3-Adic System and Chaos

Abstract

Let (𝑍(3),𝜏) be a 3-adic system. we prove in (𝑍(3),𝜏) the existence of uncountable distributional chaotic set of 𝐴(𝜏), which is an almost periodic points set, and further come to a conclusion that 𝜏 is chaotic in the sense of Devaney and Wiggins.

1. Introduction

In 1975, Li and Yorke introduced in [1] a new definition of chaos for interval maps. The central point in their definition is the existence of a scrambled set. Later, it was observed that positive topological entropy of interval map implies the existence of a scrambled set [2]. Many sharpened results come into being in succession (see [3–11]). One can find in [3, 4, 12] equivalent conditions for 𝑓 to be chaotic and in [13] or [14] a chaotic map with topological entropy zero, which showed that positive topological entropy and Li-Yorke chaos are not equivalent.

By the result, it became clear that the positive topological entropy is a much stronger notion than the definition of chaos in the sense of Li and Yorke. To remove this disadvantage, Zhou [15] introduced the notion of measure center and showed importantly dynamical properties of system on its measure center. To decide the concept of measure center, he defined weakly almost periodic point, too, showing that the closure of a set of weakly almost periodic points equals to its measure center and the set of weakly almost periodic points is a set of absolutely ergodic measure 1. These show that it is more significant to discuss problems on a set of weakly almost periodic points. On the other hand, one important extensions of Li-Yorke definition were developed by Schweizer and Smítal in [16]; this paper introduced the definition of distributional chaos and prove that this notion is equivalent to positive topological entropy for interval maps. And many scholars (such as Liao, Du, and Zhou, Wang) proved that the positive topological entropy of interval map is equivalent to the uncountable Li-Yorke chaotic set and the uncountable distributional chaotic set for 𝐴(𝑓), 𝑊(𝑓), and 𝑅(𝑓). Meanwhile Liao showed that the equivalent characterization is no longer valid when 𝑓 acts on more general compact metric spaces.

In this paper, we discuss the existence of uncountable distributional chaotic set of 𝐴(𝑓) in 3-adic system.

The main results are stated as follows.

Main Theorem 1. Let (𝑍(3),𝜏) be a 3-adic system. Then(1)𝐴(𝜏) contains an uncountable distributional chaotic set of 𝜏;(2)𝜏 is chaotic in the sense of Devaney;(3)𝜏 is chaotic in the sense of Wiggins.

2. Basic Definitions and Preparations

Throughout this paper, 𝑋 will denote a compact metric space with metric 𝑑, 𝐼 is the closed interval [0, 1].

For a continuous map 𝑓: 𝑋→𝑋, we denote the set of almost periodic points of 𝑓 by 𝐴(𝑓) and denote the topological entropy of 𝑓 by ent(𝑓), whose definitions are as usual; 𝑓^𝑛 will denote the 𝑛-fold iteration of 𝑓.

For 𝑥, 𝑦 in 𝑋, any real number 𝑡 and positive integer 𝑛, let 𝜉_𝑛𝑓,𝑥,𝑦,𝑡=#𝑖∣𝑑𝑓^𝑖(𝑥),𝑓^𝑖𝑦<𝑡,1≤𝑖≤𝑛, (2.1)

where we use #(⋅) to denote the cardinality of a set. Let 𝐹𝑓,𝑥,𝑦,𝑡=liminf_𝑛→∞1𝑛𝜉_𝑛𝑓,𝑥,𝑦,𝑡𝐹^∗𝑓,𝑥,𝑦,𝑡=limsup_𝑛→∞1𝑛𝜉_𝑛𝑓,𝑥,𝑦,𝑡. (2.2)

Definition 2.1. Call 𝑥, 𝑦∈𝑋 a pair of points displaying distributional chaos, if(1)𝐹(𝑓,𝑥,𝑦,𝑡)=0 for some 𝑡>0;(2)𝐹(𝑓,𝑥,𝑦,𝑡)=1 for any 𝑡>0.

Definition 2.2. 𝑓 is said to display distributional chaos, if there exists an uncountable set 𝐷⊂𝑋 such that any two different points in 𝐷 display distributional chaos.

