A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

Abstract

We prove that if a continuous, Lyapunov stable map 𝑓 from a compact metric space 𝑋 into itself is topologically transitive and has the asymptotic average shadowing property, then 𝑋 is consisting of one point. As an application, we prove that the identity map 𝑖_𝑋∶𝑋→𝑋 does not have the asymptotic average shadowing property, where 𝑋 is a compact metric space with at least two points.

1. Introduction

It is well known that the pseudoorbit tracing property is one of the most important notions in dynamical systems (see [1–4]). Blank [5] introduced the notion of the average shadowing property to characterize Anosov diffeomorphisms (see [6]). Yang [7] proved that if a continuous map 𝑓∶𝑋→𝑋 of a compact metric space 𝑋 has the pseudoorbit tracing property and is chain transitive, then it is topologically ergodic. Gu and Guo [8] discussed the relation between the average shadowing property and topological ergodicity for flows and showed that a Lyapunov stable flow with the average shadowing property is topologically ergodic. In a recent work, the author [9] introduced the notion of the asymptotic average shadowing property and found that the asymptotic average shadowing property is closely related with transitivity. More recently, Gu [10] showed that every Lyapunov stable map from a compact metric space onto itself is topologically ergodic, provided it has the asymptotic average shadowing property. However, in this paper, we will show that a continuous, Lyapunov stable map 𝑓 from a compact metric space 𝑋 into itself is topologically transitive and has the asymptotic average shadowing property, then 𝑋 is consisting of one point. As an application, it is shown that the identity map 𝑖_𝑋∶𝑋→𝑋 does not have the asymptotic average shadowing property, where 𝑋 is a compact metric space with at least two points. Moreover, we point out that the proof of [10, Theorem  3.1] cannot be true.

The organization of this paper is as follows. In Section 2, we recall some concepts and useful lemmas. Main results are established in Section 3.

2. Preliminaries

Firstly, we complete some notations and recall some concepts.

In this paper, by a dynamical system we mean a pair (𝑋,𝑓), where 𝑓∶𝑋→𝑋 is a continuous map and 𝑋 is a compact metric space with metric 𝑑. For 𝑥∈𝑋, we write 𝑂(𝑥,𝑓)={𝑓^𝑛(𝑥)}^∞_𝑛=0. Let |𝐴| denote the cardinality of a set 𝐴 and 𝑍⁺={0,1,…}.

A subset 𝑆⊂𝑍⁺ is called the positive upper density if limsup_𝑘→∞1𝑘+10≤𝑗≤𝑘∶𝑗∈𝑆>0. (2.1)

For any two nonempty sets 𝑈,𝑉⊂𝑋, we write 𝑁_𝑓(𝑈,𝑉)={𝑛∈𝑍⁺∶𝑈⋂𝑓^−𝑛(𝑉)≠∅}. Obviously, we have 𝑁_𝑓(𝑈,𝑉)={𝑛∈𝑍⁺∶𝑓^𝑛(𝑈)∩𝑉≠∅}.

A map 𝑓∶𝑋→𝑋 is topologically transitive if 𝑁_𝑓(𝑈,𝑉) is nonempty, for any nonempty open sets 𝑈,𝑉⊂𝑋.

A map 𝑓∶𝑋→𝑋 is topologically ergodic if 𝑁_𝑓(𝑈,𝑉) has positive upper density, for any nonempty open sets 𝑈,𝑉⊂𝑋.

A map 𝑓∶𝑋→𝑋 is topologically weak mixing if 𝑓×𝑓 is topologically transitive.

Let (𝑋,𝑓) be a dynamical system. For 𝛿>0, a sequence {𝑥_𝑖}^∞_𝑖=0 of points in 𝑋 is called a 𝛿 pseudoorbit of 𝑓 if 𝑑(𝑓(𝑥_𝑖),𝑥_𝑖+1)<𝛿 for all 𝑖≥0. A sequence {𝑥_𝑖}^∞_𝑖=0 is said to be 𝜀 traced by some point 𝑧∈𝑋 if 𝑑(𝑓^𝑖(𝑧),𝑥_𝑖)<𝜀 for every 𝑖≥0. A point 𝑥∈𝑋 is said to be a stable point of 𝑓 if, for any 𝜀>0, there exists 𝛿=𝛿(𝑥)>0 satisfying that 𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<𝜀 for any 𝑦∈𝑋 with 𝑑(𝑥,𝑦)<𝛿 and any 𝑛≥0.

