Contents lists available at ScienceDirectJournal of Functional Analysiswww.elsevier.com/locate/jfa Weak expansiveness for actions of sofic groups Nhan-Phu Chunga, ∗, Guohua Zhangb aDepart
Trang 1Contents lists available at ScienceDirectJournal of Functional Analysis
www.elsevier.com/locate/jfa
Weak expansiveness for actions of sofic groups
Nhan-Phu Chunga, ∗, Guohua Zhangb
aDepartment of Mathematics, University of Sciences, Vietnam National
University at Ho Chi Minh City, Viet Nam
b
School of Mathematical Sciences and LMNS, Fudan University and Shanghai
Center for Mathematical Sciences, Shanghai 200433, China
Measures with maximal entropy
In this paper, we shall introduceh-expansivenessand toticalh-expansivenessfor actions of sofic groups By defini- tion, each h-expansiveaction of a sofic group is asymptoti- callyh-expansive.We show that each expansive action of a sofic group ish-expansive,and, for any given asymptotically
asymp-h-expansiveaction of a sofic group, the entropy function (with respect to measures) is upper semi-continuous and hence the system admits a measure with maximal entropy.
Observe that asymptotically h-expansive property was first introduced and studied by Misiurewicz for Z-actions using the language of tail entropy And thus in the remaining part of the paper, we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of Misiurewicz’s ideas when the group is amenable.
It turns out that these two definitions are equivalent in this setting.
© 2014 Elsevier Inc All rights reserved.
Trang 2of a countable discrete sofic group admitting a generating measurable partition withfinite entropy[7,9] The main idea hereis to replace theimportant Følner sequence of
a countable discrete amenable group with a sofic approximation for a countable cretesofic group.Verysoonafter[7],inthespiritofL Bowen’ssoficmeasure-theoreticentropy, Kerrand Lidevelopedan operator-algebraicapproachto sofic entropy[32,34]
dis-which appliesnotonlytocontinuous actionsofcountablediscrete soficgroupsonpact metricspacesbutalso toallmeasure-preservingactionsofcountablediscretesoficgroups on standard probability measure spaces From then on, there are many otherpapers, presenting different but equivalent definitions of sofic entropy [31,50], extend-ing soficentropyto sofic pressure[14] andto sofic meandimension[35], anddiscussingcombinatorialindependenceforactionsofsofic groups[33]
com-Let X beacompact metricspace Any homeomorphismT : X → X generates urally atopological dynamicalsystem byconsideringthe group{T n : n ∈ Z}. EveninthecasethegivenmapT : X → X isjustacontinuousmap(maybenon-invertible),westillcallitatopologicaldynamicalsystembyconsideringthesemi-group{T n : n ∈ Z+}.
nat-A self-homeomorphismofacompactmetricspaceisexpansive if,foreachpairofdistinctpoints,someiterateofthehomeomorphismseparatesthembyadefiniteamount.Expan-sivenessisinfactamultifaceteddynamicalconditionwhichplaysaveryimportantroleintheexploitationofhyperbolicityinsmoothdynamicalsystems[39].Inthesettingofcon-sideringacontinuousmappingoveracompactmetricspace,two classesofweakexpan-siveness,theh-expansivenessandasymptoticalh-expansiveness,wereintroducedbyRu-fusBowen[6]andMisiurewicz[40],respectively.Bydefinition,eachh-expansivesystemisasymptotically h-expansive.Both ofh-expansiveness and asymptoticalh-expansiveness
turn outtobeimportantinthestudyofsmoothdynamicalsystems[12,18,21,22,37]
It isdirectto defineexpansivenessfor actionsofgroups.Thatis,letG be adiscretegroupactingonacompactmetricspaceX (withthemetricρ),thenwesaythat(X, G)
is expansive if there is δ > 0 such that for any two different points x1 and x2 in X
there exists g ∈ G with ρ(gx1, gx2) > δ. Such a δ is called an expansive constant.
Symbolic systemsarestandard examplesforexpansive actions Thisobvious extension
of the notion of expansiveness has been investigated extensively in algebraic actionsfor Zd [24,38,44,45] and for moregeneral groups[8,15,17]; and ageneral framework ofdynamics ofd ∈ N commuting homeomorphismsoveracompactmetricspace,interms
of expansive behavior along lower dimensional subspacesof Rd, was first proposed byBoyle andLind[11]
Trang 3Now,anaturalquestionrises:how to defineweakexpansivenessusing entropyniqueswhenconsideringactionsofcountablesoficgroups?
