DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi10.1155/2021/55763895576389Research ArticleThe G-Asymptotic Tracking Property and G-Asymptotic Average Tracking Property in the Inverse Limit Spaces under Group Actionhttps://orcid.org/0000-0002-0836-2968JiZhanjiang123VaidyanathanSundarapandian1School of Data Science and Software EngineeringWuzhou UniversityWuzhou 54300Chinagxuwz.edu.cn2Guangxi Colleges and Universities Key Laboratory of Image Processing and Intelligent Information SystemWuzhou UniversityWuzhou 54300Chinagxuwz.edu.cn3Guangxi Colleges and Universities Key Laboratory of Professional Software TechnologyWuzhou UniversityWuzhou 54300Chinagxuwz.edu.cn2021125202120212812021224202112520212021Copyright © 2021 Zhanjiang Ji.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.

Firstly, we introduce the definitions of G-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.

Natural Science Foundation of Guangxi Province2020JJA110021Wuzhou University2020B007
1. Introduction

LetX,d be a metric space and let f:XX be a continuous map. The sequence xii0 is called δ-pseudo orbit of f if, for any i0, we have dfxi,xi+1<δ. The sequence xii0 is said to be ε-shadowed by some point yY if, for any i0, 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 xii0 of f , there exists a point yX such that the sequence xii0 is ε-shadowed by the point y (see ).

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 . In 1980, the concept of average tracking property was introduced by Blank  and it was proved that some perturbed hyperbolic systems have the average tracking property. In Wang and Zeng , the concept of q¯-average tracking property is given and the q¯-average tracking property means chain transitivity under some conditions. Fakhari and Ghane  introduced the concept of ergodic tracking property and discussed its dynamical properties. Liang and Li  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 , we introduce the concepts of G-asymptotic tracking property. By the definition of the asymptotic average tracking property of Gu , 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 . 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  and Zhou and He . Ma  proved QWσ=limQWf,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=limQWGf,f and generalize the result of Ma .

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M46"><mml:mi>G</mml:mi></mml:math></inline-formula>-Asymptotic Tracking Property

In this section, we will prove Theorem 1. For the convenience of the reader, we give the concept used in this section. Now we start with the following definitions.

Definition 1.

(see ). Let X,d be a metric space, let G be a topological group, and let θ:G×XX be a continuous map. The triple X,G,θ is called a metric G-space if the following conditions are satisfied:

θe,x=x for all xX and e is the identity of G

θg1,θg2,x=θg1g2,x for all xX and all g1,g2G

If X,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 ). Let X,d be a metric G-space and let f:XX be a continuous map. The map f is said to be an equivariant map if we have fpx=pfx for all xX and pG.

Definition 3.

(see ). LetX,d be a metric G-space and let f:XX be a continuous map. limX,f is said to be the inverse limit space if we write limX,f=x0,x1,x2,:fxi+1=xi,i0. limX,f is denoted by Xf in this paper.

The metric d¯ in Xf is defined by d¯x¯,y¯=i=0dxi,yi/2i, where x¯=x0,x1,x2Xf and y¯=y0,y1,y2Xf. The shift mapping σ:XfXf 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 ). LetX,d be a metric G-space and let f:XX be an equivariant map. Write G¯=g,g,g:gG and G=i=0Gi, where Gi=G . The map θ:G¯×XfXf is defined by θg¯,x¯=g¯x¯=gx0,gx1,gx2,, where g¯=g,g,gG¯ and x¯=x0,x1,x2Xf. Then Xf,G¯,θ is a metric G¯-space.

Let Xf,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 ). LetX,d be a metric G-space and let f:XX be a continuous map. The sequence xii0 is called be G,δ-pseudo orbit of f if for any i0 there exists tiG such that dtifxi,xi+1<δ.

Definition 6.

(see ). Let X,d be a metric G-space and let f:XX be a continuous map. The sequence xii0 is said to be G,δ-shadowed by some point yY if for any i0 there exists tiG such that dfy,tixi<δ.

Definition 7.

(see ). Let X,d be a metric G-space and let f:XX 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 xii0 of f , there exists a point yY such that the sequence xii0 is G,ε-shadowed by the point y.

Remarks 1.

By the definitions of the G-tracking property, we will give the concept of G-asymptotic tracking property.

Definition 8.

Let X,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 xii0 of f , there exists a point yY and l0 such that the sequence xii=l is G,ε-shadowed by the point y.

Now, we start to prove Theorem 1.

Theorem 1.

Let Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f under group action. If the map f:XX 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 0im, there exists 0<δ1<ε/4 such that du,v<δ1 implies(1)dfiu,fiv<ε4.

By the definition of G-asymptotic tracking property, for δ1>0, there exists 0<δ2<δ1 and l10 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,yn2Xf. Then, for any n0, there exists g¯n=gn,gn,gnG¯ 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.