A map 𝑓∶𝑋→𝑋 is called Lyapunov stable, if every point of 𝑋 is a stable point of 𝑓.

A map 𝑓∶𝑋→𝑋 is said to have sensitive dependence on initial conditions if every point of 𝑋 is not a stable point of 𝑓.

A map 𝑓∶𝑋→𝑋 is said to have the pseudoorbit tracing property, if, for any 𝜀>0, there exists a 𝛿>0 such that every 𝛿 pseudoorbit of 𝑓 can be 𝜀 traced by some point in 𝑋.

A sequence {𝑥_𝑖}^∞_𝑖=0 of points in 𝑋 is called a 𝛿 average pseudoorbit of 𝑓 if there exists an integer 𝑁=𝑁(𝛿)>0 such that for any integer 𝑛≥𝑁 and every integer 𝑘≥0,1𝑛^𝑛−1_𝑖=0𝑑𝑓𝑥_𝑖+𝑘,𝑥_𝑖+𝑘+1<𝛿. (2.2)

A map 𝑓∶𝑋→𝑋 is said to have the average shadowing property, if for any 𝜀>0 there exists a 𝛿>0 such that every 𝛿 average pseudoorbit {𝑥_𝑖}^∞_𝑖=0 of 𝑓 is 𝜀 shadowed in average by some point 𝑧 in 𝑋, that is, limsup_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑𝑓^𝑖(𝑧),𝑥_𝑖<𝜀. (2.3)

A sequence {𝑥_𝑖}^∞_𝑖=0 of points in 𝑋 is called an asymptotic average pseudoorbit of 𝑓 iflim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑𝑓𝑥_𝑖,𝑥_𝑖+1=0. (2.4)

For a given dynamical system (𝑋,𝑓), a sequence {𝑥_𝑖}^∞_𝑖=0 of points in 𝑋 is said to be asymptotically shadowed in average by some point 𝑧∈𝑋 iflim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑𝑓^𝑖(𝑧),𝑥_𝑖=0. (2.5)

A map 𝑓∶𝑋→𝑋 is said to have the asymptotic average shadowing property, if every asymptotic average pseudoorbit of 𝑓 is asymptotically shadowed in average by some point in 𝑋.

We know from [9] that the pseudoorbit tracing property does not imply the asymptotic average shadowing property and the asymptotic average shadowing property does not imply the pseudoorbit tracing property.

3. Main Results

The following lemmas will be used in the proof of Theorem 3.5.

Lemma 3.1 (see [10]). Let 𝑋 be a compact metric space containing at least two points and 𝑓∶𝑋→𝑋 a continuous map. If 𝑓 is topologically weakly mixing, then 𝑓 has sensitive dependence on initial conditions.

Let (𝑋,𝑓) and (𝑋^′,𝑔) be dynamical systems and 𝑋×𝑋^′ the product space with metric 𝑑^′′((𝑥,𝑥^′),(𝑦,𝑦^′))=max{𝑑(𝑥,𝑦),𝑑^′(𝑥^′,𝑦^′)}, where 𝑑 and 𝑑^′ are, respectively, metrics for 𝑋 and 𝑋^′. Let the map 𝑓×𝑔 be defined by (𝑓×𝑔)(𝑥,𝑥^′)=(𝑓(𝑥),𝑔(𝑥^′))(𝑥∈𝑋,𝑥^′∈𝑋^′).

Lemma 3.2. Let (𝑋,𝑓) be a dynamical system. Then, 𝑓 has the asymptotic average shadowing property if and only if 𝑓×𝑓 has the asymptotic average shadowing property.

