# TheG-Asymptotic Tracking Property and G-Asymptotic Average Tracking Property in the Inverse Limit Spaces under Group Action

Authors: Zhanjiang Ji
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Firstly, we introduce the definitions ofG-asymptotic tracking property, G-asymptotic average tracking property, and G-quasi-weak almost-periodic point. Secondly, we study their dynamical properties and characteristics. The results obtained improve the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space and provide the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, and computer science.


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## Body

## 1. Introduction

LetX,d be a metric space and let f:X⟶X be a continuous map. The sequence xii≥0 is called δ-pseudo orbit of f if, for any i≥0, we have dfxi,xi+1<δ. The sequence xii≥0 is said to be ε-shadowed by some point y∈Y if, for any i≥0, we have dfy,xi<ε. The map f is said to have the shadowing property if for each ε>0 there exists δ>0 such that, for any δ-pseudo orbit xii≥0 of f , there exists a point y∈X such that the sequence xii≥0 is ε-shadowed by the point y (see [1]).The shadowing property plays an important role in ergodic theory and topological dynamical systems, which has attracted the attention of many scholars in recent years. The results are shown in literature [1–13]. In 1980, the concept of average tracking property was introduced by Blank [1] and it was proved that some perturbed hyperbolic systems have the average tracking property. In Wang and Zeng [2], the concept of q¯-average tracking property is given and the q¯-average tracking property means chain transitivity under some conditions. Fakhari and Ghane [3] introduced the concept of ergodic tracking property and discussed its dynamical properties. Liang and Li [4] discussed the relation between the shift mapping σ and the self-mapping f about the asymptotic tracking property in the inverse limit space. By the definitions of the G-tracking property of Ekta and Tarun [10], we introduce the concepts of G-asymptotic tracking property. By the definition of the asymptotic average tracking property of Gu [13], we give G-asymptotic average tracking property. The following conclusions are obtained: (1) The self-mapping f has G-asymptotic tracking property if and only if the shift mapping σ has G¯-asymptotic tracking property. (2) The self-mapping f has G-asymptotic average tracking property if and only if the shift mapping σ has G¯-asymptotic average tracking property. Thus, we generalize the conclusion of Liang and Li [4]. The quasi-weak almost-periodic point is an important concept in the dynamical system, which has also attracted the attention of many scholars. The relevant results are shown by Ma [14] and Zhou and He [15]. Ma [14] proved QWσ=lim←QWf,f. In this paper, we introduce the concept of G-quasi-weak almost-periodic point and study its topological structure in the inverse limit space under the action group. We obtain QWG¯σ=lim←QWGf,f and generalize the result of Ma [14].


