G-Expansibility and G-Almost Periodic Point under Topological Group Action

Ji, Zhanjiang

doi:https://doi.org/10.1155/2021/7326623

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 7326623 | https://doi.org/10.1155/2021/7326623

G-Expansibility and G-Almost Periodic Point under Topological Group Action

Zhanjiang Ji¹

Academic Editor: José Francisco Gómez Aguilar

Received21 Jun 2021

Accepted18 Oct 2021

Published08 Nov 2021

Abstract

Firstly, the new concepts of expansibility, almost periodic point, and limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map and the shift map in the inverse limit space under topological group action. The following new results are obtained. Let be a metric space and be the inverse limit space of . (1) If the map is an equivalent map, then we have . (2) If the map is an equivalent surjection, then the self-map is expansive if and only if the shift map is expansive. (3) If the map is an equivalent surjection, then the self-map has limit shadowing property if and only if the shift map has limit shadowing property. The conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li . Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.

1. Introduction

Let be a metric space and be a continuous map from to . A point is called to be an almost periodic point if for each open set containing there exists a positive integer such that for every positive integer there exists satisfying (see [1]). In recent years, there are many achievements about almost periodic points set (see [1–8]). Qiu and Zhao [2] proved that if the map has shadowing property in , then the map has shadowing property in . Liao and Wang [3] discussed the properties of almost periodic points in unilateral symbol space. In [1], it is proved that the set of almost periodic points of shift map is the inverse limit space formed by the self-map in . Inspired by the idea of Li [1], the author gave the concepts of almost periodic point according to the concept of almost periodic point. That is, let be a metric space and be a continuous map from to . A point is called to be an almost periodic point if for each open set containing there exists a positive integer such that for every positive integer there exists and satisfying . Next, we give the following theorem in the inverse limit space under topological group action.

Theorem 1. Let be a metric space, the map be an equivalent map, and be the inverse limit space of . Then, we have

According to definition, almost periodic point means almost periodic point. Otherwise, it does not hold. Hence, Theorem 1 generalizes the corresponding results given in Li [1].

The map is said to be expansive if for any there exists a positive integer such that (see [9]). The map has shadowing property if each there exists such that for any pseudo orbit of there exists a point in such that the sequence is shadowed by the point (see [10]). Expansibility and shadowing property have attracted the attention of many scholars. The relevant results are seen in [11–17]. Das and Das [11] pointed out that the map is expansive if and only if the map is expansive in locally equidistant covered space. Wang and Zeng [12] discussed the relationship between average shadowing property and average shadowing property. Das [13] proved the nonexistence of pseudo equivariant expansive homeomorphism on closed unit interval. Das and Das [14] obtained a sufficient condition for the extension of a expansive homeomorphism on a G-invariant subspace of a compact metric space with compact to be expansive on the whole space. Shah [15] gave a necessary and sufficient condition for a positively expansive map to possess the shadowing property. Wu [16] proved that the system with tracking property is chain mixed. Oprocha et al. [17] analyzed necessary and sufficient conditions for shadowing property over a set with positive density. In this paper, we gave the concepts of expansive map and limit shadowing property according to the concepts of expansive map and limit shadowing property. At last, we will give the proof of Theorems 2 and 3 in the inverse limit space under topological group action. The main results are as follows.

Theorem 2. Let be a compact metric space, the map be an equivalent surjection, and be the inverse limit space of . Then, the self-map is expansive if and only if the shift map is expansive.

Theorem 3. Let be a compact metric space, the map be an equivalent surjection, and be the inverse limit space of . Then, the self-map has limit shadowing property if and only if the shift map has limit shadowing property.

According to definition, expansibility means expansibility and limit shadowing property means limit shadowing property. Otherwise, it does not hold. Hence, Theorems 2 and 3, respectively, generalize the corresponding results given in Niu [18] and Liang and Li [10].

2. G-Almost Periodic Point under Topological Group Action

Definition 1. (see [19]). Let be a metric space, be a topological group, and be a continuous map. The triple is called to be metric space if the following conditions are satisfied:(1) where is the identity of and for all (2) for all and all ,

Remark 1. If is compact, then is also said to be compact metric space. For the convenience of writing, is usually abbreviated as .

Definition 2. (see [19]). Let be a metric space and be a continuous map from to . is said to be an equivariant map if we have for all and all .

Definition 3. (see [20]). Let be a metric space and be a continuous map from to . is said to be the inverse limit spaces of if we writeThe metric in is defined bywhere and .
The shift map is defined byThe projection map is defined byThus, is compact metric space. The shift mapping is homeomorphism, and for any , the projection map is a continuous and open map.

Definition 4. (see [20]). Let be a metric G-space and be equivariant map from to . Writewhere .
The map is defined bywhere and .
Then, is a metric space. Let and be shown as above. The space is called to be the inverse limit spaces of under group action.

Definition 5. (see [1]). Let be a metric space and be a continuous map from to . The point is called to be an almost periodic point if for each open set containing there exists a positive integer such that for every positive integer there exists satisfying . Denoted by the almost periodic point set of the map .

Remark 2. According to the concept of almost periodic point, we give the concept of almost periodic point.