Proof. Suppose that 𝑓 has the asymptotic average shadowing property. Let {(𝑥_𝑖,𝑦_𝑖)}^∞_𝑖=0 be an asymptotic average pseudoorbit of 𝑓×𝑓, that is, lim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑^′′𝑓×𝑓𝑥_𝑖,𝑦_𝑖,𝑥_𝑖+1,𝑦_𝑖+1=0. (3.1) This implies that lim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑𝑓𝑥_𝑖,𝑥_𝑖+1=0,lim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑𝑓𝑦_𝑖,𝑦_𝑖+1=0. (3.2) So {𝑥_𝑖}^∞_𝑖=0 and {𝑦_𝑖}^∞_𝑖=0 are asymptotic average pseudoorbits of 𝑓. Hence, there are two points 𝑧₁,𝑧₂∈𝑋 such that lim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑𝑓^𝑖𝑧₁,𝑥_𝑖=0,lim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑𝑓^𝑖𝑧₂,𝑦_𝑖=0. (3.3)(3.4) By Lemma  2.3 in [9] and (3.3), there is a set 𝐽₀⊂𝑍⁺ of zero density such that lim_𝑗→∞𝑑𝑓^𝑗𝑧₁,𝑥_𝑗=0, (3.5) where 𝑗∉𝐽₀. Similarly, there is a set 𝐽₁⊂𝑍⁺ of zero density such that lim_𝑗→∞𝑑𝑓^𝑗𝑧₂,𝑦_𝑗=0, (3.6) where 𝑗∉𝐽₁. Let 𝐽=𝐽₀⋃𝐽₁. Then, 𝐽 is a subset of 𝑍⁺ of zero density and lim_𝑗→∞𝑑^′′𝑓×𝑓^𝑗𝑧₁,𝑧₂,𝑥_𝑗,𝑦_𝑗=0, (3.7) where 𝑗∉𝐽. So, by Lemma  2.3 in [9], we have lim_𝑛→∞1𝑛^𝑛−1_𝑖=0𝑑^′′𝑓×𝑓^𝑗𝑧₁,𝑧₂,𝑥_𝑗,𝑦_𝑗=0. (3.8) Consequently, 𝑓×𝑓 has the asymptotic average shadowing property.
Similarly, one can easily prove that if 𝑓×𝑓 has the asymptotic average shadowing property, then so does 𝑓. Thus, the proof is ended.

Lemma 3.3. A topologically transitive map from a compact metric space into itself is a surjective map.

Proof. By the definition, the proof is easy and is omitted here.

We do not know whether a continuous self-map of a compact metric space with the asymptotic average shadowing property is topologically transitive. However, we have the following lemma which is from [11].

Lemma 3.4. Let (𝑋,𝑑) be a compact metric space. If 𝑓∶𝑋→𝑋 is a equicontinuous surjection and has the asymptotic average shadowing property, then 𝑓 is topologically transitive.

In [10], Gu proved that every Lyapunov stable map from a compact metric space onto itself is topologically ergodic, provided it has the asymptotic average shadowing property. However, we obtain the following theorem.

Theorem 3.5. If a Lyapunov stable map 𝑓 from a compact metric space 𝑋 into itself is topologically transitive and has the asymptotic average shadowing property, then 𝑋 is consisting of one point.

Proof. Let 𝑓∶𝑋→𝑋 be a Lyapunov stable map of a compact metric space 𝑋, and suppose that 𝑓 is topologically transitive and has the asymptotic average shadowing property. Assuming that 𝑋 consists of at least two points, we derive a contradiction. First of all, we note that every Lyapunov stable map is continuous by the definition and that 𝑓 is a surjective map by Lemma 3.3 since 𝑓 is topologically transitive. Since 𝑓×𝑓 also has the asymptotic average shadowing property by Lemma 3.2, 𝑓×𝑓 is topologically transitive by Lemma 3.4. Thus, 𝑓 is topologically weakly mixing by definition. Hence, by Lemma 3.1, 𝑓 has sensitive dependence on initial conditions, so that 𝑓 is not Lyapunov stable. This is a contradiction.

