𝑥₁,𝑡)∗𝑀(𝑠𝑥,𝑠𝑦,𝑡)≥𝜀^′∗𝜀^′≥𝜀₁ (16) for some 𝜀₁∈(0,1), and 𝑁(𝑠𝑥₁,𝑠𝑦,2𝑡)≤𝑁(𝑠𝑥,𝑠𝑥₁,𝑡)⋄𝑁(𝑠𝑥,𝑠𝑦,𝑡)≤(1−𝜀^′)⋄(1−𝜀^′)≤1−𝜀₂ (17) for some 𝜀₂∈(0,1).
Taking 𝜀=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. Let 𝑆 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 {𝑈_𝑛}^∞_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, 𝑀(𝑠𝑥,𝑠𝑦,𝑡)≥𝜀,𝑁(𝑠𝑥,𝑠𝑦,𝑡)≤1−𝜀. (18) Since {𝑈_𝑛}^∞_𝑛=1 is a base, there exists 𝑛₀∈ℕ such that 𝑦∈𝑈_𝑛0⊂𝑈. Without loss of generality, we can assume that 𝑠𝑈_𝑛⊂𝐵(𝑠𝑥,1/𝑛,𝑡), for all (𝑠,𝑛)∈𝑆×ℕ. Let 𝑉_𝑛=𝑆⁻¹𝑈_𝑛,𝑀=_𝑛∈ℕ𝑉_𝑛. (19) Clearly, every 𝑉_𝑛 is open and meets every open subset of 𝑋 since (𝑆,𝑋) is TT. This means that each 𝑉_𝑛 is dense in 𝑋. Since 𝑋 is second category, by Baire theorem, 𝑀=⋂_𝑛∈ℕ𝑉_𝑛 is also dense. Now it remains to show that 𝑀⊂Eq(𝑋). Given 𝑥∈𝑀, then there exists 𝑠₀∈𝑆 such that 𝑠₀𝑥∈𝑈_𝑛. Let 𝑉=𝑠₀⁻¹𝑈_𝑛. Hence for 𝑦∈𝑉 and each 𝑠=𝑠^′𝑠₀∈𝑆𝑠₀ we have 𝑀(𝑠𝑥,𝑠𝑦,𝑡)=𝑀(𝑠^′𝑠₀𝑥,𝑠^′𝑠₀1𝑦,𝑡)≥𝑛≥𝑟₁ (20) for some 𝑟₁∈(0,1) and 𝑁(𝑠𝑥,𝑠𝑦,𝑡)=𝑁(𝑠^′𝑠₀𝑥,𝑠^′𝑠₀1𝑦,𝑡)≤1−𝑛≤𝑟₂ (21) for some 𝑟₂∈(0,1).
Taking 𝑟=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, 𝑀(𝑠𝑥,𝑠𝑦,𝑡)≥𝑟,𝑁(𝑠𝑥,𝑠𝑦,𝑡)≤1−𝑟 (22) holds. Let 𝑂=𝑊∩𝑉 be an open neighborhood of 𝑥. 𝑀(𝑠𝑥,𝑠𝑦,𝑡)≥𝑟,𝑁(𝑠𝑥,𝑠𝑦,𝑡)≤1−𝑟 (23) for all 𝑠∈𝑆 and all 𝑦∈𝑂. Hence 𝑥∈Eq(𝑋). This completes the proof.

Acknowledgment

This research was partially supported by the NSF of China (Grant 10771056) and the Special Research Found for the Doctoral Program of Higher Education.