On λ-Power Distributional n-Chaos?

Heman FU Feng TAN

1 Introduction

By a dynamical system(in short,a system)we mean a pair(X,f),where X is a compact metric space with metric d and f:X→X is a continuous map.

As far as we know,the first topological Definition of chaos was introduced by Li and Yorke[6]to describe the complexity of the orbits of points in a system.The Li-Yorke chaos became one of the most discussed topics for the last several decades.Various extensions of Li-Yorke chaos were developed.In 1994,Schweizer B.and Sm′?tal J.[9]introduced distributional chaos and showed that positive topological entropy is equivalent to distributional chaos for interval selfmaps.But the equivalence is no longer valid when a general compact metric space is considered(see[4,7]).

In 2007,Li-Yorke chaos and distributional chaos have been unified by Xiong et al[14]into the frame of F-chaos,where F is a Furstenberg family.Recently,via F-chaos,Xiong et al[13]have introduced the notion of λ-power distributional chaos to strengthen distributional chaos,where λ∈[0,1].The hierarchical relation of these chaos is intensively discussed in[2].

The above Definitions of chaos are expressed in terms of dynamics of pairs.Some authors have realized that notions of chaos can also be stated by means of dynamics of tuples,e.g.,n-scrambled tuples(see[5,12]).Following this idea,we extended distributional chaos to distributional n-chaos for n≥2(see[10]).There we discussed several properties of distributional n-chaos and constructed a transitive system which is distributionally n-chaotic without any distributionally(n+1)-scrambled tuples.

In this paper,we introduce the notion of F-n-chaos generally for a given Furstenberg family F and n ≥ 2.Then we apply F-n-chaos to the study of λ-power distributional n-chaos.Our main aim is to extend some classic results on distributional chaos to be the versions of λ-power distributional n-chaos.This paper is organized as follows.Section 2 is devoted to preliminaries on Furstenberg families and on topological dynamics.In Section 3,for λ∈[0,1],λ-power distributional n-chaos is introduced as a generalization of distributional n-chaos via Furstenberg families,where 0-power distributional n-chaos is the strongest.Then we provide a simple criterion for a system to be 0-power distributionally n-chaotic.In Section 4,we present a transitive system which is 0-power distributionally n-chaotic without any distributionally(n+1)-scrambled tuples.Finally in Section 5,for each λ ∈ [0,1],we show that λ-power distributional n-chaos may exist in minimal systems with zero topological entropy.

2 Preliminaries

2.1 Furstenberg families

We review some notations related to Furstenberg families(see[1]).Denote by Z+,N the set of positive integers and the set of non-negative integers respectively.Denote by P the collection of all subsets of Z+.

F?P is called a Furstenberg family,if it is hereditary upwards,that is,F1?F2and F1∈F imply F2∈F.Obviously,the family of all in finite subsets of Z+is a Furstenberg family,denoted by B.

For a family F,denote

κF is a Furstenber family,called the dual family of F.It is easy to see that κB is the family of co finite subsets.

A subset F of Z+is thick if it contains arbitrarily long runs of positive integers.The family of all thick subset of Z+is denoted by τB.The set in κτB is said to be syndetic.So F ? Z+is syndetic if and only if it is of bounded gaps,i.e.,there is N such thatfor every i∈Z+.

where I ranges over intervals ofare said to be the upper density of J and the upper Banach density of J,respectively.The lower densityand the lower Banach density BD?(J)are defined similarly.

For every t∈[0,1],letare Furstenberg families and

2.2 Topological dynamics

A denotes the closure of the set A in X.For given δ>0,letwhere d(x,A)=inf{d(x,y):y∈A}.

Suppose that(X,f)is a system.A?X is invariant if f(A)?A.(Y,f)is a subsystem ofX is nonempty,closed and invariant.Foris called the meeting time set of U and V.Specially,if U is a singleton{x},is simply written as N(x,V),called the return time set from x to V.(X,f)or f is said to be transitive iffor any two nonempty open setsWe writeand call it the orbit of x.x ∈ X is said to be a recurrent point if x is a limit point of the set Orb(x,f).Clearly,if x is a recurrent point,thenis a transitive system.

