RELATIVE ENTROPY DIMENSION FOR COUNTABLE AMENABLE GROUP ACTIONS∗

(肖祖彪)

School of Mathematics and Statistics, Fuzhou University, Fuzhou 350116, China

E-mail: xzb2020@fzu.edu.cn

Zhengyu YIN (殷正宇)†

Department of Mathematics, Nanjing University, Nanjing 210093, China

E-mail: yzynju 20@163.com

Abstract We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups.First,for a given Følner sequence{ Fn},we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity.we also investigate the relations among these.Second,we introduce the notion of a relative dimension set.Moreover,using the method,we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions,which says that if the relative dimension sets of two extensions are different,then the extensions are disjoint.

Key words amenable groups;relative entropy dimensions;relative dimension sets

1 Introduction

A dynamical system for a group action is usually written by a pair (X,G) whereXis generally a compact metric space (called a phase space) andGis a topological group (called an acted group) which acts continuously onX.In the study of dynamical systems,entropy is an important tool for characterizing the dynamical behavior.Kolomogorov and Sinai developed the measure-theoretic entropy of Z-actions based on Shannon’s information theory in 1959.Topological entropy was first introduced by Adler,Konheim and McAndrew and defined by Bowen later on,in 1973,for a metric space.Topological entropy measures the maximal exponential growth rate of orbits for an arbitrary topological dynamical system.

Although systems with positive entropy are much more complicated than those with zero entropy,zero entropy systems possess various levels of complexity,and have recently been discussed in [3,4,7,8,11,15,20,26].The authors of these work all adopted various methods for classifying zero entropy dynamical systems.Carvalho [4]introduced the notion of the entropy dimension to distinguish the zero entropy systems and obtained some basic properties of the entropy dimension.Ferenczi and Park[11]proposed the entropy dimension for the action of Z or Zdon a probability space.Given a subsetSof Z with density 0,Dou,Huang and Park in [7]introduced the notion of the dimension ofS.Moreover,they used the dimension of a special class of sequences which were called entropy generating sequences to measure the complexity of a system,and showed that the topological entropy dimension can be computed through the dimensions of entropy generating sequences.For the case of the relative setting,Zhou [26]defined the corresponding notions for Z-actions and studied their properties.In this paper,we would like to redefine the relevant notions in the relative setting for group actions by introducing a new notion called “relative entropy generating sets”.Using this concept,we can define the relative dimensions of entropy generating sets and get the relationships among different relative dimensions.We can show that the relative upper entropy dimension of an extension is the supremum of the dimensions of the relative upper entropy generating sets (see Theorem 5.4).

Inspired by the theory of prime numbers,Furstenberg [12]first introduced the concept of disjointness to characterize the difference in the dynamics between two systems.He gave a well-known result which says that a weakly mixing system with a dense set of periodic points is disjoint from all minimal systems.Later Huang,Park and Ye [15]studied systems which are disjoint from all minimal zero entropy systems (denoted byM0).They proved that topologicalK-systems,which means every nontrivial finite open cover of the system has positive entropy,make up a proper subset of the systems which are disjoint fromM0.Dou,Huang and Park in[7]introduced the notion of the dimension set of a zero entropy topological system to measure the various levels of the topological complexity of subexponential growth rate.They investigated the property of disjointness in zero entropy systems through the dimension set and proved that under the condition of one system’s minimality,two systems with disjoint dimension sets are disjoint.This is a refinement and also a generalization of the result adhered in [2]: that uniformly positive entropy systems are disjoint from minimal entropy zero systems.Based on the above results,Zhou [26]introduced the notion of relative dimension tuples and the relative dimension set and proved that two extensions with disjoint relative dimension sets for all orders are disjoint over the same system under some conditions.This can be also regarded as a generalization of the results in [18]: that an open extension with relative uniformly positive entropy of all orders is disjoint from any minimal extension with relative zero entropy.We would like to redefine the notion of relative dimension tuples and the relative dimension set for group actions and show that an open extension is disjoint from a minimal extension if they have disjointn-th relative dimension sets,for anyn ≥2 (see Theorem 7.1).

