# G-Expansibility and G-Almost Periodic Point under Topological Group Action

Authors: Zhanjiang Ji
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Firstly, the new concepts ofG−expansibility, G−almost periodic point, and G−limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map f and the shift map σ in the inverse limit space under topological group action. The following new results are obtained. Let X,d be a metric G−space and Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f. (1) If the map f:X⟶X is an equivalent map, then we have APG¯σ=Lim←ApGf,f. (2) If the map f:X⟶X is an equivalent surjection, then the self-map f is G−expansive if and only if the shift map σ is G¯−expansive. (3) If the map f:X⟶X is an equivalent surjection, then the self-map f has G− limit shadowing property if and only if the shift map σ has G¯− limit shadowing property. The conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li . Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.


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## Body

## 1. Introduction

LetX,d be a metric space and f be a continuous map from X to X. A point x is called to be an almost periodic point if for each open set U containing x there exists a positive integer m>0 such that for every positive integer k>0 there exists r∈k,k+m satisfying frx∈U(see [1]). In recent years, there are many achievements about almost periodic points set (see [1–8]). Qiu and Zhao [2] proved that if the map f has shadowing property in X, then the map f has shadowing property in AP¯f. Liao and Wang [3] discussed the properties of almost periodic points in unilateral symbol space. In [1], it is proved that the set of almost periodic points of shift map σ is the inverse limit space formed by the self-map f in APf. Inspired by the idea of Li [1], the author gave the concepts of G−almost periodic point according to the concept of almost periodic point. That is, let X,d be a metric G−space and f be a continuous map from X to X. A point x is called to be an G−almost periodic point if for each open set U containing x there exists a positive integer m>0 such that for every positive integer k>0 there exists r∈k,k+m and gk∈G satisfying gkfrx∈U. Next, we give the following theorem in the inverse limit space under topological group action.Theorem 1.
LetX,d be a metric G−space, the map f:X⟶X be an equivalent map, and Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f. Then, we have(1)APG¯σ=Lim←ApGf,f.According to definition,G−almost periodic point means almost periodic point. Otherwise, it does not hold. Hence, Theorem 1 generalizes the corresponding results given in Li [1].The mapf is said to be expansive if for any x≠y∈X there exists a positive integer n>0 such that dfnx,fny>C (see [9]). The map f has shadowing property if each ε>0 there exists δ>0 such that for any δ−pseudo orbit of f there exists a point y in X such that the sequence xii=0∞ is ε−shadowed by the point y (see [10]). Expansibility and shadowing property have attracted the attention of many scholars. The relevant results are seen in [11–17]. Das and Das [11] pointed out that the map f is G−expansive if and only if the map g is G¯−expansive in locally equidistant covered space. Wang and Zeng [12] discussed the relationship between average shadowing property and q¯−average shadowing property. Das [13] proved the nonexistence of pseudo equivariant G−expansive homeomorphism on closed unit interval. Das and Das [14] obtained a sufficient condition for the extension of a G−expansive homeomorphism on a G-invariant subspace of a compact metric G−space with G compact to be G−expansive on the whole space. Shah [15] gave a necessary and sufficient condition for a positively G−expansive map to possess the G−shadowing property. Wu [16] proved that the system with d−tracking property is chain mixed. Oprocha et al. [17] analyzed necessary and sufficient conditions for shadowing property over a set with positive density. In this paper, we gave the concepts of G−expansive map and G−limit shadowing property according to the concepts of expansive map and limit shadowing property. At last, we will give the proof of Theorems 2 and 3 in the inverse limit space under topological group action. The main results are as follows.Theorem 2.
LetX,d be a compact metric G−space, the map f:X⟶X be an equivalent surjection, and Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f. Then, the self-map f is G−expansive if and only if the shift map σ is G¯−expansive.Theorem 3.
LetX,d be a compact metric G−space, the map f:X⟶X be an equivalent surjection, and Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f. Then, the self-map f has G−limit shadowing property if and only if the shift map σ has G¯−limit shadowing property.According to definition,G−expansibility means expansibility and G−limit shadowing property means limit shadowing property. Otherwise, it does not hold. Hence, Theorems 2 and 3, respectively, generalize the corresponding results given in Niu [18] and Liang and Li [10].