Definition 6. Let be a metric space and be a continuous map from to . The point is called to be an almost periodic point if for each open set containing there exists a positive integer such that for every positive integer there exists and satisfying . Denoted by the almost periodic point set of the map .

Theorem 4. Let be a metric space, the map be an equivalent map, and be the inverse limit space of . Then, we have

Proof. Suppose . For any , let be any open set containing . Then, is an open set containing . Hence, there exists a positive integer such that for every positive integer there exists and such thatThus, we can obtainThen, we have thatSo, . Hence, .
Suppose . Thus, we can get for every . Let be an any open set containing the point . Then, is an open set containing the point . According to , there exists a positive integer such that for every positive integer there exists and satisfyingLet . Then, we can obtainSo, . Thus, we have . This completes the proof.

3. G-Expansibility under Topological Group Action

Definition 7. (see [9]). Let be a metric space, be a continuous map from to , and be a positive constant. The map is said to be an expansive map if for any there exists a positive integer such that where the constant is called to be an expansion constant.

Remark 3. According to the concept of expansive map, we give the concept of almost expansive map.

Definition 8. Let be a metric G-space, be a continuous map from to , and be a positive constant. The map is said to be an expansive map if for any there exists an positive integer such that for any and we have that where the constant is called to be an expansion constant.
Next, we start to prove Theorem 5.

Theorem 5. Let be a compact metric space, the map be an equivalent surjection, and be the inverse limit space of . Then, the self-map is expansive if and only if the shift map is expansive.

Proof. Suppose that the self-map is expansive map with expansion constant . Thus, the map is injective. Let and with . As the map is injective, we can get . Let By the definition of expansive map , there exists an positive integer such thatBy the definition of the metric , we get thatHence, the shift map is expansive.
Suppose that the shift map is expansive map with expansion constant . Since is compact metric space, it is bounded. Write . Let such thatSince the map is surjection, for any , we can chooseIt is obvious that the point is different from the point . For any and , letAccording to that the shift map is expansive map with expansion constant , there exist a positive integer such thatAccording to that the map is equivalent, we can get thatBy (17), we have thatIf for any positive integer , we have thatThen, by (23), we can get thatIt is absurd. Hence, there exists a positive integer such thatSo, the self-map is expansive. Thus, we end the proof.

4. G-Limit Shadowing Property under Topological Group Action

Definition 9. (see [21]). Let be a metric space and be a continuous map from to . The sequence is called to be limit pseudo orbit of if for any there exists such that .

Definition 10. (see [21]). Let be a metric space and be a continuous map from to . is said to be limit shadowed by a point if for any there exists such that .

Definition 11. (see [21]). Let be a metric space and be a continuous map from to .
The map has limit shadowing property if for any limit pseudo orbit of there exists a point in such that the sequence is limit shadowed by the point .
The main result of this section is the following theorem:

Theorem 6. Let be a compact metric space, the map be an equivalent surjection, and be the inverse limit space of . Then, the self-map has limit shadowing property if and only if the shift map has limit shadowing property.

Proof. Suppose that the self-map has limit shadowing property. Since is compact, it is bounded. Write . For any , let such thatBy the continuity of the map , it follows that for given and any there exists such that which implies Let be limit pseudo orbit of the shift map with . Then, there exists with such that Hence, there exists a positive integer such that when , we have thatSo, when , we have thatThus, we can get thatHence, is limit pseudo orbit of the map . According to the hypothesis, for any , there exist and such that So, there exists a positive integer such that when , we have thatAccording to the equivalent definition of the map and (27), for any , we have thatSince the map is onto, we can chooseBy (26) and (34), when , we have thatHence, we can get thatSo, the shift map has limit shadowing property.
Suppose that the shift map has limit shadowing property. For any , there exists an positive integer such that According to the continuity of the map , it follows that for given and any there exists such that impliesLet be limit pseudo orbit of the map . Then, for any , there exists such thatHence, there exists a positive integer such that when , we have thatAccording to the equivalent definition of the map and (39), for any , we have thatSince the map is onto, we can writeBy (38) and (42), when , we have thatHence, we can get thatThus, is limit pseudo orbit of the shift map . By the hypothesis, for any , there exist and such thatHence, there exists a positive integer such that when , we have thatSo, when , we have thatThus, we have thatHence, the self-map has limit shadowing property. This completes the proof.

5. Conclusion

Firstly, the new concepts of expansibility, almost periodic point, and limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map and the shift map in the inverse limit space under topological group action. The following new results are obtained in the inverse limit space under topological group action. (1) If the map is an equivalent map, then we have . (2) If the map is an equivalent surjection, then the self-map is expansive if and only if the shift map is expansive. (3) If the map is an equivalent surjection, then the self-map has limit shadowing property if and only if the shift map has limit shadowing property.

almost periodic point, expansibility, and limit shadowing property, respectively, mean almost periodic point, expansibility, and limit shadowing property. Otherwise, it does not hold. Hence, Theorems 1–3, respectively, generalize the corresponding results given in the study by Li [1], Niu [18], and Liang and Li [10]. Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.

Data Availability

The data used to support the findings of this study are included in references [1–21] 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).

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Copyright

Copyright © 2021 Zhanjiang Ji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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