Definition 2.3. Let 𝑋 be a metric space and 𝑓∶𝑋→𝑋 be a continuous map. The dynamical system (𝑋,𝑓) is called chaotic in the sense of Devaney, if(1)(𝑋,𝑓) is transitive;(2)the periodic points are dense in 𝑋;(3)(𝑋,𝑓) is sensitive to initial conditions.

Definition 2.4. Let 𝑋 be a metric space and 𝑓∶𝑋→𝑋 be a continuous map. The dynamical system (𝑋,𝑓) is called chaotic in the sense of Wiggins, if there exists a compact invariant subset 𝑌⊂𝑋 such that(1)𝑓∣_𝑌 is sensitive to initial conditions;(2)𝑓∣_𝑌 is transitive.

Definition 2.5. Let (𝑋,𝑓) and (𝑌,𝑔) be dynamical systems; if there exists a homeomorphism ℎ∶𝑋→𝑌 such that ℎ∘𝑓=𝑔∘ℎ, then 𝑓 and 𝑔 are said to be topologically conjugate.

The notion of adic system is defined as follows.

Definition 2.6. Put 𝑍(3)=^∞_𝑖=1𝑎_𝑖3^𝑖−1∣𝑎_𝑖=0,1,2. (2.3) We use the sequence 𝑎=𝑎₁𝑎₂⋯ to denote simply the member ∑^∞_𝑖=1𝑎_𝑖3^𝑖−1 in 𝑍(3). Define 𝜌∶𝑍(3)×𝑍(3)→𝑅 as follows: for any 𝑎,𝑏∈𝑍(3), if 𝑎=𝑎₁𝑎₂⋯, 𝑏=𝑏₁𝑏₂⋯, then 𝜌(𝑎,𝑏)=10,if𝑎=𝑏,3^𝑘,if𝑎≠𝑏,𝑘=min𝑚≥1∣𝑎_𝑚≠𝑏_𝑚. (2.4) It is not difficult to check that 𝜌 is a metric on 𝑍(3) and (𝑍(3),𝜌) is a compact abelian group. Define 𝜏∶𝑍(3)→𝑍(3) by 𝜏(𝑎)=𝑎+1 for 𝑎=𝑎₁𝑎₂⋯∈𝑍(3); 𝜏 or (𝑍(3),𝜏) is called the 3-adic system. (see [17])
Call an invariant closed set 𝐴⊂𝐼  3-adic, if the restriction 𝑓∣_𝐴 is topologically conjugate to the 3-adic system.
Consider the following functional equation: 𝑓³()(𝑥),𝑓(0)(𝑥)𝜆𝑥=𝜆𝑓=1,0≤𝑓≤1, (2.5) where 𝜆∈(0,1) is to be determined, 𝑥∈[0,1] and 𝑓³ is the 3-fold iteration of 𝑓.
By ℱ we denote the set of continuous solutions of (2.5) such that any 𝑓∈ℱ satisfies: (𝑝₁) there exists 𝛼∈(𝜆,1) such that 𝑓(𝛼)=0; the restrictions 𝑓∣_[𝜆,𝛼] and 𝑓∣_[𝛼,1] are both once continuously differentiable, and 𝑓^′(𝑥)≥1 on [𝛼,1], 𝑓^′(𝑥)<−2 on [𝜆,𝛼]; (𝑝₂)𝑓(𝜆𝛼)<𝑓(𝜆).

The following Lemma can be concluded by in [18, Theorem  2.1].

Lemma 2.7. Let 0 <𝜆<1,𝛼∈(𝜆,1). Let 𝑓₀∶[𝜆,1]→[0,1] be 𝐶¹ on each of the interval [𝜆,𝛼] and [𝛼,1], and satisfy(1)𝑓₀(𝛼)=0;(2)𝑓^′₀(𝑥)<−2 on [𝜆,𝛼] and 𝑓^′₀(𝑥)≥1 on [𝛼,1];(3)there exists 𝛼₀∈(𝛼,1) such that 𝑓₀(𝛼₀)=𝛼 and 𝛼<𝑓₀(1)<𝛼₀<𝑓₀(𝜆)<1;(4)𝑓²₀(1)=𝜆,𝑓³₀(𝜆)=𝜆𝑓₀(1). Then there exists a unique 𝑓∈ℱ with 𝑓|_[𝜆,1]=𝑓₀. Conversely, if 𝑓₀ is the restriction on [𝜆,1] of some 𝑓∈ℱ, then it must satisfy (1)–(4).