The rest of the paper is organized as follows: in Section 2,we give the definition of amenable groups and some basic concepts of dynamical systems.In Section 3,we define the relative entropy dimension of an extension for an amenable group action and investigate the relevant properties.In Section 4,we consider the dimensions of the relative entropy generating sets.In Section 5,we study the interrelations among the previous defined dimensions.In Section 6,we give the notions of relative dimension tuples and dimension sets,and study inheritance and lifting properties of the uniform relative entropy dimension extension.In Section 7,we prove the disjointness theorem between the extensions with disjoint relative dimension sets.

2 Preliminaries

2.1 Amenable Groups

LetGbe a countable discrete infinite group.Denote byF(G)the set of all finite non-empty subsets ofG.ForK,F ∈F(G),we write that

whereKF={kf:k ∈K,f ∈F}.The groupGis called amenable if,for anyK ∈F(G) andδ>0,there existsF ∈F(G) such that

where|·| is the counting measure onG.Such a setFis called (K,δ)-invariant.A sequence{Fn}n∈N⊆F(G) is called a Følner sequence if,for everyK ∈F(G) andδ>0,for all large enoughnwe have thatFnis (K,δ)-invariant.A groupGis amenable if and only ifGadmits a Følner sequence{Fn}n∈N.The class of amenable groups contains,in particular,all finite groups,all abelian groups and more generally,all solvable groups;if it is closed under the operations of taking subgroups,taking quotients,taking extensions and taking inductive limits.For more details and properties of the amenable group,one can refer to [5,Chaper 4].

2.2 G-systems and Related Concepts

AG-system is a pair (X,G) whereXis a compact metric space andGis a countable discrete infinite group which acts continuously onX.Suppose thateis the identity ofG.Each elementg ∈Gwill be regarded as a homeomorphic action fromXto itself when there is no confusion.WhenXis a set consisting of a single point,we call (X,G) a trivial system.For a system (X,G) and a positive integerk ≥2,ak-productG-system of (X,G) is denoted by(Xk,G),and we assume thatg(x1,···,xk)=(gx1,···,gxk),for every (x1,···,xk)∈Xkandg ∈G.

Let (X,G) be aG-system and letx ∈X.For a subsetF ⊆G,we denote theF-orbit ofxbyFx={gx:g ∈F}.We will callGxthe orbit ofxinstead ofG-orbit ofxwhen there is no confusion.The subsetKofXis said to beG-invariant ifGK=K(equivalentlyGK ⊆K).Thus a set is invariant if and only if it is a union of orbits.

3 Relative Entropy Dimension

LetXbe a compact metric space and letGbe a countable discrete infinite amenable group.Let(X,G)be aG-system.Without loss of generality,we assume thatGadmits a strictly increasing Følner sequencewheree ∈Fnfor eachn.If not,we can take a subsequence denoted byand then we choose some sequencewithn1

We recall some notations before introducing the concepts of the relative entropy dimension.Given aG-system (X,G),letCXbe the class of finite covers ofX,and letbe the class of finite open covers ofX.LetU=(Ui)i∈I,V=(Vj)j∈Jbe finite covers ofX,and the join ofUandVis the finite cover ofXdefined by

One says thatVis finer thanUif,for anyj ∈J,there is ani ∈Isuch thatVj ⊂Ui.

Let (X,G) and (Y,G) be twoG-systems.We say that (X,G) is an extension of (Y,G) if there exists a continuous surjective mapπ: (X,G)→(Y,G) such thatπ ◦g=g ◦πfor allg ∈G.The mapπis called a factor map fromXtoY.We say thatg ∈Gis an automorphism ofπifπ=π ◦g(see [6]).

Letπ: (X,G)→(Y,G) be a factor map and let U ∈.For a closed setE ⊆X,we denote

Following the definition of the entropy dimension in [7],we give the definition of a relative upper entropy dimension ofUas

Similarly,we denote the relative lower entropy dimension ofUas

then the relative entropy dimension ofUis equal to 1.

Definition 3.1Letπ:(X,G)→(Y,G)be a factor map betweenG-systems.The relative upper (resp.lower) entropy dimension of (X,G) is

NoteIt is known that the definition of entropy for an amenable group action is independent on the choice of Følner sequences;see [17].However,we consider the definition of entropy dimensions of zero entropy systems for amenable group actions,and this is not the case as above,which is heavily dependent on the choice of Følner sequence.We give a simple example as follows: in[7,Example 2.8],a shift system(Y1,σ1)is obtained with the upper entropy dimensionand the lower entropy dimension(Y1,σ1)=(σ1,U1)=0.Let 0<α<.By the definitions of the upper and lower entropy dimensions,we have that

By the definition of lim inf,we can take a subsequenceof{n} such that

The following two propositions are basic properties of relative entropy dimension.

