# Dynamical Property of the Shift Map under Group Action

Authors: Zhanjinag Ji
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Firstly, we introduced the concept ofG‐Lipschitz tracking property, G‐asymptotic average tracking property, and G‐periodic tracking property. Secondly, we studied their dynamical properties and topological structure and obtained the following conclusions: (1) let X,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐asymptotic average tracking property; (2) let X,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐Lipschitz tracking property; (3) let X,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐periodic tracking property. The above results make up for the lack of theory of G‐Lipschitz tracking property, G‐asymptotic average tracking property, and G‐periodic tracking property in infinite product space under group action.


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## Body

## 1. Introduction

At present, shadowing property has gradually become an important theory and concept in dynamical system. Relevant results are seen in [1–11]. For example, Wang and Zeng [1] proved that if f has q‐average tracking property, then f is chain transitivity under some conditions. Wu [2] showed that f has tracking property and σ has tracking property are equivalent. Ji et al. [3] proved that f∘g has the Lipschitz shadowing property and σf∘g has the Lipschitz shadowing property are equivalent in the double inverse limit space. Ahmadi and Hosseini [4] gave that δ‐ergodic pseudo-orbit of a system means chain mixing. Fakhari and Ghane [5] introduced some kind of specification property. Niu [6] showed that the average-shadowing property and dense minimal set means weakly mixing. Oprocha et al. [7] gave equivalent conditions for shadowing. Hossein and Reza [8] investigated the relations of various shadowing. Pierre and Thibault [9] studied shadowing and periodic shadowing properties.Firstly, we introduced the concept ofG‐Lipschitz tracking property, G‐asymptotic average tracking property, and G‐periodic tracking property. Secondly, we studied their dynamical properties and topological structure and obtained the following conclusions:Theorem 1.
LetX,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐asymptotic average tracking property.Theorem 2.
LetX,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐Lipschitz tracking property.Theorem 3.
LetX,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐periodic tracking property.


## 2.G¯‐Asymptotic Average Shadowing Property of σ

The concept of metricG‐space is shown in [12]. We can find the concept that d is invariant to G in [13]. The concepts of zero density set and G‐asymptotic average tracking property are seen in [14, 15]. The concept of the shift map σ and the metric d¯ can be found in [16].Write(1)G¯=g,g,g⋯:g∈G,G∞=∏i=0∞Gi,where Gi=G,i≥0.According to [17, 18], X¯,G¯,d¯ is compact metric G¯‐space. This paper mainly studies dynamical properties of σ in compact metric G¯‐space.Theorem 4.
LetX,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐asymptotic average tracking property.Proof.
Lety¯ii≥0y¯i=y0i,y1i,y2i,⋯ be G¯‐asymptotic average pseudo orbit of σ. Then, for any nonnegative integer j≥0, there exists t¯j∈G¯ satisfying.
(2)limn⟶∞1n∑j=0j=n−1d¯t¯jσy¯j,y¯j+1=0,where t¯j=t0j,t1j,t2j,⋯.According to [14], we can choose a zero density set J satisfying
(3)limi⟶∞,i∉Jd¯t¯iσy¯i,y¯i+1=0.So for anyε>0, there exists a positive integer m>0 such that i≥m and i∉J implies
(4)d¯t¯iσy¯i,y¯i+1<ε.So, for anyi≥m, i∉J, and k≥1, it follows that
(5)dtk−1iyki,yk−1i+12k−1<ε.Thus, we can obtain the following inequality(6)dtk−1iyki,yk−1i+1<2k−1ε,dtk−2i+1yk−1i+1,yk−2i+2<2k−2ε,dtk−3i+2yk−2i+2,yk−3i+3<2k−3ε,⋯,dt1i+k−2y2i+k−2,y1i+k−1<21ε,dt0i+k−1y1i+k−1,y0i+k<20ε.Since the metricd is invariant to G, we can have that
(7)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,t0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1<2k−1ε,dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1,t0i+k−1t1i+k−2⋯tk−3i+2yk−2i+2<2k−2ε,dt0i+k−1t1i+k−2⋯tk−4i+3tk−3i+2yk−2i+2,t0i+k−1t1i+k−2⋯tk−4i+3yk−3i+3<2k−3ε,⋯,dt0i+k−1t1i+k−2y2i+k−2,t0i+k−1y1i+k−1<21ε,dt0i+k−1y1i+k−1,y0i+k<20ε.Thus, wheni≥m and i∉J for any positive integer k≥1, we can get that
(8)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,y0i+k<dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,t0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1+dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1,t0i+k−1t1i+k−2⋯tk−3i+2yk−2i+2+dt0i+k−1t1i+k−2⋯tk−4i+3tk−3i+2yk−2i+2,t0i+k−1t1i+k−2⋯tk−4i+3yk−3i+3+⋯⋯+dt0i+k−1t1i+k−2y2i+k−2,t0i+k−1y1i+k−1+dt0i+k−1y1i+k−1,y0i+k+<2k−1+2k−2+⋯+20ε<2kε.That is(9)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,y0i+k2k<ε.Lety¯=y00,y01,y02,⋯∈X¯ and g¯i=e,t0i,t0i+1t1i,t0i+2t1i+1t2i,⋯, where i≥m. Then, for any i≥m and i∉J, it follows that
(10)d¯σiy¯,g¯iy¯i<ε.Hence,(11)limi⟶∞,i∉Jd¯σiy¯,g¯iy¯i=0.According to [14], we have that
(12)limn⟶∞1n∑i=0i=n−1d¯σy¯,g¯iy¯i=0..Hence,σ has G¯‐asymptotic average tracking property.


