Some Weak Specification Properties and Strongly Mixing?

Qi YANJiandong YINTao WANG

1 Introduction

By a topological dynamical system(X,f)(a dynamical system for short),we mean that X is a compact metric space with metric d and f:X→X is continuous.

The specification property has turned out to be an important notion in the study of dynamical systems.It was firstly introduced by Bowen in[2](see also[1,3,6]for some examples with the specification property and some basic properties).Nowadays,many authors have given their attention to the study of the specification property and raised several kinds of specification properties,such as the strong specification property,the periodic specification property,the almost specification property,the weak specification property,etc.(see[4–5,7]).In this article,we will follow the terminology of[4].

Definition 1.1(see[4])We say that a surjective continuous map f:X→X has the weak specification property(briefly WSP),if for any δ>0,there is a positive integer Nδsuch that for any two points y1,y2and any sequencewiththere is a point x∈X such that,for each positive integer m=1,2 and all integers i withthe following condition holds:

WSP is one of the weakest forms of specification property.And it is known that a map with WSP is strongly mixing.This result is strongly dependent on the assumption that f is surjective,since this result may not be true if all the other conditions of WSP but the surjective property are satisfied.See the following example for details.

Example 1.1 Letn≥1.Under the usual metric of R(the real space),(X,f)is a dynamical system.Then(X,f)satisfies all the other conditions of WSP except the surjective property.

Therefore,(X,f)satisfies all the other conditions of WSP except the surjective property.

Conversely,a natural question appears:Does the strongly mixing property imply WSP?In this paper,we first show by an example that the strongly mixing property is not enough to imply WSP;the concrete example will be given in Section 3.Furthermore,we introduce two weaker concepts of specification property than WSP,which are called the quasi-weak specification property and the semi-weak specification property in this article,respectively.See the following Definitions in more details.

Definition 1.2 We say that a surjective continuous map f:X→X has the quasi-weak specification property(briefly QWSP),if for any δ>0,there is a positive integersuch that for any two points y1,y2and any n≥ Nδthere is a point x∈X such that d(x,y1)< δ and

Definition 1.3 We say that a surjective continuous map f:X→X has the semi-weak specification property(briefly SWSP),if for any δ>0,there is a positive integersuch that for any two points y1,y2and any sequencethere is a point x∈X and for each positive integer m=1,2 there exists an integer i withsuch that

On the basis of these concepts,we show the equivalence of the quasi-weak specification property,the semi-weak specification property and strongly mixing.If we note the example given in Section 3,this result shows that QWSP and SWSP are strictly weaker than WSP.

2 Preliminaries and Basic Concepts

Let(X,f)be a dynamical system.In this paper,we use Z+to denote the set of all nonnegative integers and use N to denote the set of all positive integers and denote the sets of periodic points,almost periodic points,recurrent points,and non-wandering points of f by P(f),A(f),R(f)and ?(f),respectively.Let x ∈ X,denote by orb(x,f)and ω(x,f)the orbit of x and the ω-limit set of x under f,respectively.

Denote B(x,ε)by the ε-neighborhood of x,that is

We set,for nonempty open subsets U,V of X,

f is(topologically)transitive,if for any two nonempty open sets U,V?X,N(U,V)6=?;

f is strongly mixing,if N(U,V)is co finite,namely,there exists N∈N such that for any

Let(X,f),(Y,g)be two dynamical systems with metric d,d1,respectively.The product system of(X,f)and(Y,g)is denoted by(X×Y,f×g).The metricon X×Y is de fi ned as

whenever(x1,y1),(x2,y2)∈X×Y.

If there is a continuous surjective map φ :X → Y with φ ?f=g? φ,we will say that f and g are semi-conjugate(by φ).The map φ is called a semi-conjugacy or a factor map(from f to g).The map g is called a factor of f and the map f is called an extension of g.If φ is a homeomorphism,then we call it a conjugacy(from f to g).

Next,we introduce some basic notations of symbolic dynamical systems.

Suppose that S={0,1}andis the one-sided symbolic space on S.A distance on Σ2is defined as follows:For

Thenis a compact metric space.A shift map σ :is defined as follows:for anyis called the one-sided symbolic dynamical system.