Proposition 3.2Letπ:(X,G)→(Y,G) be a factor map andU,V ∈.Then,

(1) ifUV,(G,U|π)≤(G,V|π) and(G,U|π)≤(G,V|π);

(2) ifGis an abelian group,(G,U|π)=(G,gU|π) and(G,U|π)=(G,gU|π) for anyg ∈G.Especially,the equations also hold for the case ofg ∈Gbeing an automorphism ofπ;

(4) we have that

ProofIfU,V ∈with,thenN(U|π)≤N(V|π).By the definitions of the relative upper and lower entropy dimensions,we get (1).

SinceGis a group,we have thatN(U|π)=N(g-1U|π) for allU ∈andg ∈G.Then,from the definitions of the relative upper and lower entropy dimensions and the automorphism ofπ,(2) follows.

For (4),the first inequality is obvious from (1).We now show that

From Proposition 3.2,we can get

Proposition 3.3Letπ: (X,G)→(Y,G) be a factor map.If{Un} is sequence chosen fromwith,then

ProofWe only need to consider the upper case.Sincediam(Un)=0,by (1) of Proposition 3.2,we have that

Therefore,

Let(X,G)be aG-system.A coverU={U,V}ofXis called a standard cover if it consists of two non-dense open sets ofX;we write the class of all standard covers ofXby.The next result shows that the relative upper entropy dimension of a system can be determined by the relative upper entropy dimension of standard covers.

Proposition 3.4Letπ:(X,G)→(Y,G) be a factor map betweenG-systems.Then

ProofIt is obvious that

Now we shrinkUr={Ur,Vr}into a standard coverWwith(G,W|π)≥(G,U|π).First,if=X,then,forε>0,there existsx ∈Ursuch that

SinceUis arbitrary,sup{(G,W|π):W ∈}≥(X,G|π).

4 Relative Entropy Dimension via Entropy Generating Sets

Given a strictly increasing Følner sequence{Fn}ofG,we suppose that a setS ⊆Gsatisfies that

We denote byI(G) the set of the subsetSofGsatisfying with (4.1).Forα ≥0 andS ∈I(G),we define that

Following the definition of sets with zero density in [7],we define the upper dimension ofSas

Similarly,we define the lower dimension ofSas

Next,we will investigate the dimension of a special kind of subsets ofG,which is called the relative entropy generating sets.

Letπ: (X,G)→(Y,G) be a factor map and letU ∈.S ∈I(G) is called a relative entropy generating set ofUif

Write the set of all relative entropy generating sets ofUrelevant toπbyE(G,U|π),and denote byP(G,U|π) the set ofS ∈I(G) with the property that

In other words,P(G,U|π)is the set of subsets ofGalong whichUa has positive relative upper entropy.

Definition 4.1Letπ:(X,G)→(Y,G)be a factor map betweenG-systems andU ∈.We define that

Definition 4.2Letπ:(X,G)→(Y,G) be a factor map.We define that

The next result shows the range of(G,U|π) takes only two numbers,and this is the reason that we define the entropy generating sequence as lim inf instead of lim sup.

Proposition 4.3Letπ:(X,G)→(Y,G) be a factor map andU ∈.Then

ProofWe assume thatP(G,U|π)≠∅.Then there area>0 andS ∈I(G) such that

Next we take 1≤n1

ThenF ∈I(G) and

therefore,F ∈P(G,U|π).Since|Fnj+1|≥2|Fnj ∩S| for eachj ∈N,it is easy to see that the upper density ofFis,and hence(F)=1.This implies that(G,U|π)=1.

5 Some Relationships Among Relative Entropy Dimensions

Next,we study the relationships among the relative entropy dimensions appearing in Sections 3 and 4.

Proposition 5.1Letπ:(X,G)→(Y,G) be a factor map and U ∈.Then

Therefore,we have that

we have that|S ∩Fn|≤m(n)≤|S ∩Fn|+|Fn|α,and then by (5.1),we have that

SinceS ∈E(G,U|π),we have that

which contradicts (5.3).