## 2.G-Asymptotic Tracking Property

In this section, we will prove Theorem1. For the convenience of the reader, we give the concept used in this section. Now we start with the following definitions.Definition 1.
(see [10]). Let X,d be a metric space, let G be a topological group, and let θ:G×X⟶X be a continuous map. The triple X,G,θ is called a metric G-space if the following conditions are satisfied:(1)
θe,x=x for all x∈X and e is the identity of G(2)
θg1,θg2,x=θg1g2,x for all x∈X and all g1,g2∈G
IfX,d is a compact metric space, then X,G,θ is also said to be a compact metric G-space. For the convenience of writing, θg,x is usually abbreviated as gx.Definition 2.
(see [10]). Let X,d be a metric G-space and let f:X⟶X be a continuous map. The map f is said to be an equivariant map if we have fpx=pfx for all x∈X and p∈G.Definition 3.
(see [16]). LetX,d be a metric G-space and let f:X⟶X be a continuous map. lim←X,f is said to be the inverse limit space if we write lim←X,f=x0,x1,x2,…:fxi+1=xi,i≥0. lim←X,f is denoted by Xf in this paper.
The metricd¯ in Xf is defined by d¯x¯,y¯=∑i=0∞dxi,yi/2i, where x¯=x0,x1,x2…∈Xf and y¯=y0,y1,y2…∈Xf. The shift mapping σ:Xf⟶Xf is defined by σx¯=fx0,x0,x1…. Thus, Xf,d¯ is a compact metric space and the shift mapping σ is a homeomorphism map.Definition 4.
(see [16]). LetX,d be a metric G-space and let f:X⟶X be an equivariant map. Write G¯=g,g,g…:g∈G and G∞=∏i=0∞Gi, where Gi=G . The map θ:G¯×Xf⟶Xf is defined by 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.
LetXf,G¯,d¯,σ and X,G,d,f be shown as above. The space Xf,G¯,d¯,σ is called the inverse limit space of X,G,d,f under group action.Definition 5.
(see [10]). LetX,d be a metric G-space and let f:X⟶X be a continuous map. The sequence xii≥0 is called be G,δ-pseudo orbit of f if for any i≥0 there exists ti∈G such that dtifxi,xi+1<δ.Definition 6.
(see [10]). Let X,d be a metric G-space and let f:X⟶X be a continuous map. The sequence xii≥0 is said to be G,δ-shadowed by some point y∈Y if for any i≥0 there exists ti∈G such that dfy,tixi<δ.Definition 7.
(see [10]). Let X,d be a metric G-space and let f:X⟶X be a continuous map. The map f has G-tracking property if for each ε>0 there exists δ>0 such that, for any G,δ-pseudo orbit xii≥0 of f , there exists a point y∈Y such that the sequence xii≥0 is G,ε-shadowed by the point y.Remarks 1.
By the definitions of theG-tracking property, we will give the concept of G-asymptotic tracking property.Definition 8.
LetX,d be a metric G-space and let f be a continuous map from X to X. The map f has G-asymptotic tracking property if for each ε>0 there exists δ>0 such that, for any G,δ-pseudo orbit xii≥0 of f , there exists a point y∈Y and l≥0 such that the sequence xii=l∞ is G,ε-shadowed by the point y.Now, we start to prove Theorem1.Theorem 1.
LetXf,G¯,d¯,σ be the inverse limit space of X,G,d,f under group action. If the map f:X⟶X is an equivalent surjection, we have that the self-mapping f has the G-asymptotic tracking property if and only if the shift mapping σ has the G¯-asymptotic tracking property.Proof.
⇒ Suppose that the map f has the G-asymptotic tracking property. Since X is compact, we write M=diamX. For any ε>0 , let m>0 satisfy M/2m<ε/2. According to the fact that the map f is uniformly continuous, for any 0≤i≤m, there exists 0<δ1<ε/4 such that du,v<δ1 implies(1)dfiu,fiv<ε4.
By the definition ofG-asymptotic tracking property, for δ1>0, there exists 0<δ2<δ1 and l1≥0 such that the map f satisfies the condition of the G-asymptotic tracking property. Let y¯nn=0∞ be G¯,δ2/2m-pseudo orbit, where y¯n=yn0,yn1,yn2…∈Xf. Then, for any n≥0, there exists g¯n=gn,gn,gn…∈G¯ such that(2)d¯g¯nσy¯n,y¯n+1<δ2.
Hence, we have that(3)dgnfynm,yn+1m2m<δ22m.
That is,(4)dgnfynm,yn+1m<δ2.
Soynmn=0∞ are G,δ2-pseudo orbit of the map f. Thus, for every n≥0, there exists x∈X, tn∈G, and l1≥0 such that(5)dfnx,tnyn+l1m<δ1.
According to (1) and the equivalent definition, for any n≥0 and 0≤i≤m, it follows that(6)dfn+ix,tnyn+l1m−i<ε4.
Because of the surjectivity of the mapf, we can choose x¯=fmx,fm−1x,fm−2x,…,x,…∈Xf and t¯n=tn,tn,tn…∈G¯. Then we have that(7)d¯σnx¯,t¯ny¯n+l1<∑i=0mε2i+2+∑i=m+1∞M2i<ε.
So, the mapf has the G-asymptotic tracking property.
⇐ Next we suppose that the shift mapping σ has the G¯-asymptotic tracking property. Let n0>0. For any η>0, there exists 0<δ3<η and l2≥0 such that, for any G,δ3-pseudo orbit x¯kk=1∞ of the shift mapping σ, we have that x¯kk=l2∞ is G¯,η/2n0-shadowed by the point z¯ and M/2n0<δ3/2. Because the map f is uniformly continuous, it follows that, for any 0≤i≤n0, there exists 0<δ4<δ3/4 such that du,v<δ4 implies(8)dfiu,fiv<δ34.
Now suppose thatxkk=0∞ are G,δ4-pseudo orbit of the map f. Then, for any k≥0, there exists sk∈G such that(9)dskfxk,xk+1<δ4.
According to (8) and the equivalent definition, for any k≥0 and 0≤i≤n0 , we have that(10)dskfi+1xk,fixk+1<δ34.
According to the surjectivity of the mapf, we can choose x¯k=fn0xk,fn0−1xk,…,fxk,xk,…∈Xf and s¯k=sk,sk,sk,…∈G¯. Combining GX=X and (10), when k≥0, we have(11)d¯s¯kσx¯k,x¯k+1<∑i=0n0δ32i+2+∑i=n0+1∞M2n0<δ3.
So,x¯kk=0∞ is G¯,δ3-pseudo orbit of the shift mapping σ. By the definition of G¯-asymptotic tracking property of the map σ, for any k≥0, there exists z¯=z0,z1,z2…∈Xf, l2≥0, and p¯k=pk,pk,pk…∈G¯ such that(12)d¯σkz¯,p¯kx¯k+l2<η2n0.
Thus, we have that(13)dfkzn0,pkxk+l22n0<η2n0.
Hence, we have that(14)dfkzn0,pkxk+l2<η.
So the mapf has the G-asymptotic tracking property. Thus, we end the proof.