Proposition 2.8. ℱ≠∅.

Proof. Let 𝜆=2/9, 𝛼=1/2. Define 𝑓₀∶[𝜆,1]→[0,1] by 𝑓₀(𝑥)=27−1627𝑥8,291≤𝑥≤2,4𝑥3−23,12≤𝑥≤1. (2.6)
It is not difficult to check that 𝑓₀ satisfies the condition (1)–(4) in Lemma 2.7. So ℱ≠∅.
We will be concerned in the notions of Hausdorff metric and Hausdorff dimension, whose definitions can be found in [19].

Lemma 2.9 (see [19, Theorem 8.3]). Let 𝜙₁,𝜙₂,…,𝜙_𝑚 be contractions on 𝑅^𝑛. Then there exists a unique nonempty compact set 𝐸 such that 𝐸=𝜙(𝐸)=^𝑚_𝑖=1𝜙_𝑖(𝐸), (2.7) where 𝜙=^𝑚_𝑖=1𝜙_𝑖 (2.8) is a transformation of subsets of 𝑅^𝑛. Furthermore, for any nonempty compact subset 𝐹 of 𝑅^𝑛, the iterates 𝜙^𝑘(𝐹) converge to 𝐸 in the Hausdorff metric as 𝑘→∞.

Lemma 2.10 (see [19, Theorem 8.8]). Let {𝜙_𝑖}^𝑚₁ be contractions on R for which the open set condition holds; that is, there is an open interval 𝑉 such that(1)𝜙(𝑉)=⋃^𝑚_𝑖=1𝜙_𝑖(𝑉)⊂𝑉, (2)𝜙₁(𝑉),𝜙₂(𝑉),…,𝜙_𝑚(𝑉) are pairwise disjoint.Moreover, suppose that for each 𝑖, there exists 𝑟_𝑖, such that |𝜙_𝑖(𝑥)−𝜙_𝑖(𝑦)|≤𝑟_𝑖|𝑥−𝑦| for all 𝑥,𝑦∈𝑉. Then dim𝐸≤𝑡, where dim(⋅) denotes the Hausdorff dimension and 𝑡 is defined by ^𝑚_𝑖=1𝑟^𝑡_𝑖=1. (2.9)

Lemma 2.11 (see [20, Theorem 3.2], [21]). Let 𝑓∶𝐼→𝐼 be continuous. Then the followings are equivalent:(1)ent(𝑓)>0;(2)𝐴(𝑓) contains an uncountable distributional chaotic set of 𝑓.

Lemma 2.12 (see [21]). Let 𝑓∶𝑋→𝑋, 𝑔∶𝑌→𝑌 be continuous, where 𝑋, 𝑌 are compact metric spaces. If there exists a continuous surjection ℎ∶𝑋→𝑌 such that 𝑔∘ℎ=ℎ∘𝑓, then ℎ(𝐴(𝑓))=𝐴(𝑔).

Lemma 2.13 (see [22]). Let ent(𝑓)=0 and 𝑥∈𝐼 be recurrent but not periodic such that 𝑓(𝑥)>𝑥. Then the inequality 𝑓^𝑚(𝑥)<𝑓^𝑛(𝑥) holds for all even m and all odd 𝑛.

Lemma 2.14 (see [23, Theorem 6.1.4]). Let 𝑓∶𝐼→𝐼 be an interval map. Then ent(𝑓)>0 if and only if there exists a closed invariant subset ∧⊂𝐼 such that 𝑓∣_∧ is chaotic in the sense of Devaney.