So ynmn=0 are G,δ2-pseudo orbit of the map f. Thus, for every n0, there exists xX, tnG, and l10 such that(5)dfnx,tnyn+l1m<δ1.

According to (1) and the equivalent definition, for any n0 and 0im, it follows that(6)dfn+ix,tnyn+l1mi<ε4.

Because of the surjectivity of the map f, we can choose x¯=fmx,fm1x,fm2x,,x,Xf and t¯n=tn,tn,tnG¯. Then we have that(7)d¯σnx¯,t¯ny¯n+l1<i=0mε2i+2+i=m+1M2i<ε.

So, the map f 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 l20 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 0in0, there exists 0<δ4<δ3/4 such that du,v<δ4 implies(8)dfiu,fiv<δ34.

Now suppose that xkk=0 are G,δ4-pseudo orbit of the map f. Then, for any k0, there exists skG such that(9)dskfxk,xk+1<δ4.

According to (8) and the equivalent definition, for any k0 and 0in0 , we have that(10)dskfi+1xk,fixk+1<δ34.

According to the surjectivity of the map f, we can choose x¯k=fn0xk,fn01xk,,fxk,xk,Xf and s¯k=sk,sk,sk,G¯. Combining GX=X and (10), when k0, we have(11)d¯s¯kσx¯k,x¯k+1<i=0n0δ32i+2+i=n0+1M2n0<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 k0, there exists z¯=z0,z1,z2Xf, l20, and p¯k=pk,pk,pkG¯ 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 map f has the G-asymptotic tracking property. Thus, we end the proof.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M253"><mml:mi>G</mml:mi></mml:math></inline-formula>-Asymptotic Average Tracking PropertyDefinition 9.

(see ). Let JN. If(15)limnCardJ0,1,,n1n=0,then the set J is said to be a zero density set.

Definition 10.

(see ). Let X,d be a metric space and let f:XX be a continuous map. The sequence xii0 in X is called an asymptotic average pseudoorbit of the map f if(16)limn1ni=0i=n1dfxi,xi+1=0.

Definition 11.

(see ). Let X,d be a metric space and let f:XX be a continuous map. The map f is considered to have the asymptotic average tracking property if, for any asymptotic average pseudoorbit xii0, there exists a point z in X such that(17)limn1ni=0i=n1dfiz,xi=0.

Remarks 2.

According to the definition of asymptotic average tracking property, we will give the concept of G-asymptotic average tracking property.

Definition 12.

Let X,d be a metric G-space and let f:XX be a continuous map. The sequence xii0 in X is called an G-asymptotic average pseudoorbit if there exists tiG such that(18)limn1ni=0i=n1dtifxi,xi+1=0.

Definition 13.

Let X,d be a metric G-space and let f:XX 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 xii0 there exists a point z in X and tiG such that(19)limn1ni=0i=n1dfiz,tixi=0.

Next, we give Lemma 1, which will be used in this section.

Lemma 1 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Let aii=0 be nonnegative real bounded sequence. Then the following conclusions are equivalent:

limn1/ni=0i=n1ai=0

There exists a zero density set J such that limiJ,iai=0

Now we will prove Theorem 2 by Lemma 1.

Theorem 2.

Let Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f under group action. If the map f:XX 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 map f is uniformly continuous, for any 0km1, there exists 0<δ<ε/4 such that du,v<δ implies(21)dfku,fkv<ε4.

Let y¯ii=0 be G¯-asymptotic average pseudo‐orbit, where y¯i=yi0,yi1,yi2Xf. Then there exists g¯i=gi,gi,giG¯ such that(22)limn1ni=0i=n1d¯g¯iσy¯i,y¯i+1=0.

By Lemma 1, there exists a zero density set J1 such that(23)limiJ1,id¯g¯iσy¯i,y¯i+1=0.

Then, there exists N1N+ such that when i>N1 and iJ1, 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)limiJ1,idgifyim1,yi+1m1=0.

According to Lemma 1, we have that(28)limn1ni=0i=n1dgifyim1,yi+1m1=0.

Hence, the sequence yim1i=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 zX and tiG such that(29)limn1ni=0i=n1dfiz,tiyim1=0.

According to Lemma 1, there exists a zero density set J2 such that(30)limiJ1,idfiz,tiyim1=0.

Thus, there exists N2N+ such that when i>N2 and iJ2, we have that(31)dfiz,tiyim1<δ.

According to the equivalent definition of the map f and (21), for any 0km1, we have that(32)dfk+iz,tiyim1k<ε4.

According to the surjectivity of the map f, we can choose z¯=fm1z,fm11z,fm12z,,z,Xf and t¯i=ti,ti,tiG¯. By (20) and (32), when i>N2 and iJ2, we have that(33)d¯σiz¯,t¯iy¯i<ε.

So,(34)limiJ2,id¯σiz¯,t¯iy¯i=0.

Combining with Lemma 1, we have that(35)limn1ni=0i=n1d¯σiz¯,t¯iy¯i=0.

