#### Abstract

Firstly, we introduce the concept of -chain mixing, -mixing, and -chain transitivity in metric -space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map has the -shadowing property, then the map is -chain mixed if and only if the map is -mixed. (2) The map is -chain transitive if and only if for any positive integer , the map is -chain transitive. (3) If the map is -pointwise chain recurrent, then the map is -chain transitive. (4) If there exists a nonempty open set satisfying , , and , then we have that the map is not -chain transitive. These conclusions enrich the theory of -chain mixing, -mixing, and -chain transitivity in metric -space.

#### 1. Introduction

Chain mixing, mixing, and chain transitivity are very important concepts in topological dynamical systems. Many scholars studied their dynamical properties and obtained some valuable results (see [19]). Fatehi [1] showed that chain mixing, chain transitivity, and chain recurrence properties are equivalent in iterated function systems; Ji [2] proved that if the sequence map is -mixed, then the limit map is also -mixed under -strong uniform convergence; Zeng et al. [3] gave that if the sequence map is weakly mixed, then the limit map is also weakly mixed under strong uniform convergence; Liu and Yi [4] pointed out that if the map is chain mixed, then is chain mixed for any positive integer in nonautonomous dynamical system; Diao [5] proved that the map is chain mixed if and only if there exists positive integer and such that is chain transitive.

In this paper, firstly, we introduce the concept of -chain mixing, -mixing, and -chain transitivity in metric -space. Secondly, we study their dynamical properties and obtain the following theorem.

Theorem 1. Let be a compact metric -space, be a pseudo equivalent, and isometric map and the metric be invariant to . If the map has the -shadowing property, then the map is -chain mixed if and only if the map is -mixed.

Theorem 2. Let be a compact metric -space, be a pseudo equivalent map, and topological group be compact. Then, the map is -chain transitive if and only if for any positive integer , the map is -chain transitive.

Theorem 3. Let be a compact metric -space, be a continuous map, and topological group be compact. If the map is a -pointwise chain recurrent, then the map is -chain transitive.

Theorem 4. Let be a compact metric -space, be a continuous closed map, and topological group be compact. If there exists a nonempty open set satisfying , , and , then we have that the map is not -chain transitive.

The above conclusions enrich the theory of -chain mixing, -mixing, and -chain transitivity in metric -space.

Next, we prove Theorem 1 in Section 2 and Theorems 24 in Section 3.

#### 2. -Chain Mixing and -Mixing in Metric -Space

Definition 5 (see [10]). Let be a metric -space, be a topological group, and be a continuous map. The or is called a metric -space if the following conditions are satisfied: (1) where is the identity of and for all (2) where for all and all , If is compact, then is also said to be compact metric -space. For the convenience of writing, is usually abbreviated as .

Definition 6 (see [11]). Let be a metric -space and be a continuous map from to . is said to be a pseudo equivariant map if for all and all , there exists such that

Definition 7 (see [12]). Let be a metric -space. The metric is said to be invariant to the topological group provided that for all and .

Definition 8 (see [7]). Let be a metric space and be a continuous map from to . The map is said to be an isometry map if we have for all .

Definition 9 (see [11]). Let be a metric -space and be a continuous map from to . The sequence is called to be a -pseudo orbit of if for any , there exists such that .

Definition 10 (see [11]). Let be a metric -space and be a continuous map from to . The sequence is said to be -shadowed by a point in if for any , there exists such that .

Definition 11 (see [11]). Let be a metric -space and be a continuous map from to . The map has -shadowing property if any , there exists such that for any -pseudo orbit of , there exists a point in such that the sequence is -shadowed by the point .

Definition 12. Let be a metric -space and be a continuous map from to . The map is said to be -mixed if for any nonempty open set and in , there exists a positive integer such that for any positive integer greater than or equal to , there exists such that

Definition 13 (see [13]). Let be a metric -space and be a continuous map from to . Write and . The sequence is called to be a -chain with length from to under the action of if for any , there exists such that

Definition 14. Let be a metric -space and be a continuous map from to . The map is said to be -chain mixed if for any and , there exists a positive integer such that for any positive integer greater than or equal to , there exists a -chain with length from to under the action of .

Now we start to prove Theorem 15.

Theorem 15. Let be a compact metric -space, be a pseudo equivalent and isometric map, and the metric be invariant to . If the map has the -shadowing property, then the map is -chain mixed if and only if the map is -mixed.

Proof. Suppose that the map is -chain mixed. For any , let and be any spherical field in . According to that the map has the -shadowing property, for any , there exists such that for any -pseudo orbit , there exists and such that for any , we have that Since the map is -chain mixed, there exists a positive integer such that for any positive integer greater than or equal to , there exists a -chain with length from to under the action of . By (4), there exists and such that Since the metric is invariant to , we can get that Hence, we have that Combined with the definition of isometric map , we can obtain that So we can get that Combined with the definition of the pseudo equivariant map , there exists such that Let satisfying . According to that the metric is invariant to , we have that So Thus, Hence, the map is -mixed.
Suppose that the map is -mixed. For any and , there exists a positive integer such that for any positive integer greater than or equal to , there exists such that Let Then, we have that In addition, there exists such that . Since the metric is invariant to and the map is an isometric map, we can get that So Hence, we have that Thus, is a -chain with length from to under the action of . So the map is -chain mixed. This completes the proof.