## 3.G¯‐Lipschitz Tracking Property of σ

We can find the definition ofG‐Lipschitz tracking property in [15].Theorem 5.
LetX,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐Lipschitz tracking property.Proof.
LetL=2 and ε0=1. Let y¯ii≥0y¯i=y0i,y1i,y2i,⋯ be G¯,ε‐ pseudo orbit of σ for any 0<ε<ε0. Then for any i≥0, there exists t¯i∈G¯ satisfying.
(13)d¯t¯iσy¯i,y¯i+1<ε,where t¯i=t0i,t1i,t2i,⋯.So, whenk≥1 and i≥0, we can get that
(14)dtk−1iyki,yk−1i+12k−1<ε.Thus, we can obtain the following inequality(15)dtk−1iyki,yk−1i+1<2k−1ε,dtk−2i+1yk−1i+1,yk−2i+2<2k−2ε,dtk−3i+2yk−2i+2,yk−3i+3<2k−3ε,⋯,dt1i+k−2y2i+k−2,y1i+k−1<21ε,dt0i+k−1y1i+k−1,y0i+k<20ε.Sinced is invariant to G, we can have that
(16)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,t0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1<2k−1ε,dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1,t0i+k−1t1i+k−2⋯tk−3i+2yk−2i+2<2k−2ε,dt0i+k−1t1i+k−2⋯tk−4i+3tk−3i+2yk−2i+2,t0i+k−1t1i+k−2⋯tk−4i+3yk−3i+3<2k−3ε,⋯,dt0i+k−1t1i+k−2y2i+k−2,t0i+k−1y1i+k−1<21ε,dt0i+k−1y1i+k−1,y0i+k<20ε.Thus, whenk≥1 and i≥0, we can get that
(17)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,y0i+k<dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,t0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1+dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1,t0i+k−1t1i+k−2⋯tk−3i+2yk−2i+2+dt0i+k−1t1i+k−2⋯tk−4i+3tk−3i+2yk−2i+2,t0i+k−1t1i+k−2⋯tk−4i+3yk−3i+3+⋯⋯+dt0i+k−1t1i+k−2y2i+k−2,t0i+k−1y1i+k−1+dt0i+k−1y1i+k−1,y0i+k<2k−1+2k−2+⋯+20ε<2kε.That is,(18)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,y0i+k2k<ε.Let(19)y¯=y00,y01,y02,⋯∈X¯,g¯i=e,t0i,t0i+1t1i,t0i+2t1i+1t2i,⋯.Hence, we have that(20)d¯σiy¯,g¯iy¯i<ε<Lε.So,σ has G¯‐Lipschitz tracking property.