Lemma 2.15 (see [23, Theorem 6.2.4]). Let 𝑓∶𝐼→𝐼 be an interval map. If ent(𝑓)>0, then 𝑓 is chaotic in the sense of Wiggins.

3. Proof of Main Theorem

In the sequel, we always suppose that 𝑓∈ℱ and 𝑓 take the minimum at 𝛼∈(𝜆,1).

Let 𝐼=[0,1], 𝑓₊=𝑓|_[𝛼,1]. For 𝑖=0,1,2, define 𝜙_𝑖∶𝐼→𝐼 by 𝜙₂(𝑥)=𝜆𝑥, 𝜙₁(𝑥)=𝑓₊⁻¹(𝜙₂(𝑥)), 𝜙₀(𝑥)=𝑓₊⁻¹(𝜙₁(𝑥)). Then 𝜙_𝑖 is a contraction for every 𝑖=0,1,2. Let 𝜙(𝑥)=⋃²_𝑖=0𝜙_𝑖(𝑥). By Lemma 2.9, there exists a unique nonempty compact set 𝐸 with 𝜙(𝐸)=𝐸. (3.1) For simplicity, we write 𝜙_{𝑖1⋯𝑖𝑘} for 𝜙_𝑖1∘𝜙_𝑖2∘⋯∘𝜙_𝑖𝑘.

Step 1. Prove that for any 𝑥∈𝐼, 𝑓∘𝜙₀(𝑥)=𝜙₁(𝑥), 𝑓∘𝜙₁(𝑥)=𝜙₂(𝑥), 𝑓∘𝜙₂(𝑥)=𝜙₀∘𝑓(𝑥).

Proof. Letting 𝑓 act on both sides of the equality 𝜙₀(𝑥)=𝑓₊⁻¹(𝜙₁(𝑥)), we get immediately the first equality. A similar argument yields the second equality. To show the third equality, we write (2.5) as 𝑓(𝑓(𝑓(𝜙₂(𝑥))))=𝜆𝑓(𝑥). Since 𝜙₂(𝑥)∈[0,𝜆], it follows from Lemma 2.7 that 𝑓∘𝜙₂(𝑥)∈[𝛼,1] and 𝑓²∘𝜙₂(𝑥)∈[𝛼,1]. By this and definitions of 𝜙₀ and 𝜙₁, we get 𝑓∘𝜙₂(𝑥)=𝑓₊⁻¹𝑓₊⁻¹𝜆𝑓(𝑥)=𝑓₊⁻¹𝜙₁𝑓(𝑥)=𝜙₀∘𝑓(𝑥). (3.2)

Step 2. Prove that for any subsets 𝜙_{𝑖1⋯𝑖𝑘}(𝐼) and 𝜙_{𝑗1⋯𝑗𝑘}(𝐼), there is an 𝑛>0 such that 𝑓^𝑛∘𝜙_{𝑖1⋯𝑖𝑘}(𝐼)=𝜙_{𝑗1⋯𝑗𝑘}(𝐼).

Proof. If 𝑥∈𝐼, 𝑖=0,1,2, then 𝑓³∘𝜙_𝑖(𝑥)=𝜙_𝑖∘𝑓(𝑥) by Step 1. Using this repeatedly, we get for any 𝑘>0𝑓^3𝑘∘𝜙_𝑖(𝑥)=𝜙_𝑖∘𝑓^3𝑘−1(𝑥). (3.3) If for each 𝑟=1,2,…,𝑘, we all have 𝑖_𝑟=𝑗_𝑟, then from (3.3), 𝑓^3𝑘∘𝜙_{𝑖1⋯𝑖𝑘}(𝐼)=𝜙_𝑖1∘𝑓^3𝑘−1∘𝜙_{𝑖2⋯𝑖𝑘}(𝐼)=⋯=𝜙_{𝑖1⋯𝑖𝑘}∘𝑓(𝐼)=𝜙_{𝑗1⋯𝑗𝑘}(𝐼) (3.4) (nothing that 𝑓(𝐼)=𝐼). Thus the lemma holds for this special case. Assume that there exists some 𝑟, 1≤𝑟≤𝑘, such that 𝑖_𝑞=𝑗_𝑞 for 𝑞<𝑟, but 𝑖_𝑟≤𝑗_𝑟. Then by using (3.3) repeatedly, we know that 𝑓^3𝑟−1∘𝜙_{𝑖1⋯𝑖𝑘}(𝐼) or 𝑓^{2⋅3𝑟−1}∘𝜙_{𝑖1⋯𝑖𝑘}(𝐼) has the form 𝜙_{𝑙1⋯𝑙𝑟𝑙𝑟+1⋯𝑙𝑘}(𝐼), where 𝑙_𝑞=𝑗_𝑞 for 𝑞=1,…,𝑟. Continuing this procedure, we must get some 𝑛, such that 𝑓^𝑛∘𝜙_{𝑖1⋯𝑖𝑘}(𝐼)=𝜙_{𝑗1⋯𝑗𝑘}(𝐼).
In, Steps 3, 5, and 6, we always suppose that the notation 𝐸 is as in (3.1).

