#### Abstract

According to the definition of sequence shadowing property and regularly recurrent point in the inverse limit space, we introduce the concept of sequence shadowing property and regularly recurrent point in the double inverse limit space and study their dynamical properties. The following results are obtained: (1) Regularly recurrent point sets of the double shift map are equal to the double inverse limit space of the double self-map in the regularly recurrent point sets. (2) The double self-map has sequence shadowing property if and only if the double shift map has sequence shadowing property. Thus, the conclusions of sequence shadowing property and regularly recurrent point are generalized to the double inverse limit space.

#### 1. Introduction

Inverse limit space plays an important role in topological dynamical systems, and a series of research results have been obtained (see [1–4]). With the gradual deepening of problem research, it is found that the finite symbols in the inverse limit space have been difficult to solve the practical problems in life. Therefore, scholars began to try to extend the inverse limit space to the double inverse limit space and study the dynamical properties of Lipschitz tracking property, topological transitivity, and nonwandering points in the double inverse limit space. The relevant results are shown in [5–8]. Up to now, there are few new results in the double inverse limit space and it needs to be further studied. In addition, it is known that shadowing property and regularly recurrent point have always been the focus of topological dynamical systems. At present, many scholars have studied them in the inverse limit space and obtained valuable research results (see [9–15]). But there are few research results about them in the double inverse limit space. In this paper, firstly, the new concepts of sequence shadowing property and regularly recurrent point are given in double inverse limit space. Secondly, by using the method of mutual disclosure of the double self-map and the double shift map , we can obtain the following theorem.

Theorem 1. *Let be a compact metric space, be a continuous map, be a continuous map, and . Then, we have that
*

Theorem 2. *Let be a compact metric space, be a homeomorphic map, be a homeomorphic map, and . Then, the double self-map has sequence shadowing property if and only if the double shift map has sequence shadowing property.*

The results enrich the conclusions of sequence shadowing property and regularly recurrent point in the double inverse limit space.

#### 2. Regularly Recurrent Point in the Double Inverse Limit Space

*Definition 3. *Let be a metric space and be a continuous map from to . is said to be a homeomorphic map if is one-to-one and the map and are continuous.

*Definition 4 (see [7]). *Let be a compact metric space, be a continuous map, be a continuous map and . is said to be the double inverse limit spaces of if we write
where we write .

The metric in is defined by where and.

The double shift map is defined by

Thus, is compact metric space, and the double shift map is homeomorphic. In addition, we have

For any , the projection map is defined by

Then, the map is a continuous, surjective, and open map.

*Definition 5. *Let be a metric space and be a continuous map from to . A point is called to be a regularly recurrent point if each open set , there exists positive integer such that for any positive integer , we have , denoted by the regularly recurrent point set of the map .

*Remark 6. *According to the concept of regularly recurrent point of the map , we will give the concept of regularly recurrent point of the double map .

*Definition 7. *Let be a compact metric space, be a continuous map, be a continuous map, and . A point is called to be a regularly recurrent point if each open set , there exists positive integer such that for any positive integer , we have , denoted by the regularly recurrent points set of the double map .

In order to prove the main theorem in this section, we will give two lemmas below.

Lemma 8 (see [8]). *Let be a compact metric space, be a continuous map, be a continuous map, and . Then, for any positive integer and nonnegative integer , we have
*

Lemma 9. *Let be a compact metric space, be a continuous map, be a continuous map, and . Then, we have that
*

*Proof. *Suppose . Let be an any open set containing the point . Then, is an open set containing the point . According to , there exists positive integer such that for any positive integer , we have
Thus, . So . Hence,.

*Remark 10. *The regularly recurrent point set is invariant to the double map . So we can study the double inverse limit space of the double self-map in the regularly recurrent point set . Now let us give the proof process of Theorem 11.

Theorem 11. *Let be a compact metric space, be a continuous map, be a continuous map, and . Then, we have
*

*Proof. *Suppose . For any integer , let be an any open set containing the point . Thus, is an open set containing the point . According to , there exists positive integer such that for any positive integer , we have that
Thus, we have that
By Lemma 8, we can get that
That is,
So . Hence, .