Proposition 5.2Letπ:(X,G)→(Y,G) be a factor map and let U ∈.Then

Letk ≥2 and letπ:(X,G)→(Y,G)be a factor map betweenG-systems.LetA1,A2,···,Akbeksubsets ofXandW ⊆G.We say{A1,A2,···,Ak}is independent alongWrelevant toπif there isy ∈Ysuch that,for anys ∈{1,2,···,k}W,we have that∅(see [22]).

Lemma 5.3(cf.[26]forG=Z) Letπ:(X,G)→(Y,G) be a factor map,and letA1,A2be two disjoint non-empty closed subsets ofX,For anyα ∈(0,1],0<η<αandc>0,there existsN ∈N (depending onα,η,c) such that,if there is a finite subsetBofGwith|B|≥Nsatisfying that

then there existsW ⊆Bwith|W|≥|B|ηsuch that{A1,A2} is independent alongWrelevant toπ.

ProofThe proof of the lemma is similar to the proof of [7,Lemma 3.7].

From the above lemma,we can get the following result:

Take 1=n1

Now,for eachj ∈N,there existsWj ⊆Fnj+1Fnjsuch that|Wj|≥|Fnj+1Fnj|ηjand{A1,A2}is independent alongWjrelevant toπ;that is,there existsyj ∈Ysuch that∀s ∈{1,2}Wj,so we have that

For any non-empty setB ⊆Wjands=(s(z))z∈B ∈{1,2}B,we can find

LetXB={xs:s ∈{1,2}B}.It is clear that,for anyl ∈{1,2}B,we have that

Combining this fact with|XB|=2|B|,we get

This shows thatF ∈E(G,U|π).

Noting that

Theorem 5.5Letπ:(X,G)→(Y,G) be a factor map betweenG-systems.Then,

Remark 5.6Letπ:(X,G)→(Y,G) be a factor map betweenG-systems.Then,

(1)De(X,G|π)=D(X,G|π) if one of the two values exists;

(2) if (X,G) has a generating open cover,then there exists a relative entropy generating set (of the cover)Fsuch that

6 Relative Dimension Tuples and Dimension Sets

In this section,we will localize the relative entropy dimension to obtain the notion of relative dimension tuples and dimension sets.Letπ:(X,G)→(Y,G)be a factor map betweenG-systems and letn ≥2.Denote by ∆n(X)={(xi)∈Xn:x1=···=xn} then-th diagonal ofX.Through out this section,B(x,∊) denotes the open ball center atxwith radius∊.

6.1 Relative Dimension Tuples

Definition 6.1Letπ: (X,G)→(Y,G) be a factor map and let (xi)∈Xn ∆n(X).The relative entropy dimension of (xi)relevant toπis

Lemma 6.2Letπ: (X,G)→(Y,G) be a factor map and letU={U1,···,Un} ∈.Then there existsxi ∈,1≤i ≤nsuch that(x1,···,xn|π)≥(G,U|π).

where the last equality comes from Proposition 3.2(3).Thus there existsi∗∈{1,···,u}such thatWe denote the set.We apply the same argument forUi,and obtain,1

After repeating the same arguments,one can getndecreasing sequences of non-empty closed setssuch that

Then we obtain that

Proposition 6.3Letπ:(X,G)→(Y,G) be a factor map betweenG-systems.Then,

(2) moreover,ifGis an abelian group,(X,G|π)∪∆nXisG-invariant.

where the second equality comes from Proposition 3.2(2).Therefore,(gx1,···,gxn)∈(G,X|π),and so(G,X|π)∪∆nXisG-invariant.

Proposition 6.4In a commutative diagram

let (X,G),(Y,G) and (Z,G) beG-systems,and suppose thatπ,πXandπYare factor maps.

(1) If(x1,···,xn)∈(X,G|πX),and supposing thatyi=π(xi)with(yi)∈Y n∆n(Y),then(y1,···,yn|πY)≥α.

(2) If (y1,···,yn)∈(Y,G|πY),then there exists (x1,···,xn)∈(X,G|πX) andπ(xi)=yi,i=1,···,n.