(X,f)or f is said to be minimal if there is no proper subsystem of(X,f).If a subsystem(Y,f)of(X,f)is minimal,then we say that Y is a minimal set of X.Each point in a minimal set is called a minimal point.It is well known that x∈X is a minimal point if and only if N(x,U)is syndetic for any neighborhood U of x.

For a finite open cover U of X define

where N(C)is the minimal cardinality among all cardinalities of subcovers of C.The topological entropy ofwhere the supremum is taken over all finite open covers of X.

Consider the setendowed with the discrete topology.Letwith the product topology.Then Σnis a compact metric space,called a symbolic space(on n symbols).A compatible metric on Σnis given by,where,otherwise 0 when x=y.Define σ :any.It is obvious that σ is continuous.is called the full shift(on n symbols).Any subsystem ofis called a subshift.

Eachis called a word over E,whereis the set of all k-words.Ifis an l-word,then we call that the length of A is l,denotedis an m-word,the catenation of A and B is denoted by.Then AB is an(l+m)-word.If A1,A2,···is a sequence of words,then·is regarded as a point of.For simplicity,denote A···A(k times),AA···byandrespectively.We say that A occurs in B,denoted,if there issuch thatholds for each j=1,2,···,l.For a point,we get the Definition ofsimilarly and say that A occurs in x.

Lemma 2.1 comes from[4].

Lemma 2.1 LetIf for any k ≥ 1,there exists K>0 such thatholds for each i≥ 1,then x is a minimal point of σ.

Let Y?Σn.For k≥1,denote

The following Lemma 2.2 is well known,for example see[11].

Lemma 2.2 Suppose thatis a subshift.Then

3 λ-Power Distributional n-Chaos

Suppose that(X,f)is a system and F is a Furstenberg family.

Let A ? X and δ>0.x∈X is said to be an F-attaching point of A if N(x,A)∈F;an F-adherent point of A if x is an F-attaching point of[A]εfor any ε>0;an F-δ-escape point of A if x is an F-attaching point of the setan F-escape point of A if x is an F-δ′-escape point of A for some δ′>0.

3.1 F-n-chaos

Let n≥2.Similar to the Definition of F-chaos in[14],we introduce F-n-chaos.

Denote by(Xn,f(n))the n-fold product systemPutandfor some i 6=j}.

Let δ>0.A tupleis said to be F-δ-n-scrambled ifis an F-adherent point of?nand an F-δ-escape point ofin the product system.A subset C of X is said to be F-δ-n-scrambled if each tupleis F-δ-n-scrambled.A system(X,f)is said to be uniformly F-n-chaotic if there exists an uncountable-n-scrambled set for some

In the same manner,we get the Definitions of F-n-scrambled tuples,F-n-scrambled sets and F-n-chaos.

3.2 λ-power distributional n-chaos

Then Dλis a Furstenberg family for each λ ∈ [0,1].It is easy to see that D1=M(1)? τB and that Dλ1? Dλ2for any 0≤ λ1≤ λ2≤ 1.

Recall that a system is distributionally chaotic if and only if it is D1-chaotic(see[14]),which inspires us to introduce the following intuitive synonyms.

Let λ ∈ [0,1],n ≥ 2 and δ>0.By a λ-power distributionally δ-n-scrambled tuple(or set),we mean a Dλ-δ-n-scrambled tuple(or set respectively).Likewise,(uniformly) λ-power distributionally n-chaotic systems means(uniformly)Dλ-n-chaotic systems.

