The <i>d</i>-Shadowing Property and Average Shadowing Property for Iterated Function Systems

Jiang, Jie; Wang, Lidong; Zhao, Yingcui

doi:https://doi.org/10.1155/2020/4374508

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Research Article | Open Access

Volume 2020 | Article ID 4374508 | https://doi.org/10.1155/2020/4374508

The d-Shadowing Property and Average Shadowing Property for Iterated Function Systems

Jie Jiang,¹Lidong Wang,²and Yingcui Zhao³

Academic Editor: Mojtaba Ahmadieh Khanesar

Received25 Aug 2019

Revised04 Dec 2019

Accepted25 Jan 2020

Published29 Apr 2020

Abstract

In this paper, we introduce the definitions of -shadowing property, -shadowing property, topological ergodicity, and strong ergodicity of iterated function systems . Then, we show the following: if has the -shadowing property (respectively, -shadowing property), then has the -shadowing property (respectively, -shadowing property) for any ; if has the -shadowing property (respectively, -shadowing property) for some , then has the -shadowing property (respectively, -shadowing property); if has the -shadowing property or -shadowing property, and or is surjective, then is chain mixing; let be open maps. For with the -shadowing property (respectively, -shadowing property), if is dense in X, and s is a minimal point of or for any , then is strongly ergodic, and hence, is strongly ergodic; and for with the average shadowing property, if is dense in X, and s is a quasi-weakly almost periodic point of or for any , then is ergodic.

1. Introduction

In this paper, let and . Suppose that X is a compact metric space and f: a continuous map. The set is a syndetic set if there is such that for each . For any , let denote the ε-neighborhood of x. is a minimal point of f if for any neighborhood U of x, the set is syndetic, and the set of all minimal points of f is denoted by . is called a quasi-weakly almost periodic point of f if for any neighborhood U of x, the set has positive upper density.

The shadowing property is a very important notion in dynamical systems. Many researchers have found some relationship among various shadowing properties, chain transitivity, transitivity, and ergodicity. Gu [1] proved that if has the asymptotic average shadowing property, and f is surjective, then is chain transitive. For more recent results about various shadowing properties, one can refer to [2–9] and references therein.

A δ-ergodic-pseudo-orbit of f is a sequence such that for any ,where represents the cardinality.

f is said to have the -shadowing property if for any , there is such that every δ-ergodic-pseudo-orbit is ε-shadowed by a true orbit in a way such thatwhere f is said to have the -shadowing property if for any , there is such that every -ergodic-pseudo-orbit is ε-shadowed by a true orbit in a way such that

Let X be a metric space and be continuous maps on X. The iterated function system is the action of the semigroup generated by on X. In this paper, we introduce the definitions of the -shadowing property and -shadowing property for .

An orbit of is a sequence , where , and for any ,

A sequence is called a δ-ergodic-pseudo-orbit for if there is such that

is said to have the -shadowing property if for any , there is such that every δ-ergodic-pseudo-orbit is ε-shadowed by a true orbit in a way such that

is said to have the -shadowing property if for any , there is such that every -ergodic-pseudo-orbit is ε-shadowed by a true orbit in a way such that

Denote , where .

In this paper, the definitions of the shadowing property and average shadowing property for are introduced by Bahabadi [10], and the definition of the asymptotic average shadowing property for is introduced by Nia [11]. Let be open maps. For with the -shadowing property (respectively, -shadowing property), if is dense in X, and s is a minimal point of or for any , then is strongly ergodic. Under similar conditions, Niu [2] researched that has the average shadowing property and then the conclusion is true. Wang and Niu showed that, for with the average shadowing property, if is dense in X, and s is a quasi-weakly almost periodic point of f for any , then f is transitivity (see [12]). However, under similar conditions, we will prove that is ergodic. Then, we will come up with a situation that does not have the asymptotic average shadowing property.

According to Bahabadi [10], a finite sequence is called a δ-chain of if for any , there is such that .