## 2. G-Almost Periodic Point under Topological Group Action

Definition 1.
(see [19]). Let X,d be a metric G−space, G be a topological group, and φ:G×X⟶X be a continuous map. The triple X,G,θorX is called to be metric G−space if the following conditions are satisfied:(1)
φe,x=x where e is the identity of G and for all x∈X(2)
φg1,φg2,x=φg1g2,x for all x∈X and all g1, g2∈GRemark 1.
IfX is compact, then X is also said to be compact metric G−space. For the convenience of writing, φg,x is usually abbreviated as gx.Definition 2.
(see [19]). Let X,d be a metric G−space and f be a continuous map from X to X. f is said to be an equivariant map if we have fpx=pfx for all x∈X and all p∈G.Definition 3.
(see [20]). Let X,d be a metric space and f be a continuous map from X to X. Xf is said to be the inverse limit spaces of X if we write(2)Xf=x0,x1,x2,…:fxi+1=xi,∀i≥0.
The metricd¯ in Xf is defined by(3)d¯x¯,y¯=∑i=0∞dxi,yi2i,where x¯=x0,x1,x2,…∈Xf and y¯=y0,y1,y2,…∈Xf.
The shift mapσ:Xf⟶Xf is defined by(4)σx¯=fx0,x0,x1,….
The projection mapπi:Xf⟶X is defined by(5)πix¯=xi.
Thus,Xf,d¯ is compact metric space. The shift mapping σ is homeomorphism, and for any i≥0, the projection map πi is a continuous and open map.Definition 4.
(see [20]). Let X,d be a metric G-space and f be equivariant map from X to X. Write(6)G¯=g,g,g,…:g∈G,G∞=∏i=0∞Gi,where Gi=G.
The mapθ:G¯×Xf⟶Xf is defined by(7)θg¯,x¯=g¯⋅x¯=gx0,gx1,gx2,…,where g¯=g,g,g,…∈G¯ and x¯=x0,x1,x2,…∈Xf.
Then,Xf,G¯,θ is a metric G−space. Let X,G,d,f and Xf,G¯,d¯,σ be shown as above. The space Xf,G¯,d¯,σ is called to be the inverse limit spaces of X,d,f,G under group action.Definition 5.
(see [1]). Let X,d be a metric space and f be a continuous map from X to X. The point x is called to be an almost periodic point if for each open set U containing x there exists a positive integer m>0 such that for every positive integer k>0 there exists r∈k,k+m satisfying frx∈U. Denoted by APf the almost periodic point set of the map f.Remark 2.
According to the concept of almost periodic point, we give the concept ofG−almost periodic point.Definition 6.
LetX,d be a metric space and f be a continuous map from X to X. The point x is called to be an G−almost periodic point if for each open set U containing x there exists a positive integer m>0 such that for every positive integer k>0 there exists r∈k,k+m and gk∈G satisfying gkfrx∈U. Denoted by APGf the G−almost periodic point set of the map f.Theorem 4.
LetX,d be a metric G−space, the map f:X⟶X be an equivalent map, and Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f. Then, we have(8)APG¯σ=Lim←ApGf,f.Proof.
⟹ Suppose x¯=x0,x1,x2,…∈APG¯σ. For any i≥0, let U be any open set containing xi. Then, πi−1U is an open set containing x¯. Hence, there exists a positive integer m1>0 such that for every positive integer k>0 there exists p∈k,k+m1 and g¯k=gk,gk,gk,…∈G¯ such that(9)g¯kσpx¯∈πi−1U.
Thus, we can obtain(10)πig¯kσpx¯∈U.
Then, we have that(11)gkfpxi∈U.
So,xi∈APGf. Hence, APG¯σ⊂Lim←APGf,f.
⟸ Suppose y¯=y0,y1,y2,…∈Lim←APGf,f. Thus, we can get yi∈APGf for every i≥0. Let V be an any open set containing the point y¯. Then, πiV is an open set containing the point yi. According to yi∈APGf, there exists a positive integer m2>0 such that for every positive integer n>0 there exists r∈n,n+m2 and tn∈G satisfying(12)tnfryi∈πiV,i≥0.
Lett¯n=tn,tn,tn,…∈G¯. Then, we can obtain(13)t¯nσry¯∈V.
So,y¯∈APG¯σ. Thus, we have Lim←APGf,f⊂APG¯σ. This completes the proof.


