The -Asymptotic Tracking Property and -Asymptotic Average Tracking Property in the Inverse Limit Spaces under Group Action

Abstract

Firstly, we introduce the definitions of -asymptotic tracking property, -asymptotic average tracking property, and -quasi-weak almost-periodic point. Secondly, we study their dynamical properties and characteristics. The results obtained improve the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space and provide the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, and computer science.

1. Introduction

Let be a metric space and let be a continuous map. The sequence is called -pseudo orbit of if, for any , we have . The sequence is said to be -shadowed by some point if, for any , we have . The map is said to have the shadowing property if for each there exists such that, for any -pseudo orbit of , there exists a point such that the sequence is -shadowed by the point (see [1]).

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 [1–13]. In 1980, the concept of average tracking property was introduced by Blank [1] and it was proved that some perturbed hyperbolic systems have the average tracking property. In Wang and Zeng [2], the concept of -average tracking property is given and the -average tracking property means chain transitivity under some conditions. Fakhari and Ghane [3] introduced the concept of ergodic tracking property and discussed its dynamical properties. Liang and Li [4] discussed the relation between the shift mapping and the self-mapping about the asymptotic tracking property in the inverse limit space. By the definitions of the -tracking property of Ekta and Tarun [10], we introduce the concepts of -asymptotic tracking property. By the definition of the asymptotic average tracking property of Gu [13], we give -asymptotic average tracking property. The following conclusions are obtained: (1) The self-mapping has -asymptotic tracking property if and only if the shift mapping has -asymptotic tracking property. (2) The self-mapping has -asymptotic average tracking property if and only if the shift mapping has -asymptotic average tracking property. Thus, we generalize the conclusion of Liang and Li [4]. The quasi-weak almost-periodic point is an important concept in the dynamical system, which has also attracted the attention of many scholars. The relevant results are shown by Ma [14] and Zhou and He [15]. Ma [14] proved . In this paper, we introduce the concept of -quasi-weak almost-periodic point and study its topological structure in the inverse limit space under the action group. We obtain and generalize the result of Ma [14].

2. -Asymptotic Tracking Property

In this section, we will prove Theorem 1. For the convenience of the reader, we give the concept used in this section. Now we start with the following definitions.

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

Definition 2. (see [10]). Let be a metric -space and let be a continuous map. The map is said to be an equivariant map if we have for all and .

Definition 3. (see [16]). Let be a metric -space and let be a continuous map. is said to be the inverse limit space if we write . is denoted by in this paper.
The metric in is defined by , where and . The shift mapping is defined by . Thus, is a compact metric space and the shift mapping is a homeomorphism map.

Definition 4. (see [16]). Let be a metric -space and let be an equivariant map. Write and , where . The map is defined by , where and . Then is a metric -space.
Let and be shown as above. The space is called the inverse limit space of under group action.

Definition 5. (see [10]). Let be a metric -space and let be a continuous map. The sequence is called be -pseudo orbit of if for any there exists such that .

Definition 6. (see [10]). Let be a metric -space and let be a continuous map. The sequence is said to be -shadowed by some point if for any there exists such that .

Definition 7. (see [10]). Let be a metric -space and let be a continuous map. The map has -tracking property if for each there exists such that, for any -pseudo orbit of , there exists a point such that the sequence is -shadowed by the point .

Remarks 1. By the definitions of the -tracking property, we will give the concept of -asymptotic tracking property.

Definition 8. Let be a metric -space and let be a continuous map from to . The map has -asymptotic tracking property if for each there exists such that, for any -pseudo orbit of , there exists a point and such that the sequence is -shadowed by the point .

Now, we start to prove Theorem 1.

Theorem 1. Let be the inverse limit space of under group action. If the map is an equivalent surjection, we have that the self-mapping has the -asymptotic tracking property if and only if the shift mapping has the -asymptotic tracking property.

Proof. Suppose that the map has the -asymptotic tracking property. Since is compact, we write . For any , let satisfy . According to the fact that the map is uniformly continuous, for any , there exists such that impliesBy the definition of -asymptotic tracking property, for , there exists and such that the map satisfies the condition of the -asymptotic tracking property. Let be -pseudo orbit, where . Then, for any , there exists such thatHence, we have thatThat is,So are -pseudo orbit of the map . Thus, for every , there exists , , and such thatAccording to (1) and the equivalent definition, for any and , it follows thatBecause of the surjectivity of the map , we can choose and . Then we have thatSo, the map has the -asymptotic tracking property.
Next we suppose that the shift mapping has the -asymptotic tracking property. Let . For any , there exists and such that, for any -pseudo orbit of the shift mapping , we have that is -shadowed by the point and . Because the map is uniformly continuous, it follows that, for any , there exists such that impliesNow suppose that are -pseudo orbit of the map . Then, for any , there exists such thatAccording to (8) and the equivalent definition, for any and , we have thatAccording to the surjectivity of the map , we can choose and . Combining and (10), when , we haveSo, is -pseudo orbit of the shift mapping . By the definition of -asymptotic tracking property of the map , for any , there exists , , and such thatThus, we have thatHence, we have thatSo the map has the -asymptotic tracking property. Thus, we end the proof.