## 3.G-Asymptotic Average Tracking Property

Definition 9.
(see [17]). Let J⊂N. If(15)limn⟶∞CardJ∩0,1,…,n−1n=0,then the set J is said to be a zero density set.Definition 10.
(see [13]). Let X,d be a metric space and let f:X⟶X be a continuous map. The sequence xii≥0 in X is called an asymptotic average pseudoorbit of the map f if(16)limn⟶∞1n∑i=0i=n−1dfxi,xi+1=0.Definition 11.
(see [13]). Let X,d be a metric space and let f:X⟶X be a continuous map. The map f is considered to have the asymptotic average tracking property if, for any asymptotic average pseudoorbit xii≥0, there exists a point z in X such that(17)limn⟶∞1n∑i=0i=n−1dfiz,xi=0.Remarks 2.
According to the definition of asymptotic average tracking property, we will give the concept ofG-asymptotic average tracking property.Definition 12.
LetX,d be a metric G-space and let f:X⟶X be a continuous map. The sequence xii≥0 in X is called an G-asymptotic average pseudoorbit if there exists ti∈G such that(18)limn⟶∞1n∑i=0i=n−1dtifxi,xi+1=0.Definition 13.
LetX,d be a metric G-space and let f:X⟶X be a continuous map. The map f is considered to have the G-asymptotic average tracking property if for any G-asymptotic average pseudo‐orbit xii≥0 there exists a point z in X and ti∈G such that(19)limn⟶∞1n∑i=0i=n−1dfiz,tixi=0.Next, we give Lemma 1, which will be used in this section.Lemma 1 (see [17]).
Letaii=0∞ be nonnegative real bounded sequence. Then the following conclusions are equivalent:(1)
limn⟶∞1/n∑i=0i=n−1ai=0(2)
There exists a zero density setJ such that limi∉J,i⟶∞ai=0Now we will prove Theorem2 by Lemma 1.Theorem 2.
LetXf,G¯,d¯,σ be the inverse limit space of X,G,d,f under group action. If the map f:X⟶X is an equivalent surjection, we have that the self-mapping f has the G-asymptotic average tracking property if and only if the shift mapping σ has G¯-asymptotic average tracking property.Proof.
⇒ Suppose that the map f has the G-asymptotic average tracking property. Since X is compact, we write M=diamX. For any ε>0 , let m1>0 satisfy(20)M2m1<ε2.
According to the fact that the mapf is uniformly continuous, for any 0≤k≤m1, there exists 0<δ<ε/4 such that du,v<δ implies(21)dfku,fkv<ε4.
Lety¯ii=0∞ be G¯-asymptotic average pseudo‐orbit, where y¯i=yi0,yi1,yi2…∈Xf. Then there exists g¯i=gi,gi,gi…∈G¯ such that(22)limn⟶∞1n∑i=0i=n−1d¯g¯iσy¯i,y¯i+1=0.
By Lemma1, there exists a zero density set J1 such that(23)limi∉J1,i⟶∞d¯g¯iσy¯i,y¯i+1=0.
Then, there existsN1∈N+ such that when i>N1 and i∉J1, we have that(24)d¯g¯iσy¯i,y¯i+1<ε2m1.
Thus, we have that(25)dgifyim1,yi+1m12m1<ε2m1.