## 3. G-Expansibility under Topological Group Action

Definition 7.
(see [9]). Let X,d be a metric space, f be a continuous map from X to X, and C be a positive constant. The map f is said to be an expansive map if for any x≠y∈X there exists a positive integer n>0 such that dfnx,fny>C where the constant C is called to be an expansion constant.Remark 3.
According to the concept of expansive map, we give the concept ofG−almost expansive map.Definition 8.
LetX,d be a metric G-space, f be a continuous map from X to X, and C be a positive constant. The map f is said to be an G−expansive map if for any x≠y∈X there exists an positive integer n such that for any g and p∈G we have that dfngx,fnpy>C where the constant C is called to be an G−expansion constant.
Next, we start to prove Theorem5.Theorem 5.
LetX,d be a compact metric G−space, the map f:X⟶X be an equivalent surjection, and Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f. Then, the self-map f is G−expansive if and only if the shift map σ is G¯−expansive.Proof.
⟹ Suppose that the self-map f is G−expansive map with expansion constant C1>0. Thus, the map f is injective. Let x¯=x0,x1,x2,…∈Xf and y¯=y0,y1,y2,…∈Xf with x¯≠y¯. As the map f is injective, we can get x0≠y0. Let (14)g¯=g,g,g,…∈G¯,l¯=l,l,l,…∈G¯.
By the definition ofG−expansive map f, there exists an positive integer n1>0 such that(15)dfn1gx0,fn1ly0>C1.
By the definition of the metricd¯, we get that(16)d¯σn1g¯⋅x¯,σn1l¯⋅y¯>C1.
Hence, the shift mapσ is G¯−expansive.
⟸ Suppose that the shift map σ is G¯−expansive map with expansion constant 4C2>0. Since X is compact metric space, it is bounded. Write M=DiamX. Let m>0 such that(17)∑i=m+1∞M2i<2C2.
Since the mapf is surjection, for any u≠v∈X, we can choose(18)u¯=fmu,fm−1u,…,fu,u,…∈Xf,v¯=fmv,fm−1v,…fv,v,…∈Xf.
It is obvious that the pointu¯ is different from the point v¯. For any s and t∈G, let(19)s¯=s,s,s,…∈G¯,t¯=t,t,t,…∈G¯.
According to that the shift mapσ is G¯−expansive map with expansion constant 4C2>0, there exist a positive integer n2>0 such that(20)d¯σn2s¯⋅u¯,σn2t¯⋅v¯>4C2.
According to that the mapf is equivalent, we can get that(21)∑i=0mdsfn2+m−iu,tfn2+m−iv2i+∑i=m+1∞M2i>4C2.
By (17), we have that(22)∑i=0mdsfn2+m−iu,tfn2+m−iv2i>2C2.
If for any positive integerk>0, we have that(23)dsfku,tfkv≤C2.
Then, by (23), we can get that(24)∑i=0mdsfn2+m−iu,tfn2+m−iv2i≤∑i=0mC22i<2C2.
It is absurd. Hence, there exists a positive integern3 such that(25)dsfn3u,tfn3v>C2.
So, the self-mapf is G−expansive. Thus, we end the proof.