Remark 3.6. Theorem 3.5 shows that [10, Theorem  3.1] cannot be true. Indeed, let 𝑋 be a compact metric space containing at least two points, and let 𝑓∶𝑋→𝑋 be a continuous Lyapunov stable map. Suppose that 𝑓 has the asymptotic average shadowing property. If [10, Theorem  3.1] is correct, then 𝑓 is topologically ergodic, so that 𝑓 is topologically transitive by definition. Thus, by Theorem 3.5 proved above, 𝑋 must be consisting one point. This is a contradiction, and, thus, [10, Theorem  3.1] cannot be true.
To prove Theorem  3.1 in [10], the author first construct a sequence {𝑤_𝑖}^∞_𝑖=0 as follows.
For any two nonempty open subsets 𝑈,𝑉⊂𝑋 and any 𝜀>0, choose 𝑥∈𝑈,𝑦∈𝑉 with {𝑧∈𝑋∶𝑑(𝑥,𝑧)<𝜀}⊂𝑈 and {𝑧∈𝑋∶𝑑(𝑦,𝑧)<𝜀}⊂𝑉. Let 𝑤₀=𝑥, 𝑤₁=𝑦, 𝑤₂=𝑥, 𝑤₃=𝑦, 𝑤₄=𝑥₋₁, 𝑤₅=𝑥, 𝑤₆=𝑦₋₁, 𝑤₇=𝑦, 𝑤₂^𝑘_+𝑗=𝑥₋₂^𝑘−1_+1−𝑗 for 𝑗∈{0,1,…,2^𝑘−1−1}, and 𝑤₂^𝑘_+𝑗=𝑦_{−(2𝑘−1−1)+𝑗−2𝑘−1} for 𝑗∈{2^𝑘−1,2^𝑘−1+1,…,2^𝑘−1}, where 𝑓(𝑥_−𝑡)=𝑥_−𝑡+1 for every 𝑡>0,𝑥₀=𝑥 and 𝑓(𝑦_−𝑙)=𝑦_−𝑙+1 for every 𝑙>0,𝑦₀=𝑦. And then the author showed that the sequence {𝑤_𝑖}^∞_𝑖=0 is an asymptotic average pseudoorbit of 𝑓. Finally, the author defined 𝐽_𝑧={𝑖∶𝑤_𝑖∈{𝑧₋₂^𝑖−1₊₁,𝑧₋₂^𝑖−1₊₂,…,𝑧₋₁,𝑧}and𝑑(𝑓^𝑖(𝑤),𝑤_𝑖)<𝛿} for 𝑧∈{𝑥,𝑦} with the positive constant 𝛿 satisfying that for any 𝑢,𝑣∈𝑋, 𝑑(𝑢,𝑣)<𝛿 implies 𝑑(𝑓^𝑛(𝑢),𝑓^𝑛(𝑣))<𝜀 for every integer 𝑛≥0 and 𝐽_𝑚𝑦=𝑖∈𝐽_𝑦∶𝑤_𝑖=𝑦_−𝑚 (3.9) for every 𝑚≥0 and showed that there is an integer 𝑚₀≥0 such that 𝐷(𝐽_𝑚0(𝑦))>0 which leads to the fact that 𝑓 is topologically ergodic, where 𝐷(𝐽_𝑚0(𝑦)) is the upper density of the set 𝐽_𝑚0.

Remark 3.7. In fact, for a given integer 𝑚≥0 and any 𝑛≥𝑚+1, let 𝑘=[(ln𝑛)/(ln2)], where [(ln𝑛)/(ln2)] is the greatest integer less than or equal to ln𝑛/ln2. Then, by the definition of the sequence {𝑤_𝑖}^∞_𝑖=0,𝐷(𝐽_𝑚(𝑦))≤((𝑘+1)/2^𝑘), which implies 𝐷(𝐽_𝑚(𝑦))=0. Consequently, the proof of Theorem  3.1 from [10] is not correct.

As an application, we obtain the following theorem which follows from Theorem  3.1 in [12]. For completeness, we now give a different proof here.

Theorem 3.8. Let (𝑋,𝑑) be a compact metric space with at least two points. Then the identity map 𝑖_𝑋∶𝑋→𝑋 does not have the asymptotic average shadowing property.

Proof. Clearly, the identity map 𝑖_𝑋 is Lyapunov stable. Assume that the identity map 𝑖_𝑋∶𝑋→𝑋 has the asymptotic average shadowing property. By Lemma 3.4, the identity map 𝑖_𝑋 is topologically transitive. Since (𝑋,𝑑) is a compact metric space with at least two points, by Theorem 3.5, the identity map 𝑖_𝑋 does not have the asymptotic average shadowing property. Thus, the proof is ended.

Remark 3.9. Theorem 3.8 extends Example  5.1 from [9], and the proof of Theorem 3.8 is simpler than that of Example  5.1 from [9].

Acknowledgments

The authors thank the (unknown) referee of the paper for many valuable suggestions, in particular for giving a clear proof of Theorem 3.5 and a good description of Remark 3.6. This research was supported by the NSF of Guangdong Province (Grant no. 10452408801004217) and the Key Scientific and Technological Research Project of Science and Technology Department of Zhanjiang City (Grant no. 2010C3112005).