Call V a tuple of S,if V is a finite arrangement of some elements of S.Ifwhere vi∈ S for i=1,···,r,then we call r the length of V,denoted by|V|.Denote by S?the set of all the tuples of S.Let W=w1w2···wsbe another tuple of S,denote

Then V W is also a tuple of S.V is said to occur in W,denoted by,if there is p≥0 such thatOtherwise,denoted by.Letandbe a tuple of S.We say that V is a tuple of x,if there exists i ≥ 1 such thatwe say that V occurs in x in finite times,if there exists a positive integer sequencesuch thatj=1,2,···,r for any i≥ 1.

3 Main Results and Proofs

Firstly,we present some properties of WSP.

Proposition 3.1 Let(X,f)and(Y,g)be two dynamical systems and f,g be semi-conjugate.If(X,f)has WSP,so does(Y,g).

Proof Let φ :X → Y be the semi-conjugate map from f to g.For any ε>0,there exists δ>0 such that d1(φ(x),φ(y))< ε for all x,y ∈ X with d(x,y)< δ.Let Nδbe such a positive integer corresponding to δ as in the Definition of WSP.For the above ε,take Nε=Nδ>0.Then for any y1,y2∈ Y,there exist x1,x2∈ X such that φ(x1)=y1, φ(x2)=y2.For any sequence 0=j1≤k1

Let w=φ(z),then

Thus(Y,g)has WSP.

Proposition 3.2 Let(X,f)and(Y,g)be two dynamical systems.If(X,f)and(Y,g)have WSP,then(X×Y,f×g)has WSP.

Proof For any ε>0,letbe such positive integers given by WSP of(X,f)and(Y,g),respectively.TakeFor anyand any sequencewiththere existsuch that for each positive integer m=1,2 and all integers i with,the following conditions hold:

for all integers j withThereforehas WSP.

Remark 3.1 Propositions 3.1–3.3 are also true for QWSP and SWSP.

Next we list two lemmas,which are helpful for the proofs of our main results.

Lemma 3.1(see[8])Suppose that there exists x ∈ X such that ω(x,f)=X,then f is strongly mixing if and only if for any ε>0 there is N>0 such that

for all n≥N.

Lemma 3.2 Let(X,f)be a dynamical system.If ?(f)=X,then f is surjective.

Proof The proof is simple,so we omit it.

As is well known that a dynamical system with WSP is strongly mixing,how about the converse?The following Theorem 3.1 shows that the converse may not be true.

Lemma 3.3 Let(X,f)be a dynamical system,then(X,f)has QWSP if and only if f is strongly mixing.

Proof First,we prove the necessity.

Let U,V ?X be any nonempty open sets,then there existsuch that B(x,δ0) ? U andLet Nδ0be such a positive integer corresponding to δ0as appears in the Definition of QWSP.Since f is surjective,for anythere exists z∈X such that y=fn(z).By the Definition of QWSP,there is r∈ X such that d(r,x)< δ0andThuswhich implies that f is strongly mixing.

Next,we prove the sufficiency.

Since f is strongly mixing,?(f)=X.By Lemma 3.2,f is surjective.

Theorem 3.2 Let(X,f)be a dynamical system,then the following statements are equivalent:

(1)f is strongly mixing;

(2)(X,f)has QWSP;

(3)(X,f)has SWSP.

Proof By Lemma 3.3,(X,f)has QWSP if and only if f is strongly mixing,and by Lemma 3.4,we have that if(X,f)has SWSP,then f is strongly mixing.Thus,we only need to prove that(X,f)has SWSP if(X,f)has QWSP.

Suppose that(X,f)has QWSP.For any δ>0,let Nδbe such a positive integer corresponding to δ as appears in the Definition of QWSP.For any two points y1,y2and any sequenceby the surjective property of f,there exist y3,y4such thatLetSince(X,f)has QWSP,there exists z∈ X such thatδ andNote that f is surjective,then there is z1∈X such thatThus

Hence(X,f)has SWSP.

Remark 3.3 By the main results of this paper,one can deduce that both QWSP and SWSP are strictly weaker than WSP.

Remark 3.4 In particular,WSP,QWSP,SWSP and strongly mixing are equivalent for the case of interval maps.And we believe that WSP,QWSP,SWSP and strongly mixing are equivalent for the case of sub-shifts of finite type.

AcknowledgementThe authors are grateful to the referee for his(her)critical remarks leading to improvement of the presentation of the work.

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