## 4. G-Limit Shadowing Property under Topological Group Action

Definition 9.
(see [21]). Let X,d be a metric G−space and f be a continuous map from X to X. The sequence xii≥0 is called to be G−limit pseudo orbit of f if for any i≥0 there exists ti∈G such that limi⟶∞dtifxi,xi+1=0.Definition 10.
(see [21]). Let X,d be a metric G−space and f be a continuous map from X to X. xii≥0 is said to be G−limit shadowed by a point y if for any i≥0 there exists ti∈G such that limi⟶∞dfiy,tixi=0.Definition 11.
(see [21]). Let X,d be a metric G−space and f be a continuous map from X to X.
The mapf has G−limit shadowing property if for any G−limit pseudo orbit xii=0∞ of f there exists a point y in X such that the sequence xii=0∞ is G−limit shadowed by the point y.
The main result of this section is the following theorem:Theorem 6.
LetX,d be a compact metric G−space, the map f:X⟶X be an equivalent surjection, and Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f. Then, the self-map f has G−limit shadowing property if and only if the shift map σ has G¯−limit shadowing property.Proof.
⟹ Suppose that the self-map f has G−limit shadowing property. Since X is compact, it is bounded. Write M=DiamX. For any ε>0, let m>0 such that(26)M2m<ε2.
By the continuity of the mapf, it follows that for given ε/4>0 and any 0≤i≤m there exists 0<δ1<ε/4 such that dx,y<δ1 which implies (27)dfix,fiy<ε4.
Lety¯nn=0∞ be G¯−limit pseudo orbit of the shift map σ with y¯n=yn0,yn1,yn2,…∈Xf. Then, there exists g¯n∈G¯ with g¯n=gn,gn,gn,…∈G¯ such that (28)limn⟶∞d¯g¯nσy¯n,y¯n+1=0.
Hence, there exists a positive integerN1∈N+ such that when n>N1, we have that(29)d¯g¯nσy¯n,y¯n+1<ε2m.
So, whenn>N1, we have that(30)dgnmfynm,yn+1m<ε.
Thus, we can get that(31)limn⟶∞dgnmfynm,yn+1m=0.
Hence,ynmn=0∞ is G−limit pseudo orbit of the map f. According to the hypothesis, for any n≥0, there exist z∈X and tn∈G such that (32)limn⟶∞dfnz,tnynm=0.
So, there exists a positive integerN2∈N+ such that when n>N2, we have that(33)dfnz,tnynm<δ1.
According to the equivalent definition of the mapf and (27), for any 0≤i≤m, we have that(34)dfn+iz,tnynm−i<ε4.
Since the mapf is onto, we can choose(35)z¯=fmz,fm−1z,fm−2z,…,fz,z…∈Xf,t¯n=tn,tn,tn,…,tn,…∈G¯.
By (26) and (34), when n>N2, we have that(36)d¯σnz¯,t¯ny¯n<∑i=0m12i⋅ε4+M2m<ε.
Hence, we can get that(37)limn⟶∞d¯σnz¯,t¯ny¯n=0.
So, the shift mapσ has G¯−limit shadowing property.
⟸ Suppose that the shift map σ has G¯−limit shadowing property. For any η>0, there exists an positive integer m′>0 such that (38)M2m′<η2.
According to the continuity of the mapf, it follows that for given η/4>0 and any 0≤i≤m′ there exists 0<δ2<η/4 such that dx,y<δ2 implies(39)dfix,fiy<η4.
Letxkk=0∞ be G−limit pseudo orbit of the map f. Then, for any k≥0, there exists tk∈G such that(40)limk⟶∞dtkfxk,xk+1=0.
Hence, there exists a positive integerN3∈N+ such that when k>N3, we have that(41)dtkfxk,xk+1<δ3.
According to the equivalent definition of the mapf and (39), for any 0≤i≤m′, we have that(42)dtkfi+1xk,fixk+1<η4.
Since the mapf is onto, we can write(43)x¯k=fm′xk,fm′−1xk,fm′−2xk,…,fxk,xk,…∈Xf,t¯k=tk,tk,tk,…,tk,…∈G¯.
By (38) and (42), when k>N3, we have that(44)d¯t¯kσx¯k,x¯k+1<∑i=0m12i⋅η4+M2m<η.
Hence, we can get that(45)limk⟶∞d¯t¯kσx¯k,x¯k+1=0.
Thus,x¯kk=0∞ is G¯−limit pseudo orbit of the shift map σ. By the hypothesis, for any k≥0, there exist s¯=s0,s1,s2,…∈Xf and p¯k=pk,pk,pk,…∈G¯ such that(46)limk⟶∞d¯σks¯,p¯k⋅x¯k=0.
Hence, there exists a positive integerN4∈N+ such that when k>N4, we have that(47)d¯σks¯,p¯k⋅x¯k<η2m′.
So, whenk>N4, we have that(48)dfksm′,pk⋅xk<η.
Thus, we have that(49)limk⟶∞dfksm′,pkxk=0.
Hence, the self-mapf has G−limit shadowing property. This completes the proof.


## 5. Conclusion

Firstly, the new concepts ofG−expansibility, G−almost periodic point, and G−limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map f and the shift map σ in the inverse limit space under topological group action. The following new results are obtained in the inverse limit space under topological group action. (1) If the map f:X⟶X is an equivalent map, then we have APG¯σ=Lim←ApGf,f. (2) If the map f:X⟶X is an equivalent surjection, then the self-map f is G−expansive if and only if the shift map σ is G¯−expansive. (3) If the map f:X⟶X is an equivalent surjection, then the self-map f has G−limit shadowing property if and only if the shift map σ has G¯−limit shadowing property.G−almost periodic point, G−expansibility, and G−limit shadowing property, respectively, mean almost periodic point, expansibility, and limit shadowing property. Otherwise, it does not hold. Hence, Theorems 1–3, respectively, generalize the corresponding results given in the study by Li [1], Niu [18], and Liang and Li [10]. Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.



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