Step 3. Prove that 𝐸=^∞_𝑘=0𝜙^𝑘(𝐼). (3.5)

Proof. Since 𝜙(𝐼)⊂𝐼, we have 𝜙^𝑘+1(𝐼)=𝜙^𝑘∘𝜙(𝐼)⊂𝜙^𝑘(𝐼) for any 𝑘>0. So from Lemma 2.9 we get ^∞_𝑘=0𝜙^𝑘(𝐼)=lim_𝑘→∞𝜙^𝑘(𝐼)=𝐸. (3.6)

Step 4. Prove that for any 𝑘>0, 𝜙^𝑘(𝐼)=⋃²_{𝑖1⋯𝑖𝑘=0}(𝐼) is an invariant set of 𝑓, that is, 𝑓(𝜙^𝑘(𝐼))⊂𝜙^𝑘(𝐼).

Proof. Note that each 𝜙_{𝑖1⋯𝑖𝑘} has the form 𝜙_22⋯2 or 𝜙_{22⋯20𝑖𝑟⋯𝑖𝑘} or 𝜙_{22⋯21𝑖𝑟⋯𝑖𝑘}. Then, by using Step 1 repeatedly, we have 𝑓∘𝜙_22⋯2=𝜙_00⋯0∘𝑓,𝑓∘𝜙_{22⋯20𝑖𝑟⋯𝑖𝑘}=𝜙_{00⋯01𝑖𝑟⋯𝑖𝑘},𝑓∘𝜙_{22⋯21𝑖𝑟⋯𝑖𝑘}=𝜙_{00⋯2𝑖𝑟⋯𝑖𝑘}. (3.7) Thus by 𝑓(𝐼)⊂𝐼, we have 𝑓∘𝜙_{𝑖1⋯𝑖𝑘}(𝐼)⊂𝜙^𝑘(𝐼). Moreover, 𝑓𝜙^𝑘(𝐼)⊂²_{𝑖1⋯𝑖𝑘=0}𝑓∘𝜙_{𝑖1⋯𝑖𝑘}(𝐼)⊂𝜙^𝑘(𝐼). (3.8)

Step 5. Prove that the restriction 𝑓|_𝐸 is topologically conjugate to 𝜏, where 𝜏 is the 3-adic system as defined in Section 1.

