^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

Firstly, we introduce the definitions of

Let

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 this section, we will prove Theorem

(see [

If

(see [

(see [

The metric

(see [

Let

(see [

(see [

(see [

By the definitions of the

Let

Now, we start to prove Theorem

Let

By the definition of

Hence, we have that

That is,

So

According to (

Because of the surjectivity of the map

So, the map

Now suppose that

According to (

According to the surjectivity of the map

So,

Thus, we have that

Hence, we have that

So the map

(see [

(see [

(see [

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

Let

Let

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

Let

There exists a zero density set

Now we will prove Theorem

Let

According to the fact that the map

Let

By Lemma

Then, there exists

Thus, we have that

That is,

Hence, we have that

According to Lemma

Hence, the sequence

According to Lemma

Thus, there exists

According to the equivalent definition of the map

According to the surjectivity of the map

So,

Combining with Lemma

So, the shift mapping

Because the map

Now suppose that

By Lemma

Hence, there exists

According to the equivalent definition of the map

According to the surjectivity of the map

Hence, we have

Combining with Lemma

So,

By Lemma

Thus, we have that

By Lemma

So, the map

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

(see [

According to the concept of quasi-weak almost-periodic point, we will give the concept of

Let

Now, we start to prove Theorem

Let

Write

Now, suppose that

Hence, we have that

Thus, we can obtain

So,

According to

Let

Suppose that

Thus, we have that

Thus, we can obtain

Hence,

In this paper, we study dynamical properties and characteristics of

No data were used to support this study.

The authors declare that they have no conflicts of interest.

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