Surely,1-power distributional n-chaos is just distributional n-chaos defined in[10].λ power distributional n-chaos gets stronger and stronger as λ varies from 1 to 0.In[2],Fu et al constructed examples to demonstrate that λ1-power distributional chaos and λ2-power distributional chaos are not equivalent for any different λ1,λ2∈ [0,1].The examples there,in fact,also show that λ1-power distributional n-chaos and λ2-power distributional n-chaos are not equivalent for any different λ1,λ2∈ [0,1].So all of λ-power distributional n-chaos,where λ∈[0,1],form a hierarchy of distributional n-chaos.

Corollary 4.4 in[13]has offered a criterion for a system to be uniformly 0-power distributionally chaotic.It can be modified slightly into a version for uniformly 0-power distributionally n-chaotic systems.For proof of Theorem 4.1 here,we merely present a simplified and practical criterion as follows.

Proposition 3.1 Suppose that(X,f)is a system and n≥2.If f has n distrinct fixed points pj,j=0,1,···,n ? 1,such thatis dense in X for each j,then(X,f)is uniformly 0-power distributionally n-chaotic.

In fact,Proposition 3.1 implies that(X,f)is generically uniformly 0-power distributionally n-chaotic,that is,the set of 0-power distributionally δ-n-scrambled tuples is residual in Xnfor some fixed δ>0.

We take the most known systemas an example.Note that all ofΣn,0≤ j≤ n?1 are fi xed points of σ and they satisfy the condition stated in Proposition 3.1.Consequently,we give the following example.

Example 3.1 The full shiftis uniformly 0-power distributionally n-chaotic.

4 0-Power Distributionally n-Chaotic Systems Without Distributionally(n+1)-Scrambled Tuples

For n≥2,we have constructed a transitive system which is distributionally n-chaotic without any distributionally(n+1)-scrambled tuples(see[10]).Analogously,a transitive system is constructed,which is 0-power distributionally n-chaotic without any distributionally(n+1)-scrambled tuples.Before doing it,we need some preparations.

4.1 Technical preparations

The following Lemma 4.1 comes from[1,8].

Lemma 4.1 Suppose that(X,f)is a system,and A?X is nonempty and closed.Then x ∈ X is a τ B-adherent point of A if and only if A contains a minimal set of

Proposition 4.1 Suppose that(X,f)is a system and n ≥ 2.If there is a τB-escape point of?(n)in the product system,then(X,f)has at least n distrinct minimal points.

Proof Let∈ Xnbe a τ B-escape point of?(n).Thenis a τ B-attaching point of the setfor some δ>0.By Lemma 4.1,contains a minimal point ofTherefore,y1,···,ynare n distrinct minimal points of(X,f).

Corollary 4.1 Suppose that(X,f)is a system and n≥2.If(X,f)has a distributionally n-scrambled tuple,then(X,f)has at least n distrinct minimal points.

Proof Letbe a distributionally n-scrambled tuple.It follows thatis a D1-escape point ofin the product systemthe corollary holds by Proposition 4.1.

Corollary 4.1 tells us,if a system does not admit n distrinct minimal points,then it has no distributionally n-scrambled tuple.Needless to say,it is not distributionally n-chaotic.

4.2 Construction of examples

Suppose n ≥ 2.Below,we de fi ne a sequence of wordsinductively.

Let A1=1.For k ≥ 2,suppose that Ak?1is de fi ned,and denotethe length of Ak?1.If k=ni+j,where i∈N,0≤j≤ n?1,de fi ne

For any given M∈Z+,there exists m such that

5 Minimal Systems

Liao et al[4]constructed in a symbolic space a minimal and distributionally chaotic system with topological entropy 0.Oprocha[7]obtained an uncountable family of such systems in a symbolic space.Following the ideas in[4,7],for each λ ∈[0,1]and n≥ 2,we construct in a symbolic space a uniformly λ-power distributionally n-chaotic and minimal system with zero topological entropy.

Consequently,whenever

On the one hand,for given pk>1,when

AcknowledgementThe authors would like to thank the referee for the careful reading of this paper and many valuable comments.

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