3. -Asymptotic Average Tracking Property

Definition 9. (see [17]). Let . Ifthen the set is said to be a zero density set.

Definition 10. (see [13]). Let be a metric space and let be a continuous map. The sequence in is called an asymptotic average pseudoorbit of the map if

Definition 11. (see [13]). Let be a metric space and let be a continuous map. The map is considered to have the asymptotic average tracking property if, for any asymptotic average pseudoorbit , there exists a point in such that

Remarks 2. According to the definition of asymptotic average tracking property, we will give the concept of -asymptotic average tracking property.

Definition 12. Let be a metric -space and let be a continuous map. The sequence in is called an -asymptotic average pseudoorbit if there exists such that

Definition 13. Let be a metric -space and let be a continuous map. The map is considered to have the -asymptotic average tracking property if for any -asymptotic average pseudo‐orbit there exists a point in and such that

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

Lemma 1 (see [17]). Let be nonnegative real bounded sequence. Then the following conclusions are equivalent:(1)(2)There exists a zero density set J such that

Now we will prove Theorem 2 by Lemma 1.

Theorem 2. Let be the inverse limit space of under group action. If the map is an equivalent surjection, we have that the self-mapping has the -asymptotic average tracking property if and only if the shift mapping has -asymptotic average tracking property.

Proof. Suppose that the map has the -asymptotic average tracking property. Since is compact, we write . For any , let satisfyAccording to the fact that the map is uniformly continuous, for any , there exists such that impliesLet be -asymptotic average pseudo‐orbit, where . Then there exists such thatBy Lemma 1, there exists a zero density set such thatThen, there exists such that when and , we have thatThus, we have thatThat is,Hence, we have thatAccording to Lemma 1, we have thatHence, the sequence is -asymptotic average pseudoorbit of the map . By the definition of -asymptotic average tracking property of the map , there exist and such thatAccording to Lemma 1, there exists a zero density set such thatThus, there exists such that when and , we have thatAccording to the equivalent definition of the map and (21), for any , we have thatAccording to the surjectivity of the map , we can choose and . By (20) and (32), when and , we have thatSo,Combining with Lemma 1, we have thatSo, the shift mapping has the -asymptotic average tracking property.
Next we suppose that the shift mapping has the -asymptotic average tracking property. For any , let such thatBecause the map is uniformly continuous, it follows that, for any , there exists such that impliesNow suppose that is -asymptotic average pseudoorbit of the map . Then there exists such thatBy Lemma 1, there exists a zero density set such thatHence, there exists such that when and , we have thatAccording to the equivalent definition of the map and (37), for any , we have thatAccording to the surjectivity of the map , we can choose and , where . By (36) and (41), when and , we have thatHence, we haveCombining with Lemma 1, we can get thatSo, is -average pseudoorbit. By the definition of -asymptotic tracking property, there exist and such thatBy Lemma 1, there exists a zero density set such thatThus, we have thatBy Lemma 1, we have thatSo, the map has the G-asymptotic average tracking property. Thus, we end the proof.

4. -Quasi-Weak Almost-Periodic Point

First, we give the definition that is used in this section. Second, we give the proof process of Theorem 3.

Definition 14. (see [18]). Let be a metric space and let be a continuous map. A point is considered to be a quasi-weak almost-periodic point if, for any , there exists and nonnegative increasing integers sequence such that . The quasi-weak almost-periodic point set of the map f is denoted by .

Remarks 3. According to the concept of quasi-weak almost-periodic point, we will give the concept of -quasi-weak almost-periodic point.

Definition 15. Let be a metric -space and let be a continuous map. A point is considered to be a -quasi-weak almost-periodic point if, for any , there exists , nonnegative increasing integers sequence , and such that . the -quasi-weak almost-periodic point set of the map f is denoted by .

Now, we start to prove Theorem 3.

Theorem 3. Let be the inverse limit space of under group action. If the map is an equivalent surjection, we have that .

Proof. Suppose that . According to the definition of the -quasi-weak almost-periodic point, for any , there exists , nonnegative integer sequence , and such thatWriteNow, suppose that . Then we have thatHence, we have thatThus, we can obtain . So, we get thatSo, . Hence, .
Suppose that . Then, for any , we have . Since is compact, we write . For any , let satisfyAccording to , there exists , nonnegative integer sequence , and such thatLet . WriteSuppose that . Then we have thatThus, we have thatThus, we can obtain . So we get thatHence, . Then . This completes the proof.

5. Conclusions

In this paper, we study dynamical properties and characteristics of -asymptotic tracking property, -asymptotic average tracking property, and -quasi-weak almost-periodic point. The results obtained can generalize the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space. Most importantly, the paper provides the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, nature, and society.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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