That is,(26)dgifyim1,yi+1m1<ε.
Hence, we have that(27)limi∉J1,i⟶∞dgifyim1,yi+1m1=0.
According to Lemma1, we have that(28)limn⟶∞1n∑i=0i=n−1dgifyim1,yi+1m1=0.
Hence, the sequenceyim1i=0∞ is G-asymptotic average pseudoorbit of the map f. By the definition of G-asymptotic average tracking property of the map f, there exist z∈X and ti∈G such that(29)limn⟶∞1n∑i=0i=n−1dfiz,tiyim1=0.
According to Lemma1, there exists a zero density set J2 such that(30)limi∉J1,i⟶∞dfiz,tiyim1=0.
Thus, there existsN2∈N+ such that when i>N2 and i∉J2, we have that(31)dfiz,tiyim1<δ.
According to the equivalent definition of the mapf and (21), for any 0≤k≤m1, we have that(32)dfk+iz,tiyim1−k<ε4.
According to the surjectivity of the mapf, we can choose z¯=fm1z,fm1−1z,fm1−2z,…,z,…∈Xf and t¯i=ti,ti,ti…∈G¯. By (20) and (32), when i>N2 and i∉J2, we have that(33)d¯σiz¯,t¯iy¯i<ε.
So,(34)limi∉J2,i⟶∞d¯σiz¯,t¯iy¯i=0.
Combining with Lemma1, we have that(35)limn⟶∞1n∑i=0i=n−1d¯σiz¯,t¯iy¯i=0.
So, the shift mappingσ has the G¯-asymptotic average tracking property.
⇐ Next we suppose that the shift mapping σ has the G¯-asymptotic average tracking property. For any η>0, let m2>0 such that(36)M2m2<η2.
Because the mapf is uniformly continuous, it follows that, for any 0≤k≤m2, there exists 0<δ′<ε/4 such that du,v<δ′ implies(37)dfku,fkv<η4.
Now suppose thatxii=0∞ is G-asymptotic average pseudoorbit of the map f. Then there exists pi∈G such that(38)limn⟶∞1n∑i=0i=n−1dpifxi,xi+1=0.
By Lemma1, there exists a zero density set J3 such that(39)limi∉J3,i⟶∞dpifxi,xi+1=0.
Hence, there existsN3∈N+ such that when i>N3 and i∉J3, we have that(40)dpifxi,xi+1<δ′.
According to the equivalent definition of the mapf and (37), for any 0≤k≤m2, we have that(41)dpifk+1xi,fkxi+1<η4.
According to the surjectivity of the mapf, we can choose x¯i=fm2xi,fm2−1xi,…,fxi,xi,…∈Xf and p¯i=pi,pi,pi…∈G¯, where i≥0. By (36) and (41), when i>N3 and i∉J3, we have that(42)d¯p¯iσx¯i,x¯i+1<∑i=0m2η2i+2+∑i=m2+1∞M2i<η.
Hence, we have(43)limi∉J3,i⟶∞d¯p¯iσx¯i,x¯i+1=0.
Combining with Lemma1, we can get that(44)limn⟶∞1n∑i=0i=n−1d¯p¯iσx¯i,x¯i+1=0.
So,x¯ii=0∞ is G¯-average pseudoorbit. By the definition of G¯-asymptotic tracking property, there exist y¯=y0,y1,y2,…∈Xf and s¯i=si,si,si,…∈G¯ such that(45)limn⟶∞1n∑i=0i=n−1d¯σiy¯,s¯ix¯i=0.
By Lemma1, there exists a zero density set J4 such that(46)limi∉J4,i⟶∞d¯σiy¯,s¯ix¯i=0.
Thus, we have that(47)limi∉J4,i⟶∞dfiym2,sixi=0.
By Lemma1, we have that(48)limn⟶∞1n∑i=0i=n−1dfiym2,sixi=0.
So, the mapf has the G-asymptotic average tracking property. Thus, we end the proof.