Suppose . Then, for any integer , we have
Let be an any open set containing the point . Then, is an open set containing the point . Thus, there exists positive integer such that for any positive integer , we have that
Thus, we have that
By Lemma 8, we can get that
Thus, we have that
So . Hence, . This completes the proof.

#### 3. Sequence Shadowing Property in the Double Inverse Limit Space

*Definition 12 (see [4]). *Let be a metric space and be a continuous map from to . The sequence is called -pseudo orbit of if for any , we have .

*Definition 13 (see [4]). *Let be a metric space and be a continuous map from to . The sequence is said to be -shadowed by the point in if for any , we have .

*Definition 14 (see [4]). *Let be a metric space and be a continuous map from to. The map has a sequence shadowing property if each , there exists such that any -pseudo orbit of , there exists a point in and nonnegative integer sequence such that the sequence is -shadowed by the point.

*Remark 15. *By Definitions 12–14, we will give the concept of sequence shadowing property of the double map .

*Definition 16. *Let be a compact metric space, be a continuous map, be a continuous map, and . The sequence is called -pseudo orbit of the double map if for any integer , we have .

*Definition 17. *Let be a compact metric space, be a continuous map, be a continuous map, and . The sequence is said to be -shadowed by the point in if for any positive integer , we have

*Definition 18. *Let be a compact metric space, be a continuous map, be a continuous map, and . The double map has sequence shadowing property if each , there exists such that for any -pseudo orbit of , there exists a point in and increasing positive integer sequence and such that the sequence is -shadowed by the point .

Now we begin to prove Theorem 19.

Theorem 19. *Let be a compact metric space, be a homeomorphic map, be a homeomorphic map, and . Then, the double map has sequence shadowing property if and only if the double shift map has sequence shadowing property.*

*Proof. *Suppose the double map has the sequence shadowing property. Since is compact metric space, it is bounded. Write . Then, for any , there exists positive integer and such that
Since the map and is uniformly continuous, it follows that for any and , there exists such that implies
Note that the double map has the sequence shadowing property, and it follows that there exists such that for any -pseudo orbit of , there exists a point and increasing positive integer sequence and such that the sequence is -shadowed by the point . Let be -pseudo orbit of the double map where . Then, for any integer , we have that
According to the definition of the metric , for any integer , we can get that
So we have that
Thus, is -pseudo orbit of the double map in . Hence, there exists and increasing positive integer sequence and such that for any integer , we have that
By (22), for any and , we can get
Let and . For any and , we have that
So the point is in . It is easy to know that
When or , we can, respectively, get that
When and , for any and , we have that
where and .

By (27), we can obtain that
Thus, for any we can get that
So for any integer , we have that
Hence, the double shift map has the sequence shadowing property.

Suppose the double shift map has the sequence shadowing property. For each , there exists such that for any -pseudo orbit of , there exists a point and increasing positive integer sequence and such that the sequence is -shadowed by the point . Let and such that
Since the map and is uniformly continuous, it follows that for any and, there exists such that implies
Let be -pseudo orbit of the double map . Then, for any , we have that
By (36), when and , we have that
For every , let . It is easy to know that When or , we can, respectively, get that
When and , for any and , we have that
where and .

By (38), we can get that
So for any integer , we have that
Thus for any integer , we have that
Hence, is -pseudo orbit of the double map in . So there exists a point and increasing positive integer sequence and such that for any integer , we have that
According to the definition of the metric , we can get that
Hence, the map has the sequence shadowing property. Thus, we end the proof.

#### 4. Conclusion

Firstly, we give the new concepts of sequence shadowing property and regularly recurrent point of double inverse limit space in this paper. Secondly, let be a compact metric space, be a homeomorphic map, be a homeomorphic map, and . By using the method of mutual disclosure of the double self-map and the double shift map , we can obtain the following results: (1) ; (2) the double self-map has sequence shadowing property if and only if the double shift map has sequence shadowing property. These results enrich the conclusions of sequence has shadowing property and regularly recurrent point in double inverse limit space. Most importantly, it provided the theoretical basis and scientific foundation for the application of sequence shadowing property in computational mathematics and biological mathematics.

#### Data Availability

The data used to support the findings of this study are included within references [1–15] in the article.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

#### Acknowledgments

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