Definition 1. (see [10]). is as follows:(1)Chain transitive if for any and any , there is a δ-chain of from x to y(2)Transitive if for any nonempty open sets , there are and such that (3)Mixing if for any nonempty open sets , there are and such that for any (4)Chain mixing if for any and any , there is such that for any , there is a δ-chain of from x to y consisting of exactly n elementsFor any , define the positive upper density of A byDefine the lower density of A byIf , is the complementary set of A.
ω in the pseudo-orbit of the shadowing property,-shadowing property, and -shadowing property for is the same as the one chosen in the shadowing orbit, while the two in the definitions of the average shadowing property and asymptotic average shadowing property may be different.

2. The -Shadowing Property and -Shadowing Property for

Bahabadi [10] introduced the definition of the shadowing property for . In this paper, we will introduce the definitions of the -shadowing property and -shadowing property for and give some results.

Definition 2. A sequence is called a δ-pseudo-orbit for if there is such that for any ,

Definition 3. has the shadowing property if for any , there is such that every δ-pseudo-orbit is ε-shadowed by a point , i.e., there is such that for any ,

Example 1. Suppose that . For any , there is . Let be a δ-ergodic-pseudo-orbit for . Therefore, there is such thatThen, there is such thatObviously, has the -shadowing property and -shadowing property.

Lemma 1. For any , and any , there is such that there is any δ-chain of length for , i.e., there is such that satisfies

Proof. Fix and let . Since and are uniformly continuous, for any , is also uniformly continuous, where . Then for , there is such that for any satisfies . Since is a δ-chain of , there is such thatPut . Then,where .

Theorem 1. (1)If has the -shadowing property, then has the -shadowing property for any (2)If has the -shadowing property, then has the -shadowing property for any

Proof. Fix and let . We can find a number satisfying Lemma 1. We will show that is true. We can find a number such that each -ergodic-pseudo-orbit of is δ-shadowed by a true orbit along a set with positive lower density, and . Let be a -ergodic-pseudo-orbit of . That is to say, there is such thatLet Obviously, is a -ergodic-pseudo-orbit of . Then, there is such thatPut .
Obviously, , and then . Now, to prove the theorem, we need to show the following claim.

Claim 1. Put , then .

Proof. Without loss of generality, suppose that , then the claim is true when . Assume , then we will show that . For any , where , we have , and the sequence is a δ-chain. According to Lemma 1, . So, . Therefore, and
Obviously,Then, . So is true. Now suppose that has the -shadowing property. In this case, the proof is the same as for except that . Then, we can see that .

Theorem 2. (1)If has the -shadowing property for some , then has the -shadowing property(2)If has the -shadowing property for some , then has the -shadowing property

Proof. Suppose that has the -shadowing property for some . Let , and we can find a number satisfying Lemma 1. Since has the -shadowing property, there is such that any -ergodic-pseudo-orbit of is -shadowed by a true orbit along a set with positive lower density, and For , we can get by Lemma 1. Let be a δ-ergodic-pseudo-orbit of , and . Therefore, there is such thatLet , thenTherefore, if , then the sequence is a δ-chain. According to Lemma 1, , so is a -ergodic-pseudo-orbit of . There is such thatPut and , then . There is such that , and then the sequence is a -chain. According to Lemma 1, we have , where . Now, to prove the theorem, we need to show the following claim.

Claim 2. Put , then .

Proof. Without loss of generality, suppose that and , then the claim is true when . Assume that , then we will show that . Obviously, or . If , then , and we can see that , where . So and . If , then and .
Then, . So is true. Now suppose that has the -shadowing property for some . In this case, the proof is the same as for except that . Then, we can see that .

Corollary 1. Let be a iterated function system, then the following statements are equivalent:(1) has the -shadowing property (respectively, -shadowing property)(2) has the -shadowing property (respectively, -shadowing property) for some (3) has the -shadowing property (respectively, -shadowing property) for any

Lemma 2. Let ; if and , then ; if and , then . Here, .

Proof. Firstly, we will show . Suppose on the contrary that . Then, . Therefore, we can see . But , which contradicts the hypothesis. Hence, . The proof for the second case is the same as for .

