-Chain Mixing and -Chain Transitivity in Metric G-Space

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 [1–9]). 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 2–4 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 [1–14] 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).