Abstract
In this paper, we introduce the concepts of
1 Introduction
The tracking property has an important application in topological dynamical systems. In recent years, more and more scholars pay attention to it, and the relevant research results are shown in [1,2,3, 4,5,6, 7,8,9, 10,11,12, 13,14,15, 16,17]. Liang and Li [1] proved that the selfmap
The map
The map
Theorem 1.1
Let
Theorem 1.2
Let
2 The equivalent condition of
G
asymptotic tracking property
In this section, we present some concepts that may be used in the following. The concept of metric
Definition 2.1
[19] Let
Definition 2.2
[20] Let
Definition 2.3
Let (
Lemma 2.4
Let
Proof
By continuity of the map
Suppose that for any
According to the equivalent definition of the map
Then,
Since the metric
Therefore,
Theorem 2.5
Let
Proof
(Necessity) Suppose that the map
Let
By (2) and the equivalent definition of the map
Noting that the metric
and thus,
So for any
Since the map
Since the metric
By (4) and (5), for any
Together with the fact that the metric
Hence, the sequence
(Sufficiency) Suppose that for any
Let
In addition, for any
By Lemma 2.9, for any
Since the metric
and
Since
Hence, the map
3
G
Lipschitz tracking property
The concept of the inverse limit spaces in this section under group action can be found in [21].
Definition 3.1
[5] Let
Definition 3.2
[18] Let
Now, we give the proof of Theorem 3.3.
Theorem 3.3
Let
Proof
(Necessity) Suppose that the map
For any
From the definition of the metric
Thus,
Since the map
By the definition of the equivalent map
According to (8) and the definition of Lipschitz map
Then, for any
Therefore, the shift map
(Sufficiency) Next, we suppose that the shift map
For any
Since the map
Then, for any
According to (9), the definition of Lipschitz map
So for any
Hence,
From the definition of the metric
Thus,
So the map
4 Conclusion
In this paper, we studied dynamical properties of
Acknowledgements
This work was partially supported by the NSF of Guangxi Province (2020JJA110021) and construction project of Wuzhou University of China (2020B007).

Conflict of interest: The author states no conflict of interest.
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