#### 3. -Chain Transitive in Metric -Space

Definition 16 (see [13]). Let be a metric space and be a continuous map from to . The is called to be a -chain equivalent set of the point if we write

Definition 17. Let be a metric -space and be a continuous map from to . The map is said to be -chain transitive if for any and , there exists a -chain from to under the action of .

Definition 18 (see [14]). Let be a metric -space and be a continuous map from to . A point is called to be a -chain recurrent point if for any , there exists -chain from to under the action of , denoted by the -chain recurrent point set of the map .

Definition 19. Let be a metric -space and be a continuous map from to . if , then is said to be -pointwise chain recurrent.

Definition 20 (see [12]). Let be a metric -space and . Write

Now we give Lemma 21 in order to prove Theorems 2 and 3 in this section.

Lemma 21 (see [12]). Let be a compact metric -space and be compact topological group. Then, for any , there exists such that implies for any .

Theorem 22. Let be a compact metric -space, be a pseudo equivalent map, and topological group be compact. Then, the map is -chain transitive if and only if for any positive integer , the map is -chain transitive.

Proof. Suppose that the map is -chain transitive. By Lemma 21, for any , there exists a positive integer greater than or equal to such that implies According to the uniform continuity of the map , for , there exists a positive integer greater than or equal to such that implies Since the map is -chain transitive, for any , there exists -chain from to under the action of . So we have that and . Now we have two cases.

Case 1. If is a factor of , then there exists a positive integer such that . According to that is -chain under the action of , for any , there exists such that

Hence, for any , we have that

Combined with the definition of pseudo equivariant map and (23), there exists such that

By (22), we can obtain that

Hence, we can get that so is a -chain from to under the action of . Hence, the map is -chain transitive.

Case 2. If is not a factor of , then there exists a positive integer such that

By the above same way, we can get that is a -chain under the action of . In addition, we have that

According to the definition of pseudo equivariant map and (23), there exists such that

By (22), we can obtain that.

Hence, we can get that so is a -chain from to under the action of . Hence, the map is -chain transitive.

Suppose that the map is -chain transitive. For any and , there exists -chain from to under the action of . Let Then, is a -chain from to under the action of . Hence, is -chain transitive. Thus, we end the proof.

Theorem 23. Let be a compact metric -space, be a continuous map, and topological group be compact. If the map is a -pointwise chain recurrent, then the map is -chain transitive.

Proof. Suppose that the map is -pointwise chain recurrent. Let . According to [13], the is a closed set. Suppose . By Lemma 21, for any , there exists such that implies According to , there exists -chain under the action of where and . Hence, there exists such that According to the continuity of the map , there exists such that implies Let . By (36) and (40), we have that According to and (38), we have that By (39) and (41), we can get that Hence, is a -chain under the action of . So we have that Thus, Hence, the is an open set. Since is -pointwise chain recurrent, we can get that Hence, is not an empty set. Because the metric space is connected, so we have that Hence, Thus, there exists a -chain from to under the action of . So the map is -chain transitive. Thus, we end the proof.

Theorem 24. Let be a compact metric -space, be a continuous closed map, and topological group be compact. If there exists a nonempty open set satisfying , , and , then we have that the map is not -chain transitive.

Proof. Let and . Since is compact metric space, is greater than . By Lemma 21, for the above , there exists such that implies Let . It remains to show that there is no -chain from to under the action of . If there exists a -chain from to under the action of , then for any , there exists such that By (49), we have that In particular, we have that According to and , we can obtain that Combined with , we can get that According to , we have that the point is in . So By the same way, we can get for any . Hence, . This contradicts that the point is in . So there is no -chain from to under the action of . Then, the map is not -chain transitive. Thus, we end the proof.

#### 4. Conclusion

Firstly, we introduce the concept of -chain mixing, -mixing, and -chain transitivity in metric -space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map has the -shadowing property, then the map is -chain mixed if and only if the map is -mixed. (2) The map is -chain transitive if and only if for any positive integer , the map is -chain transitive. (3) If the map is -pointwise chain recurrent, then the map is -chain transitive. (4) If there exists a nonempty open set satisfying , , and , then we have that the map is not -chain transitive. These conclusions enrich the theory of -chain mixing, -mixing, and -chain transitivity in metric -space.

Most importantly, it provided the theoretical basis and scientific foundation for the application of -chain mixing, -mixing, and -chain transitivity in computational mathematics and biological mathematics.

#### Data Availability

The data used to support the findings of this study are included within references [114] 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).