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 Meanwhile Liao showed that the equivalent characterization is no longer valid when f acts on more general compact metric spaces.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

A(τ) contains an uncountable distributional chaotic set of τ;

τ is chaotic in the sense of Devaney;

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; fn will denote the n-fold iteration of f.

For x, y in X, any real number t and positive integer n, let ξn(f,x,y,t)=#{i∣d(fi(x),fi(y))<t,1≤i≤n},

where we use #(·) to denote the cardinality of a set. Let F(f,x,y,t)=liminfn→∞1nξn(f,x,y,t)F*(f,x,y,t)=limsupn→∞1nξn(f,x,y,t).

Definition 2.1.

Call x, y∈X a pair of points displaying distributional chaos, if

F(f,x,y,t)=0 for some t>0;

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

(X,f) is transitive;

the periodic points are dense in X;

(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

f∣Y is sensitive to initial conditions;

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=1∞ai3i-1∣ai=0,1,2}.
We use the sequence a=a1a2⋯ to denote simply the member ∑i=1∞ai3i-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)={0,ifa=b,13k,ifa≠b,k=min{m≥1∣am≠bm}.
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(λx)=λf(x),f(0)=1,0≤f(x)≤1,
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

f0(α)=0;

f0′(x)<-2 on [λ,α] and f0′(x)≥1 on [α,1];

there exists α0∈(α,1) such that f0(α0)=α and α<f0(1)<α0<f0(λ)<1;

f02(1)=λ,f03(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)={2716-27x8,29≤x≤12,4x3-23,12≤x≤1.

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 [<xref ref-type="bibr" rid="B19">19</xref>, Theorem 8.3]).

Let ϕ1,ϕ2,…,ϕm be contractions on Rn. Then there exists a unique nonempty compact set E such that
E=ϕ(E)=⋃i=1mϕi(E),
where
ϕ=⋃i=1mϕi
is a transformation of subsets of Rn. Furthermore, for any nonempty compact subset F of Rn, the iterates ϕk(F) converge to E in the Hausdorff metric as k→∞.

Lemma 2.10 (see [<xref ref-type="bibr" rid="B19">19</xref>, Theorem 8.8]).

Let {ϕi}1m be contractions on R for which the open set condition holds; that is, there is an open interval V such that

ϕ(V)=⋃i=1mϕi(V)⊂V,

ϕ1(V),ϕ2(V),…,ϕm(V) are pairwise disjoint.

Moreover, suppose that for each i, there exists ri, such that |ϕi(x)-ϕi(y)|≤ri|x-y| for all x,y∈V¯. Then dimE≤t, where dim(·) denotes the Hausdorff dimension and t is defined by
∑i=1mrit=1.Lemma 2.11 (see [<xref ref-type="bibr" rid="B20">20</xref>, Theorem 3.2], [<xref ref-type="bibr" rid="B21">21</xref>]).

Let f:I→I be continuous. Then the followings are equivalent:

ent(f)>0;

A(f) contains an uncountable distributional chaotic set of f.

Lemma 2.12 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

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 [<xref ref-type="bibr" rid="B22">22</xref>]).

Let ent(f)=0 and x∈I be recurrent but not periodic such that f(x)>x. Then the inequality fm(x)<fn(x) holds for all even m and all odd n.

Lemma 2.14 (see [<xref ref-type="bibr" rid="B23">23</xref>, Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M165"><mml:mn>6.1</mml:mn><mml:mo>.</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>]).

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 [<xref ref-type="bibr" rid="B23">23</xref>, Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M170"><mml:mn>6.2</mml:mn><mml:mo>.</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>]).

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 ϕi:I→I by ϕ2(x)=λx, ϕ1(x)=f+-1(ϕ2(x)), ϕ0(x)=f+-1(ϕ1(x)). Then ϕi is a contraction for every i=0,1,2. Let ϕ(x)=⋃i=02ϕi(x). By Lemma 2.9, there exists a unique nonempty compact set E with ϕ(E)=E.
For simplicity, we write ϕi1⋯ik for ϕi1∘ϕi2∘⋯∘ϕik.

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+-1(f+-1(λf(x)))=f+-1(ϕ1(f(x)))=ϕ0∘f(x).

Step 2.

Prove that for any subsets ϕi1⋯ik(I) and ϕj1⋯jk(I), there is an n>0 such that fn∘ϕi1⋯ik(I)=ϕj1⋯jk(I).

Proof.

If x∈I, i=0,1,2, then f3∘ϕi(x)=ϕi∘f(x) by Step 1. Using this repeatedly, we get for any k>0f3k∘ϕi(x)=ϕi∘f3k-1(x).
If for each r=1,2,…,k, we all have ir=jr, then from (3.3),
f3k∘ϕi1⋯ik(I)=ϕi1∘f3k-1∘ϕi2⋯ik(I)=⋯=ϕi1⋯ik∘f(I)=ϕj1⋯jk(I)
(nothing that f(I)=I). Thus the lemma holds for this special case. Assume that there exists some r, 1≤r≤k, such that iq=jq for q<r, but ir≤jr. Then by using (3.3) repeatedly, we know that f3r-1∘ϕi1⋯ik(I) or f2·3r-1∘ϕi1⋯ik(I) has the form ϕl1⋯lrlr+1⋯lk(I), where lq=jq for q=1,…,r. Continuing this procedure, we must get some n, such that fn∘ϕi1⋯ik(I)=ϕj1⋯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=0∞ϕk(I).