Proof. By the definition of 𝜙, we have 𝜙(𝐼)=⋃²_𝑖=0𝜙_𝑖(𝐼) with this union disjoint. Then transforming by 𝜙_{𝑖1⋯𝑖𝑘}, ²_𝑖=0𝜙_{𝑖1⋯𝑖𝑘𝑖}(𝐼)⊂𝜙_{𝑖1⋯𝑖𝑘}(𝐼) (3.9) again with a disjoint union. Thus the sets {𝜙_{𝑖1⋯𝑖𝑘}(𝐼)} (with 𝑘 arbitrary) form a net in the sense that any pair of sets from the collection are either disjoint or such that one is included in the other. It follows from Step 3 that for any 𝑎=𝑎₁𝑎₂⋯∈𝑍(3), if let 𝜙_𝑎(𝐼)=^∞_𝑘=1𝜙_{𝑎1⋯𝑎𝑘}(𝐼), (3.10) then 𝜙_𝑎(𝐼)⊂𝐸 is nonempty, and if 𝑥∈𝐸, then there exists a unique 𝑎∈𝑍(3) with 𝑥∈𝜙_𝑎(𝐼).
We now define a map 𝐻 of 𝐸 onto 𝑍(3) by setting 𝐻(𝑥)=𝑎 if 𝑥∈𝜙_𝑎(𝐼). Then 𝐻 is well defined. It is easy to see that for each 𝑖=0,1,2, the contraction ratio of 𝜙_𝑖≤𝜆, so the contraction ratio of 𝜙_{𝑖1⋯𝑖𝑘}≤𝜆^𝑘. It follows that diam 𝜙_{𝑖1⋯𝑖𝑘}(𝐼) converges to zero uniformly for 𝑖_𝑟∈{0,1,2} as 𝑘→∞ (where diam denotes diameter). Thus 𝜙_𝑎(𝐼) is a single point for each 𝑎∈𝑍(3). And so 𝐻 is injective. Moreover the map 𝐻 is continuous. Let 𝛿_𝑘>0 be the least distance between any two of the 3^𝑘 interval 𝜙_{𝑎1⋯𝑎𝑘}(𝐼). If 𝑥∈𝜙_𝛼(𝐼), 𝑦∈𝜙_𝛽(𝐼), and |𝑥−𝑦|<𝛿, then 𝜌(𝛼,𝛽)<3^−𝑘. Finally, since 𝑓(𝜙_𝑎(𝐼))=𝜙_𝜏(𝑎)(𝐼) by (3.7), we have 𝐻∘𝑓(𝑥)=𝜏∘𝐻(𝑥) for each 𝑥∈𝐸.

Step 6. Prove that if 𝑓 has an 𝑛-adic set and the 𝑛 is not a power of 2, then ent(𝑓)>0.