Theorem 3. or is surjective,(1)If has the -shadowing property, then is chain transitive(2)If has the -shadowing property, then is chain transitive

Proof. Without loss of generality, suppose that is surjective. Then, there is a sequence such that for any , . Let be arbitrary, and . Firstly, we will prove that (1) is true. We can find a number as in the definition of the -shadowing property for . Put . Construct a sequence as follows: . Then, is a δ-ergodic-pseudo-orbit for , i.e., there is such thatwherePut . Then, . Since has the -shadowing property, there is such thatAccording to Lemma 2, there are such thatSo is an ε-chain from x to y. Hence, is chain transitive.
Then, we will prove that (2) is true. Construct a sequence as follows:Then, is a δ-ergodic-pseudo-orbit for , i.e., there is such thatwhereThen, . Since has the -shadowing property, there is such thatThen, the proof of this case is the same as for .

Theorem 4. If has the -shadowing property or -shadowing property, and or is surjective, then is chain mixing.

Proof. The result is obtained by Corollary 1, Theorem 3, and Theorem 2.3 in [13].

Lemma 3 (see [10]). is chain transitive and has the shadowing property, and then is transitive.

Corollary 2. has the -shadowing property (respectively, -shadowing property) and the shadowing property, one of is surjective, and then is mixing.

Proof. The result is obtained by Theorem 4 and Lemma 3.
is ergodic if for any pair of nonempty open subsets , there is such that has the positive upper density. is strongly ergodic if for any pair of nonempty open subsets , there is such that is a syndetic set.
A map is an open map if for any open set , is also an open set.

Theorem 5. Let be open maps. For with the -shadowing property (respectively, -shadowing property), if is dense in X, and s is a minimal point of or for any , then is strongly ergodic.

Proof. Suppose that U and V are two nonempty open subsets of X, then there are such that and are syndetic sets, where . Since are syndetic sets, we can find such that for any . Firstly, we will prove if has the -shadowing property, then the conclusion is true. Put . PutWe can see that . For , we can find a number satisfying Lemma 1, where . There is such that every δ-ergodic-pseudo-orbit of is -shadowed by a true orbit along a set with positive lower density, and . Put as follows:That is to say, . Then, is a δ-ergodic-pseudo-orbit for , i.e., there is such thatwhereSince has the -shadowing property, there is such thatLet , then . According to Lemma 2, both are infinite. Therefore, we can find such that , where . Suppose on the contrary that for any , , where . We can seewhich is in contrast with . Another proof can be obtained similarly. Since and , and , where and . Owing to , there are and such that and . Obviously, and are -chains. According to Lemma 1, . Therefore, . Then, and . Obviously, . Put and , then . Hence, .
We write . As are open sets, is an open set. Therefore, W is an open set. Then, there are and such that is a syndetic set. For any , . There is such that . Since , is a syndetic set. As U and V are arbitrary, is strongly ergodic.
Then, we will prove if has the -shadowing property, then the conclusion is true. Put .We can see that . Then, the proof of this case is the same as for case one.

Theorem 6. Let be open maps. For with the -shadowing property (respectively, -shadowing property), if is dense in X, and s is a minimal point of or for any , then is strongly ergodic.

Proof. According to the condition, for any , s is a minimal point of or . For any , it is well known that and then s is a minimal point of or . We can combine Corollary 1 and Theorem 5. Then, is strongly ergodic.
A point is called a stable point of if for any , there is satisfying for any with and any . is called Lyapunov stable if any point of X is a stable point of .

Theorem 7. For with the -shadowing property (respectively, -shadowing property), if is Lyapunov stable, and or is surjective, then is transitive. Hence, is transitive for any

Proof. Suppose that U and V are two nonempty open subsets of X, then there are such that , where are stable points of . There is such that for any ,Without loss of generality, suppose that is surjective. Then, there is a sequence such that for all and . Firstly, we will prove if has the -shadowing property, then the conclusion is true. Put Putthen we can see that . For , we can find a number as in the definition of the -shadowing property for . Construct a sequence as follows:That is to say, . Then, is a δ-ergodic-pseudo-orbit for , i.e., there is such thatwhereSince has the -shadowing property, there is such thatLet , then . According to Lemma 2, both are infinite. Therefore, there are such that , where ; then, and , where and . According to , and .
Then, . Put and , then . Hence, is transitive. According to Corollary 1, is transitive for any .
Then, we will prove if has the -shadowing property, then the conclusion is true. Put .We can see that . Then, the proof of this case is the same as for case one.