Proof(1) If (x1,···,xn)∈(X,G|πX) andyi=π(xi),(yi)∈Y n∆n(Y),then

Then,by Lemma 6.2,there exists

for some (x1,···,xn)∈Xn.Clearlyπ(x1)=y1,···,π(xn)=yn,so (x1,···,xn)∆n(X).Now on the one hand,By Proposition 6.3(1),we have (x1,···,xn)∈(X,G|πX);that is,(x1,···,xn|πX)≥α.On the other hand,by (1),we have that

6.2 Relative Dimension Sets and Uniform Relative Dimension Systems

Definition 6.5Letπ: (X,G)→(Y,G) be a factor map betweenG-systems.We call the subset{α ≥0 :(X,G|π)≠∅} of [0,1]then-th relative dimension set of (X,G),and denote it byDn(X,G|π).If 0∉Dn(X,G|π),we will say that (X,G) has a strictly positiven-th relative entropy dimension.Letingα ∈(0,1],we call (X,G) anα-uniformn-th relative entropy dimension system (n-thα-u.r.d.system for short,and when there is no confusion,we omit “n-th”) ifDn(X,G|π)={α},and call (X,G) ann-thα+-relative dimension system (n-thα+-r.d.system for short,there is no confusion,we omit the “n-th”),ifDn(X,G|π)⊆[α,1].If(X,G) is the trivial system,we letDn(X,G|π)=∅.

Proposition 6.6Letα ∈(0,1].Then,

(1) a nontrivialG-space (X,G) is an-thα-u.r.d.system if and only if(G,U|π)=αfor any open coverUofXwithU={U1,···,Un};

(2) a commutative diagram

if (X,G) is anα-u.r.d.system and (Y,G) is a nontrivial system,then (Y,G) is also anα-u.r.d.system.

(3) in the commutative diagram of(2),if(X,G)is anα+-r.d.system and(Y,G)a nontrivial system,then (Y,G) is also anα+-r.d.system.

ProofBy Proposition 6.4,we have(2)and(3).It is sufficient to show(1).Suppose that(G,U|π)=αfor any open coverUofXwithU={U1,···,Un}.Then,by Definition 6.1,for (x1,···,xn)∈Xn ∆nX,we have(x1,···,xn|π)=α;that is,(X,G) isn-thα-u.r.d.system.

7 The Relative Disjointness Between G-systems

LetπX: (X,G)→(Z,G) and letπY: (Y,G)→(Z,G) be two factor maps,andπ1:X×Y →X,π2:X×Y →Ybe two projections.J ⊆X×Yis called a joining if (J,G) is aG-subsystem of (X×Y,G) with

Define that

Clearly,X×Z Yis a joining of (X,G) and (Y,G) over (Z,G).A joiningJof (X,G) and(Y,G) over (Z,G) is said to be proper ifJ≠X×Z Y.We say that (X,G) and (Y,G) are disjoint over (Z,G),ifX×Z Ycontains no proper joining of (X,G) and (Y,G) over (Z,G).

Letπ:(X,G)→(Y,G)be a factor map between twoG-systems.We say thatπis minimal ifM=Xfor any closed andG-invariant subsetMofXwithπ(M)=Y,and thatπis open ifπ(U) is open for any open subsetU ⊂X.

The following result is a generalization of the theorem that uniformly positive entropy systems are disjoint from minimal entropy zero systems [2]:

Theorem 7.1LetπX: (X,G)→(Z,G) andπY: (Y,G)→(Z,G) be two factor maps withπXopen andπYminimal.If,for anyn ≥2,Dn(G,X|πX)>Dn(G,Y|πY) (i.e.for anyα ∈Dn(G,X|πX) andβ ∈Dn(G,Y|πY),α>β),then (X,G) and (Y,G) are disjoint over(Z,G).

ProofLetJ ⊆X×Z Ybe any given joining of (X,G) and (Y,G) over (Z,G).

which is a contradiction to the assumption thatDn(G,X|πX)>Dn(G,Y|πY).

In order to prove that (X,G) and (Y,G) are disjoint over (Z,G),it is sufficient to prove thatJ=X×Z Y.Set

AcknowledgementsThe authors would like to thank Professor Dou Dou for his many helpful comments on the first version of this paper.

Conflict of InterestThe authors declare no conflict of interest.