CHEN ZHI-ZHI,LIAO LIAND WANG WEI

(1.Institute of Mathematics,Jilin University,Changchun,130012) (2.Institute of Statistics,Jilin University of Finance and Economics,Changchun,130017) (3.Institute of Applied Physics and Computational Mathematics,Beijing,100094)

Communicated by Lei Feng-chun

A Devaney Chaotic System Which Is Neither Distributively nor Topologically Chaotic

CHEN ZHI-ZHI1,2,LIAO LI3AND WANG WEI1

(1.Institute of Mathematics,Jilin University,Changchun,130012) (2.Institute of Statistics,Jilin University of Finance and Economics,Changchun,130017) (3.Institute of Applied Physics and Computational Mathematics,Beijing,100094)

Communicated by Lei Feng-chun

Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos.In this paper,by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic,we give a unif i ed proof for the results of Weiss and Oprocha.

Devaney chaos,distributional chaos,topological entropy

1 Introduction

Devaney,distributional and topological chaos are a few of dif f erent versions of chaos.Let us f i rst recall their concrete def i nitions.

Let(X,d)be a metric space,and f:X→X continuous(sometimes f is said to be a system).We call f Devaney chaotic,brief l y DevC,if it possesses the three properties as def i ned in[1]:

(1)transitivity,i.e.,there exists a point x∈X such that the orbit orb(x,f)={x,f(x), f2(x),···}is dense in X;

(2)periodic density,i.e.,the set of periodic points of f is dense in X;

(3)sensitive dependence(on initial conditions).

To determine if a map is chaotic,it is sufficient to consider whether it possesses the transitivity and the periodic density,since the properties(1)and(2)in the def i nition ofDevaney imply(3)for the case that f is inf i nite(see[2–3]).

The notion of distributional chaos was given in[4](where,however,distributional chaos is called“strong chaos”).We call f distributively chaotic,brief l y DC,if there exists an uncountable set D?X such that any dif f erent points x,y∈D form a distributively chaotic pair,brief l y DC pair,i.e.,there exists an ε＞0 such that

and for any ε＞0,

where

(#denotes the cardinality).

The def i nition of topological entropy was introduced in[5].For more detail discussion we refer the readers to[6].In this note,the topological entropy of f is denoted by ent(f). f is said to be topologically chaotic,brief l y PTE,if ent(f)＞0.

Many researchers gave their attention to the relations among Devaney,distributional and topological chaos(see[7–13]).

By the def i nitions,Devaney chaos is a global characteristic,but distributional and topological chaos are not.One can easily give an example which is either distributional or topological chaos but not Devaney chaos.However,the inverse implications are not so evident.In 1971,Weiss[7]found that the transitivity and the periodic density do not imply PTE,and he had proved essentially that DevC does not imply PTE.Recently,the conclusion of Weiss was restated in[8].To show that DevC does not imply DC,Oprocha[9]constructed a Devaney chaotic subshift without DC pairs where.However,he did not give a strict proof.

In the present paper,by forming a simple example which is Devaney chaotic but neither distributively nor topologically chaotic,we give a unif i ed proof for the results of Weiss and Oprocha.

2 Symbolic Space,Shift and Subshift

Let S={0,1},Σ={x=x0x1···|xi∈S,i=0,1,2,···},and def i ne ρ:Σ×Σ→R as: for any x=x0x1···,y=y0y1···∈Σ,

where i is the minimal integer such that xi/=yi.It is not difficult to check that ρ is a metric on Σ.(Σ,ρ)is compact(see[6])and called the one-sided symbolic space(with two symbols).Def i ne σ:Σ→Σ by

σ is continuous(see[6])and is called the shift on Σ.If X?Σ is closed and σ(X)?X,we call(X,σ|X)or σ|Xa subshift of σ.

Call A a word,if it is a f i nite arrangement of the elements in S.If A=a0···an-1,where ai∈S for i=0,1,···,n-1,then n is said to be the length of A,denoted by|A|=n.Let B=b0···bm-1be another word.Denote

Then AB is also a word.Furthermore,if A0,A1,···are all words,then A0A1···is an element in Σ.We use An(n may be∞)to denote the word arranged by n A's.We say that A occurs in B,denoted by A?B,if there is an i≥0 such that

A word A occurs in a point x∈Σ if it occurs in some initial word of x.For any x∈Σ, i≥0 and n＞0,we use Qn(A)to denote the number of subwords of length n occurring in the word A,and for any W?Σ,

For any x∈Σ,i≥0 and n＞0,x[i,i+n]denotes the word xi···xi+n. We give the following lemma(for the proof see[6]).