## 4.G¯‐Periodic Tracking Property of σ

We can find the concepts ofG‐periodic point and G‐periodic tracking property in [19, 20].Theorem 6.
LetX,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐periodic tracking property.Proof.
Let0<δ<ε for any ε>0 and y¯ii≥0y¯i=y0i,y1i,y2i,⋯ be G¯,δ‐ pseudo orbit of σ. Thus, there exists n>0satisfying.
(21)y¯kn+j=y¯j,0<k,0≤j<n.Writey¯=y00,y01,y02,⋯∈X¯. It follows that
(22)e¯σny¯=y¯.Hence,y¯∈PG¯σ. In addition, for any i≥0, there exists t¯i∈G¯ satisfying
(23)d¯t¯iσy¯i,y¯i+1<δ,where t¯i=t0i,t1i,t2i,⋯.So, whenk≥1 and i≥0, we can get that
(24)dtk−1iyki,yk−1i+12k−1<δ.Thus, we can obtain the following inequality(25)dtk−1iyki,yk−1i+1<2k−1δ,dtk−2i+1yk−1i+1,yk−2i+2<2k−2δ,dtk−3i+2yk−2i+2,yk−3i+3<2k−3δ,⋯,dt1i+k−2y2i+k−2,y1i+k−1<21δ,dt0i+k−1y1i+k−1,y0i+k<20δ,Sinced is invariant to G, we can have that
(26)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,t0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1<2k−1δ,dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1,t0i+k−1t1i+k−2⋯tk−3i+2yk−2i+2<2k−2δ,dt0i+k−1t1i+k−2⋯tk−4i+3tk−3i+2yk−2i+2,t0i+k−1t1i+k−2⋯tk−4i+3yk−3i+3<2k−3δ,⋯,dt0i+k−1t1i+k−2y2i+k−2,t0i+k−1y1i+k−1<21δ,dt0i+k−1y1i+k−1,y0i+k<20δ,Thus, whenk≥1 and i≥0, we can get that
(27)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,y0i+k<dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,t0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1+dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1yk−1i+1,t0i+k−1t1i+k−2⋯tk−3i+2yk−2i+2+dt0i+k−1t1i+k−2⋯tk−4i+3tk−3i+2yk−2i+2,t0i+k−1t1i+k−2⋯tk−4i+3yk−3i+3+⋯⋯+dt0i+k−1t1i+k−2y2i+k−2,t0i+k−1y1i+k−1+dt0i+k−1y1i+k−1,y0i+k<2k−1+2k−2+⋯+20δ<2kδ.That is,(28)dt0i+k−1t1i+k−2⋯tk−3i+2tk−2i+1tk−1iyki,y0i+k2k<δ.Letg¯i=e,t0i,t0i+1t1i,t0i+2t1i+1t2i,⋯ for any i≥0. Hence, we have that
(29)d¯σiy¯,g¯iy¯i<δ<ε.So, the shift mapσ has G¯‐periodic tracking property.


## 5. Conclusion

Firstly, we introduced the concept ofG‐Lipschitz tracking property, G‐asymptotic average tracking property, and G‐periodic tracking property. Secondly, we studied their dynamical properties and topological structure in the infinite product space under group action and obtained the following conclusions: (1) let X,d be compact metric G−space and the metric d be invariant to G. Then, σ has G¯‐asymptotic average tracking property; (2) let X,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐Lipschitz tracking property; (3) let X,d be compact metric G‐space and the metric d be invariant to G. Then, σ has G¯‐periodic tracking property. The above results make up for the lack of theory of G‐Lipschitz tracking property, G‐asymptotic average tracking property, and G‐periodic tracking property in infinite product space under group action.



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