3. The Average Shadowing Property for

Bahabadi [10] introduced the definition of the average shadowing property for . In this section, we will research the relationship among the average shadowing property, ergodicity, and strong ergodicity.

Definition 4. A sequence is called a δ-average-pseudo-orbit for if there are and such that for every ,

Definition 5. has the average shadowing property if for any , there is such that every δ-average-pseudo-orbit is ε-shadowed on average by a point , i.e., there is such that

Theorem 8. For with the , if is dense in X, and s is a quasi-weakly almost periodic point of or for any , then is ergodic.

Proof. Suppose that are two nonempty open sets. We can find two points , , and such that . There is such that , where . Therefore, and have the positive upper density, where . We can find such that . Since has the , there is such that every δ-average-pseudo-orbit of is -shadowed on average by a point in X. We can find a number such that , and , where , . Define the sequence as follows: , .
We can choose. Obviously, for ,So is a δ-average-pseudo-orbit for . Hence, there are such thatSince , we have , where , . Put . Then, . Suppose on the contrary that , thenwhich is in contrast with (50). So . Therefore, . According to Lemma 2, and . There are such that , and , , where . Then, . Therefore, and . Let , and , then . Obviously,Therefore, . As are arbitrary, is ergodic.

Theorem 9. Let be open maps. For with the , if is dense in X, and s is a minimal point of or for any , then is strongly ergodic.

Proof. Suppose that U and V are two nonempty open subsets of X. Obviously, the syndetic set has the positive upper density. By Theorem 8, there are such that . We write . As are open sets, is an open set. Therefore, W is an open set. Then, there are , such that is a syndetic set. If , then . There is such that . Since , is a syndetic set. As U and V are arbitrary, is strongly ergodic.

Lemma 4 (Theorem 3.1 in [13]). If has the , then has the for any .

Theorem 10. Let be open maps. For with the , if is dense in X, and s is a minimal point of or for any , then is strongly ergodic.

Proof. According to the condition, for any , s is a minimal point of or . For any , it is well known that and then s is a minimal point of or . We can combine Lemma 4 and Theorem 9. Then, is strongly ergodic.

4. A Remark on the Asymptotic Average Shadowing Property for

The definition of the asymptotic average shadowing property for is introduced by Nia [11]. Here, we will come up with a situation to show that an example does not have the asymptotic average shadowing property.

Definition 6. A sequence is called an asymptotic average pseudo-orbit for if there is such that

Definition 7. has the asymptotic average shadowing property if every asymptotic average pseudo-orbit is shadowed on average by a point , i.e., there is such thatFor any ,

Theorem 11. If is an open set, then U is invariant under . Assume that there is such that or . Suppose that is a point whose orbit is metrically separated from U (for any , ). Then, does not have the .

Proof. Without loss of generality, suppose . Construct as follows: ,
For any , we can find withWe can choose. We can see thatThis implies thatSo is an asymptotic average pseudo-orbit of . We will prove that does not have the . Suppose that has the . For asymptotic average pseudo-orbit , there are such that is asymptotically shadowed on average by the point z. Meanwhile, the orbit of z has to enter U at some point. Otherwise,So does not have the . Therefore, there is such that . Then, for any . Therefore, for n that is large enough,which contradicts with the hypothesis. does not have the .
The following example comes from the study in [13]. It is a vitally important example of . It is controversial whether it has the . We can use the above theorem to prove that it does not have the .

Example 2. Let : be two continuous maps such that if and only if and . Obviously, we can see that , , where . And for any , . According to Theorem 11, does not have the .

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 supported by the Foundation of Zhuhai College of Jilin University, Autonomous Research Foundation for Colleges and Universities of the Party Central Committee, National Young Science Foundation of China (no. 11701066), and Key Natural Science Foundation of Universities in Guangdong Province (no. 2019KZDXM027).

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Copyright © 2020 Jie Jiang et al. 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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