Lemma 2.1Let(X,σ|X)be a subshift.Then

3 The Main Results and Their Proofs

Example 3.1Let q1=1,p1=1∞.For each i≥1,def i ne inductively

Put

Lemma 3.1For each i＞0,there exist inf i nite sequences〈Aj〉and〈Bj〉of words satisfying

(P1)Aj∈Pi,Bj∈i,and

for any j≥1 such that u=A1B1A2B2···,which is called an i-representation of u. Proof.For given i＞0,by the def i nition,there exist f i nite sequencesandof words satisfying(P1)such that

where Aj∈Pi(1≤j≤k),Bj∈i(1≤j≤k-1).It follows by induction that for any l≥i+1 and any m＞0,may be written as the form(3.1).In particular, where k≥1 and for each(1≤j≤k),Aj∈Pi,Bj∈i.Since every qlis an initial word of u by the def i nition,we see that for given i＞0 there must exist inf i nite sequences〈Aj〉and〈Bj〉of words satisfying(P1)such that To complete the proof of the lemma,it suffices to show that the inf i nite sequences〈Aj〉and〈Bj〉also satisfy(P2).

Notice that rewriting u cannot contract any gap of zeros.It follows that for any j≥1, if t=maxthenandby the def i nition of u. We then have

Whatever happens,the sequences〈Aj〉and〈Bj〉satisfy(P2).

In the sequel,X always denotes the space which is def i ned in Example 3.1.Put

Lemma 3.2Let x=x0x1···∈X.If x∈E,then for any ε＞0,

(B0denotes nothing,if lt-1=0).Thus for each t,by(P2),one has

Lemma 3.3Let x=x0x1···∈X.If x∈F,then there exists an l≥0 such that σl(x) is a periodic point not to be 0∞.

Proof.By the hypotheses,there exists an i＞0 such that for each k≥i,0kdoes not occur in x.Let

Then 0≤j≤i-1(we prescribe that n0=0).There exists an l≥0 such that xl=1 and 0kdoes not occur infor all k＞j.Since x?u,it follows that for any n＞l withand,there exists an m＞0 such that.Letting n→∞gives

The lemma then follows(noting that for any j≥0,by the def i nition,pj+1is a periodic point not to be).

Theorem 3.1σ|X is a Devaney chaotic system without a DC pair.

Proof.It is evident from the def i nition that σ|X is transitive and has a dense set of periodic points.So σ|X is Devaney chaotic.It remains to show that σ|X contains no DC pairs.For this we let x,y∈X and ε＞0 be given.We prove successively for three possible cases that {x,y}is not a DC pair.

Case 1.{x,y}?E.

In this case,noting that if bothandare less thanthen

we have

Thus,by Lemma 3.2,

Hence

Case 2.{x,y}?F.

By Lemma 3.3,we know that x and y are both eventually periodic points.So for any,if σl(x)=σl(y)for some l≥0,then

if for all l≥0,σl(x)/=σl(y),then there must exist some ε＞0 such that

Case 3.One point is in E and another in F.

We may assume that x∈F,y∈E.By Lemma 3.3,there exists a δ＞0 such that for all i≥0,

Since

it follows that for any i≥0,

Thus we have

provided that

Then for each n,

By Lemma 3.2,

Whatever happens,{x,y}is not a DC pair by the def i nition.

Theorem 3.2ent(σ|X)=0,i.e.,σ|X is not topologically chaotic.

Proof.For given i＞0,let A be a word of length|qi|.From Lemma 3.1 we may observe any i-representation of u,from which we see that A?u if and only iffor some 1≤j≤i and some 0≤m≤2|qi|.Since for 1≤j≤i,0≤m≤2|qi|,it follows that

From Theorems 3.1 and 3.2,we obtain

Corollary 3.1There is a Devaney chaotic system which is neither Distributively chaotic nor topologically chaotic.

This is a unif i ed statement for some known results.

AcknowledgmentThe authors would like to thank Professors Piotr Oprocha and Wen Huang for of f ering us valuable references.

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tion:54H20,58F03,58F08

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1674-5647(2013)02-0148-07

Received date:Sept.30,2011.

2013 Jilin's universities science and technology project during the 12th f i ve-year plan,and the f i nancial special funds for projects of higher education of Jilin province.

E-mail address:chenzz77@163.com(Chen Z Z).