## 4.G-Quasi-Weak Almost-Periodic Point

First, we give the definition that is used in this section. Second, we give the proof process of Theorem3.Definition 14.
(see [18]). Let X,d be a metric space and let f:X⟶X be a continuous map. A point x∈X is considered to be a quasi-weak almost-periodic point if, for any ε>0, there exists N>0 and nonnegative increasing integers sequence nii≥0 such that Cardr:frx∈Bx,ε,0≤r<niN≥ni. The quasi-weak almost-periodic point set of the map f is denoted by QWf.Remarks 3.
According to the concept of quasi-weak almost-periodic point, we will give the concept ofG-quasi-weak almost-periodic point.Definition 15.
LetX,d be a metric G-space and let f:X⟶X be a continuous map. A point x∈X is considered to be a G-quasi-weak almost-periodic point if, for any ε>0, there exists N>0, nonnegative increasing integers sequence nii≥0, and ti∈G such that Cardr:tifrx∈Bx,ε,0≤r<niN≥ni. the G-quasi-weak almost-periodic point set of the map f is denoted by QWGf.Now, we start to prove Theorem3.Theorem 3.
LetXf,G¯,d¯,σ be the inverse limit space of X,G,d,f under group action. If the map f:X⟶X is an equivalent surjection, we have that QWG¯σ=lim←QWGf,f.Proof.
⇒ Suppose that x¯=x0,x1,x2…∈QWG¯σ. According to the definition of the G-quasi-weak almost-periodic point, for any ε>0, there exists N>0, nonnegative integer sequence nii≥0, and g¯i=g0,g1,g2…∈G¯ such that(49)Cardr:g¯iσrx¯∈Bx¯,ε2i,0≤r<niN≥ni.
Write(50)Ani=r:g¯iσrx¯∈Bx¯,ε2i,0≤r<niN,Bni=r:gifrxi∈Bxi,ε,0≤r<niN.
Now, suppose thatr∈Ani. Then we have that(51)d¯g¯iσrx¯,x¯<ε2i.
Hence, we have that(52)dgifrxi,xi<ε.
Thus, we can obtainr∈Bni. So, we get that(53)Cardr:gifrxi∈Bxi,ε,0≤r<niN≥ni.
So,xi∈QWGf. Hence, QWG¯σ⊂lim←QWGf,f.
⇐ Suppose that y¯=y0,y1,y2,…∈lim←QWGf,f. Then, for any i≥0 , we have yi∈QWGf. Since X is compact, we write M=diamX. For any η>0, let m>0 satisfy(54)M2m<η2.
According toyi∈QWGf, there exists N>0, nonnegative integer sequence mii≥0, and ti∈G such that(55)Cardr:tifryi∈Byi,η,0≤r<miN≥mi.
Lett¯=t0,t1,t2,…. Write(56)Cmi=r:tifryi∈Byi,η4,0≤r<miN,Dmi=r:t¯σry¯∈By¯,η,0≤r<miN.
Suppose thatr∈Cmi. Then we have that(57)dtifryi,yi<η4.
Thus, we have that(58)d¯t¯σry¯,y¯=∑i=0mdtifryi,yi2i+∑i=m+1∞dtifryi,yi2i<∑i=0m12i⋅η4+∑i=m+1∞M2i<η2+η2=η.
Thus, we can obtainr∈Dmi. So we get that(59)Cardr:t¯σry¯∈By¯,η,0≤r<miN≥mi.
Hence,y¯∈QWG¯σ. Then lim⟶QWGf,f⊂QWG¯σ. This completes the proof.


## 5. Conclusions

In this paper, we study dynamical properties and characteristics ofG-asymptotic tracking property, G-asymptotic average tracking property, and G-quasi-weak almost-periodic point. The results obtained can generalize the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space. Most importantly, the paper provides the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, nature, and society.



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