#### Abstract

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

#### 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 has average tracking property, then is chain transitivity under some conditions. Wu [2] showed that has tracking property and has tracking property are equivalent. Ji et al. [3] proved that has the Lipschitz shadowing property and has the Lipschitz shadowing property are equivalent in the double inverse limit space. Ahmadi and Hosseini [4] gave that 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 of Lipschitz tracking property, asymptotic average tracking property, and periodic tracking property. Secondly, we studied their dynamical properties and topological structure and obtained the following conclusions:

Theorem 1. *Let be compact metric space and the metric be invariant to . Then, has asymptotic average tracking property.*

Theorem 2. *Let be compact metric space and the metric be invariant to . Then, has Lipschitz tracking property.*

Theorem 3. *Let be compact metric space and the metric be invariant to . Then, has periodic tracking property.*

#### 2. Asymptotic Average Shadowing Property of

The concept of metric space is shown in [12]. We can find the concept that is invariant to in [13]. The concepts of zero density set and asymptotic average tracking property are seen in [14, 15]. The concept of the shift map and the metric can be found in [16].

Write

where .

According to [17, 18], is compact metric space. This paper mainly studies dynamical properties of in compact metric space.

Theorem 4. *Let be compact metric space and the metric be invariant to . Then, has asymptotic average tracking property.*

*Proof. *Let be asymptotic average pseudo orbit of . Then, for any nonnegative integer , there exists satisfying.
where .

According to [14], we can choose a zero density set satisfying

So for any , there exists a positive integer such that and implies

So, for any , , and , it follows that

Thus, we can obtain the following inequality

Since the metric is invariant to , we can have that

Thus, when and for any positive integer , we can get that

That is

Let and , where . Then, for any and , it follows that

Hence,

According to [14], we have that

Hence, has asymptotic average tracking property.

#### 3. Lipschitz Tracking Property of

We can find the definition of Lipschitz tracking property in [15].

Theorem 5. *Let be compact metric space and the metric be invariant to . Then, has Lipschitz tracking property.*

*Proof. *Let and . Let be pseudo orbit of for any . Then for any , there exists satisfying.
where .

So, when and , we can get that

Thus, we can obtain the following inequality

Since is invariant to , we can have that

Thus, when and , we can get that

That is,

Let

Hence, we have that

So, has Lipschitz tracking property.

#### 4. Periodic Tracking Property of

We can find the concepts of periodic point and periodic tracking property in [19, 20].

Theorem 6. *Let be compact metric space and the metric be invariant to . Then, has periodic tracking property.*

*Proof. *Let for any and be pseudo orbit of . Thus, there exists satisfying.

Write . It follows that

Hence, . In addition, for any , there exists satisfying where .

So, when and , we can get that

Thus, we can obtain the following inequality

Since is invariant to , we can have that

Thus, when and , we can get that

That is,

Let for any . Hence, we have that

So, the shift map has periodic tracking property.

#### 5. Conclusion

Firstly, we introduced the concept ofLipschitz tracking property, asymptotic average tracking property, and 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 be compact metric space and the metric be invariant to . Then, has asymptotic average tracking property; (2) let be compact metric space and the metric be invariant to . Then, has Lipschitz tracking property; (3) let be compact metric space and the metric be invariant to . Then, has periodic tracking property. The above results make up for the lack of theory of Lipschitz tracking property, asymptotic average tracking property, and periodic tracking property in infinite product space under group action.

#### Data Availability

The data used to support the findings of this study are included within references [1–20] in the article.

#### Conflicts of Interest

The author declares that he has no conflicts of interest.

#### Acknowledgments

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