Proof. Write 𝑛=𝑘⋅2^𝑚, where 𝑘≥3 is odd and 𝑚≥0 is an integer. Let 𝐴 be the 𝑛-adic set of 𝑓 and 𝑝=min𝐴. There exists a homeomorphism 𝐻∶𝐴→𝑍(𝑛) such that for 𝑥∈𝐴, 𝜏∘𝐻(𝑥)=𝐻∘𝑓(𝑥). We may assume without loss of generality that 𝐻(𝑝)=𝑎=0𝑎₂𝑎₃⋯. Put 𝑉=𝑧∈𝑍(𝑛)∣𝑧₁=0. (3.11) Then 𝑉⊂𝑍(𝑛) is an open neighborhood of the sequence 𝑎. There exists an 𝜀>0, such that for any 𝑞∈𝐴, if 𝑞−𝑝<𝜀, there 𝐻(𝑞)∈𝑉. Note that for 𝑙→∞,𝜏^𝑛𝑙(𝑎)→𝑎 and furthermore 𝑓^𝑛𝑙(𝑝)→𝑝, we have that there exists an 𝑙≥0 such that 𝑓^𝑛𝑙𝑝−𝑝<𝜀. (3.12) Let 𝑔=𝑓^2𝑙𝑚. Since we easily see that 𝐻(𝑓^𝑠(𝑝))=𝜏^𝑠(𝐻(𝑝))∈𝑉 if and only if 𝑛 divides 𝑠, it follows that 𝐻(𝑓^2𝑙𝑚(𝑝))∉𝑉, since 𝑛 can not divide 2^𝑙𝑚. And so 𝑔(𝑝)=𝑓^2𝑙𝑚(𝑝)≥𝑝+𝜀. In particular, 𝑔(𝑝)>𝑝. By the same argument, we also have 𝑔²(𝑝)=𝑓^2𝑙𝑚+1(𝑝)≥𝑝+𝜀. In particular, 𝑔²(𝑝)>𝑝. Since 𝑛^𝑙=(𝑘⋅2^𝑚)^𝑙=𝑘^𝑙⋅2^𝑙𝑚, from (3.12), 𝑔^𝑘𝑙(𝑝)−𝑝=𝑓^𝑛𝑙(𝑝)−𝑝<𝜀, that is, 𝑔^𝑘𝑙(𝑝)<𝑝+𝜀. Thus we have for the odd 𝑘^𝑙, 𝑔^𝑘𝑙𝑝<𝑔²𝑝. (3.13) Note that 𝑎 is current and nonperiodic for 𝜏^2𝑙𝑚, and so is 𝑝 for 𝑔. By Lemma 2.13 we get ent(𝑔)>0. Moreover ent(𝑓)>0.
Finally, we prove that 𝐴(𝜏) contains an uncountable distributional chaotic set of 𝜏. By Step 5, the restriction 𝑓∣_𝐸 is topologically conjugate to 𝜏. Thus there is a homeomorphism ℎ∶𝑍(3)→𝐸 such that for any 𝑥∈𝑍(3), 𝑓∘ℎ(𝑥)=ℎ∘𝜏(𝑥). (3.14)
According to Lemma 2.11, there is an uncountable set ∧⊂𝐴(𝑓), which is distributional chaotic. By Lemma 2.12 for any 𝑦∈∧, there exists 𝑥∈𝐴(𝜏) such that ℎ(𝑥)=𝑦. Let 𝐷=𝑥∣𝑥∈𝐴(𝜏),ℎ(𝑥)=𝑦,𝑦∈∧. (3.15)Then 𝐷 is an uncountable set.
To complete the proof, it suffices to show that 𝐷 is a distributional chaotic set for 𝜏.
First of all, we prove that for any 𝑥₁,𝑥₂∈𝐷, if 𝐹(𝑓,ℎ(𝑥₁),ℎ(𝑥₂),𝑡)=0 for some 𝑡>0, then 𝐹(𝜏,𝑥₁,𝑥₂,𝑠)=0 for some 𝑠>0.
For given  𝑡>0, by uniform continuity of ℎ, there exists 𝑠>0, such that for any 𝑝,𝑞∈𝐷, |ℎ(𝑝)−ℎ(𝑞)|<𝑡, provided 𝜌(𝑝,𝑞)<𝑠. Since we easily see that ℎ∘𝜏^𝑖=𝑓^𝑖∘ℎ, it follows that if 𝜌(𝜏^𝑖(𝑥₁),𝜏^𝑖(𝑥₂))<𝑠, then 𝑓^𝑖∘ℎ𝑥₁−𝑓^𝑖∘ℎ𝑥₂<𝑡. (3.16) This implies 𝜉_𝑛𝜏,𝑥₁,𝑥₂,𝑠≤𝜉_𝑛𝑓,ℎ𝑥₁,ℎ𝑥₂,𝑡 (3.17) for any 𝑛≥0. Thus by the definition of 𝐹, we immediately have the following result: 𝐹𝜏,𝑥₁,𝑥₂,𝑠=0. (3.18)
Secondly, we prove that if 𝐹^∗(𝑓,ℎ(𝑥₁),ℎ(𝑥₂),𝑠)=1 for all 𝑠>0, then 𝐹^∗(𝜏,𝑥₁,𝑥₂,𝑡)=1 for all 𝑡>0. Since ℎ is homeomorphism, ℎ⁻¹∶𝐸→𝑍(𝑛) is a surjective continuous map. By the first proof, we have 𝜉_𝑛𝑓,ℎ𝑥₁,ℎ𝑥₂,𝑠≤𝜉_𝑛𝜏,𝑥₁,𝑥₂,𝑡, (3.19) which gives 𝐹^∗𝜏,𝑥₁,𝑥₂,𝑡=1. (3.20) By (3.18), (3.20), and the arbitrariness of 𝑥₁ and 𝑥₂, we conclude that 𝐷 is an uncountable distributional chaotic set of 𝜏.
The proofs of (2) and (3) of the Main Theorem are obvious.

Acknowledgments

This work is supported by the major basic research fund of Department of Education of Liaoning Province no. 2009A141 and the NSFC no. 10971245 and the independent fund of central universities no. 10010101.