So, the shift mapping σ 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 map f is uniformly continuous, it follows that, for any 0km2, there exists 0<δ<ε/4 such that du,v<δ implies(37)dfku,fkv<η4.

Now suppose that xii=0 is G-asymptotic average pseudoorbit of the map f. Then there exists piG such that(38)limn1ni=0i=n1dpifxi,xi+1=0.

By Lemma 1, there exists a zero density set J3 such that(39)limiJ3,idpifxi,xi+1=0.

Hence, there exists N3N+ such that when i>N3 and iJ3, we have that(40)dpifxi,xi+1<δ.

According to the equivalent definition of the map f and (37), for any 0km2, we have that(41)dpifk+1xi,fkxi+1<η4.

According to the surjectivity of the map f, we can choose x¯i=fm2xi,fm21xi,,fxi,xi,Xf and p¯i=pi,pi,piG¯, where i0. By (36) and (41), when i>N3 and iJ3, we have that(42)d¯p¯iσx¯i,x¯i+1<i=0m2η2i+2+i=m2+1M2i<η.

Hence, we have(43)limiJ3,id¯p¯iσx¯i,x¯i+1=0.

Combining with Lemma 1, we can get that(44)limn1ni=0i=n1d¯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)limn1ni=0i=n1d¯σiy¯,s¯ix¯i=0.

By Lemma 1, there exists a zero density set J4 such that(46)limiJ4,id¯σiy¯,s¯ix¯i=0.

Thus, we have that(47)limiJ4,idfiym2,sixi=0.

By Lemma 1, we have that(48)limn1ni=0i=n1dfiym2,sixi=0.

So, the map f has the G-asymptotic average tracking property. Thus, we end the proof.

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M400"><mml:mi>G</mml:mi></mml:math></inline-formula>-Quasi-Weak Almost-Periodic Point

First, we give the definition that is used in this section. Second, we give the proof process of Theorem 3.

Definition 14.

(see ). Let X,d be a metric space and let f:XX be a continuous map. A point xX is considered to be a quasi-weak almost-periodic point if, for any ε>0, there exists N>0 and nonnegative increasing integers sequence nii0 such that Cardr:frxBx,ε,0r<niNni. 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 of G-quasi-weak almost-periodic point.

Definition 15.

Let X,d be a metric G-space and let f:XX be a continuous map. A point xX is considered to be a G-quasi-weak almost-periodic point if, for any ε>0, there exists N>0, nonnegative increasing integers sequence nii0, and tiG such that Cardr:tifrxBx,ε,0r<niNni. the G-quasi-weak almost-periodic point set of the map f is denoted by QWGf.

Now, we start to prove Theorem 3.

Theorem 3.

Let Xf,G¯,d¯,σ be the inverse limit space of X,G,d,f under group action. If the map f:XX is an equivalent surjection, we have that QWG¯σ=limQWGf,f.

Proof.

Suppose that x¯=x0,x1,x2QWG¯σ. According to the definition of the G-quasi-weak almost-periodic point, for any ε>0, there exists N>0, nonnegative integer sequence nii0, and g¯i=g0,g1,g2G¯ such that(49)Cardr:g¯iσrx¯Bx¯,ε2i,0r<niNni.

Write(50)Ani=r:g¯iσrx¯Bx¯,ε2i,0r<niN,Bni=r:gifrxiBxi,ε,0r<niN.

Now, suppose that rAni. Then we have that(51)d¯g¯iσrx¯,x¯<ε2i.

Hence, we have that(52)dgifrxi,xi<ε.

Thus, we can obtain rBni. So, we get that(53)Cardr:gifrxiBxi,ε,0r<niNni.

So, xiQWGf. Hence, QWG¯σlimQWGf,f.

Suppose that y¯=y0,y1,y2,limQWGf,f. Then, for any i0 , we have yiQWGf. Since X is compact, we write M=diamX. For any η>0, let m>0 satisfy(54)M2m<η2.

According to yiQWGf, there exists N>0, nonnegative integer sequence mii0, and tiG such that(55)Cardr:tifryiByi,η,0r<miNmi.

Let t¯=t0,t1,t2,. Write(56)Cmi=r:tifryiByi,η4,0r<miN,Dmi=r:t¯σry¯By¯,η,0r<miN.

Suppose that rCmi. Then we have that(57)dtifryi,yi<η4.

Thus, we have that(58)d¯t¯σry¯,y¯=i=0mdtifryi,yi2i+i=m+1dtifryi,yi2i<i=0m12iη4+i=m+1M2i<η2+η2=η.

Thus, we can obtain rDmi. So we get that(59)Cardr:t¯σry¯By¯,η,0r<miNmi.

Hence, y¯QWG¯σ. Then limQWGf,fQWG¯σ. This completes the proof.

5. Conclusions

In this paper, we study dynamical properties and characteristics of G-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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the NSF of Guangxi Province (2020JJA110021) and construction project of Wuzhou University of China (2020B007).