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).