Proof.

Since ϕ(I)⊂I, we have ϕk+1(I)=ϕk∘ϕ(I)⊂ϕk(I) for any k>0. So from Lemma 2.9 we get
⋂k=0∞ϕk(I)=limk→∞ϕk(I)=E.

Step 4.

Prove that for any k>0, ϕk(I)=⋃i1⋯ik=02(I) is an invariant set of f, that is, f(ϕk(I))⊂ϕk(I).

Proof.

Note that each ϕi1⋯ik has the form ϕ22⋯2 or ϕ22⋯20ir⋯ik or ϕ22⋯21ir⋯ik. Then, by using Step 1 repeatedly, we have
f∘ϕ22⋯2=ϕ00⋯0∘f,f∘ϕ22⋯20ir⋯ik=ϕ00⋯01ir⋯ik,f∘ϕ22⋯21ir⋯ik=ϕ00⋯2ir⋯ik.
Thus by f(I)⊂I, we have f∘ϕi1⋯ik(I)⊂ϕk(I). Moreover,
f(ϕk(I))⊂⋃i1⋯ik=02f∘ϕi1⋯ik(I)⊂ϕk(I).

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)=⋃i=02ϕi(I) with this union disjoint. Then transforming by ϕi1⋯ik,
⋃i=02ϕi1⋯iki(I)⊂ϕi1⋯ik(I)
again with a disjoint union. Thus the sets {ϕi1⋯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
ϕa(I)=⋂k=1∞ϕa1⋯ak(I),
then ϕa(I)⊂E is nonempty, and if x∈E, then there exists a unique a∈Z(3) with x∈ϕa(I).

We now define a map H of E onto Z(3) by setting H(x)=a if x∈ϕa(I). Then H is well defined. It is easy to see that for each i=0,1,2, the contraction ratio of ϕi≤λ, so the contraction ratio of ϕi1⋯ik≤λk. It follows that diam ϕi1⋯ik(I) converges to zero uniformly for ir∈{0,1,2} as k→∞ (where diam denotes diameter). Thus ϕa(I) is a single point for each a∈Z(3). And so H is injective. Moreover the map H is continuous. Let δk>0 be the least distance between any two of the 3k interval ϕa1⋯ak(I). If x∈ϕα(I), y∈ϕβ(I), and |x-y|<δ, then ρ(α,β)<3-k. Finally, since f(ϕa(I))=ϕτ(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·2m, 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}.
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→∞,τnl(a)→a and furthermore fnl(p)→p, we have that there exists an l≥0 such that
fnl(p)-p<ε.
Let g=f2lm. Since we easily see that H(fs(p))=τs(H(p))∈V if and only if n divides s, it follows that H(f2lm(p))∉V, since n can not divide 2lm. And so g(p)=f2lm(p)≥p+ε. In particular, g(p)>p. By the same argument, we also have g2(p)=f2lm+1(p)≥p+ε. In particular, g2(p)>p. Since nl=(k·2m)l=kl·2lm, from (3.12), gkl(p)-p=fnl(p)-p<ε, that is, gkl(p)<p+ε. Thus we have for the odd kl,
gkl(p)<g2(p).
Note that a is current and nonperiodic for τ2lm, 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).

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∈∧}.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∘τi=fi∘h, it follows that if ρ(τi(x1),τi(x2))<s, then
|fi∘h(x1)-fi∘h(x2)|<t.
This implies
ξn(τ,x1,x2,s)≤ξn(f,h(x1),h(x2),t)
for any n≥0. Thus by the definition of F, we immediately have the following result:
F(τ,x1,x2,s)=0.

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
ξn(f,h(x1),h(x2),s)≤ξn(τ,x1,x2,t),
which gives
F*(τ,x1,x2,t)=1.
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.

LiT. Y.YorkeJ. A.Period three implies chaosBlanchardF.GlasnerE.KolyadaS.MaassA.On Li-Yorke pairsBlockL. S.CoppelW. A.ItoS.TanakaS.NakadaH.On unimodal linear transformations and chaos. IJankováK.SmítalJ.A characterization of chaosLiT. Y.MisiurewiczM.PianigianiG.YorkeJ. A.No division implies chaosLiT. Y.MisiurewiczM.PianigianiG.YorkeJ. A.Odd chaosLiaoG. F.A note on a chaotic map with topological entropy 0LiaoG. F.ω-limit sets and chaos for maps of the intervalOsikawaM.OonoY.Chaos in 0-endomorphism of intervalSmítalJ.Chaotic functions with zero topological entropyLiaoG. F.Chain recurrent orbits of mapping of the intervalMisiurewiczM.SmítalJ.Smooth chaotic maps with zero topological entropyXiongJ. C.A chaotic map with topological entropy 0ZhouZ. L.Weakly almost periodic point and measure centreSchweizerB.SmítalJ.Measures of chaos and a spectral decomposition of dynamical systems on the intervalCovenE. M.KeaneM. S.The structure of substitution minimal setsLiaoG. F.On the Feigenbaum's functional equation fp(λx)=λf(x)FalconerK. J.LiaoG. F.WangL. D.Almost periodicity and the SS scrambled setsLiaoG.WangL.Almost periodicity, chain recurrence and chaosNiteckiZ.Maps of the interval with closed periodic setRuetteS.Chaos for continuous interval maps2002http://www.math.u-psud.fr/~ruette/