tech-The problem is addressed in this paper When considering a continuous mappingoveracompactmetricspace,R Bowenintroducedh-expansivenessusingseparatedandspanningsubsetsbyconsideringtopologicalentropyofspecialsubsets[6],andthenMisi-urewiczintroducedasymptoticalh-expansiveness,whichisweakerthanh-expansiveness,
using open covers by introducing tailentropy [40] Observe thatthe notion of tailtropywasfirstintroducedbyMisiurewiczin[40]wherehecalledittopologicalconditionalentropy,and thenBuzzicalled thisquantitylocal entropyin[13].Here,wefollowDow-narowicz and Serafin [23] and Downarowicz[20] It was Li who first used open coversforactionsofsoficgroupstoconsidersofic meandimension[35],andthenthis ideawasused in[50] to consider equivalently the entropyfor actions ofsofic groups.To fix theproblem of defining weak expansiveness naturally for actions of sofic groups, we shalluseopen covers againto introduce thepropertiesof h-expansiveness and asymptotical
en-h-expansiveness inthe spirit of Misiurewicz [40] The idea turns out to be successful.From definitioneachh-expansiveaction ofasofic groupis asymptoticallyh-expansive;
weshallprovethateachexpansiveactionofasoficgroupish-expansive(Theorem 3.1),andhence, h-expansivenessand asymptoticalh-expansiveness areindeedtwo classesofweakexpansiveness.Additionally,similartothesettingofconsideringacontinuousmap-pingover acompact metricspace,forany givenasymptoticallyh-expansiveactionofasofic group, the entropy function (with respect to measures) is upper semi-continuous(Theorem 3.5)andhencethesystemadmits ameasure withmaximalentropy
Observethattheasymptoticallyh-expansivepropertywasfirstintroducedandstudied
byMisiurewiczforZ-actionsusingthelanguageoftailentropy.Wecandefinetailentropyfor actions of amenable groupsin the same spirit, and so it is quite natural to ask if
wecould defineasymptotical h-expansiveness foractionsof amenablegroupsalongtheline of tail entropy The answer turns out to be true, that is, our definitions of weakexpansiveness for actions of sofic groups are equivalent to the definitions given in thesame spirit of Misiurewicz’s ideas of using tail entropy when the group is amenable(Theorem 6.1) In [40] Misiurewicz provided a typical example of an asymptotically
h-expansivesystem, thatis, any continuous endomorphism of acompact metricgroupwithfiniteentropyisasymptoticallyh-expansive.Weshallshowthatinfactthisholdsin
amoregeneralsettingwiththehelpofTheorem 6.1,precisely,anyactionofacountablediscreteamenablegroupactingonacompactmetricgroupbycontinuousautomorphisms
isasymptoticallyh-expansiveifandonlyiftheactionhasfiniteentropy(Theorem 7.1).Thepaperisorganizedasfollows.InSection2weprovethateachexpansiveactionofasoficgroupadmitsameasurewithmaximalentropybasedonthesoficmeasure-theoreticentropy introduced in [7] In Section 3 we introduce h-expansive and asymptotically
h-expansiveactionsofsoficgroupsinthespiritofMisiurewicz[40].Eachh-expansivetionofasoficgroupisasymptoticallyh-expansivebythedefinitions.Weshowthateachexpansive action of a sofic group is h-expansive, and each asymptotically h-expansive
ac-action of a sofic group admits a measure with maximal entropy (in fact, its entropy
Trang 4function is upper semi-continuous with respect to measures) In Section 4 we presentourfirstinterestingnon-trivialexampleof anh-expansiveactionof asofic groupwhich
is infact theprofinite actionof acountable group.In order to understandfurtherourintroduced weak expansiveness for actions of sofic groups in the setting of amenablegroups, inSection5we define tailentropy foractions of amenablegroupsinthe samespirit ofMisiurewicz AndtheninSection6wecompareourdefinitionsofweakexpan-sivenessforactionsofsoficgroupswiththedefinitionsgiveninSection5whenthegroup
is amenable It turns outthatthese two definitions areequivalent inthis setting AndtheninSection7weshowthatanyactionofacountablediscreteamenablegroupacting
onacompactmetricgroupbycontinuousautomorphismsisasymptoticallyh-expansive
ifandonlyiftheactionhasfiniteentropy
2 Expansiveactionsofsoficgroups
Let G beacountablediscrete group.Foreach d ∈ N, denoteby Sym(d) thetation groupof{1, · · · , d }.WesaythatG is sofic ifthere isasequenceΣ = {σ i : G →
permu-Sym(d i ), g → σ i,g , d i ∈ N} i ∈N suchthat
d ia ∈ {1, · · · , d i } : σ i,s (a) = σ i,t (a)= 1 for all distinct s, t ∈ G.
Here,by| • | wemeanthecardinalityofaset•.SuchasequenceΣ withlimi→∞ d i =∞
is referred as a sofic approximation of G. Observethat the condition limi→∞ d i = ∞
is essential for the variational principle concerning entropy of actions of sofic groups(see [32] and [50] forthe globaland local variationalprinciples, respectively),and itisautomaticifG isinfinite
Throughout the paper, G will be a countable discrete sofic group, with a fixed sofic approximation Σ as above and G acts on a compact metric space (X, ρ).
In this section, based on the sofic measure-theoretic entropy introduced in [7], wemainlyprovethat,foranexpansiveactionofasoficgroup,theentropyfunctionisuppersemi-continuous withrespect tomeasures,andhencetheactionadmitsameasure withmaximal entropy.Additionally,weshow thatingeneraltheentropyfunctionofafiniteopen coverisalsouppersemi-continuouswithrespect tomeasures
Denote byM (X) thesetof allBorelprobabilitymeasuresonX,whichisacompactmetric spaceifendowed with thewell-knownweakstar topology;and byM (X, G) the
set ofallG-invariantelements μ in M (X),i.e.,μ(A) = μ(g −1 A) for eachg ∈ G andall
A ∈ B X, whereBX is theBorelσ-algebraof X.NotethatifM (X, G) = ∅ then itis acompact metricspace
Trang 5ForasetY ,wedenotebyFY thesetofallnon-emptyfinitesubsetsofY Byacover
ofX wemeanafamilyofsubsetsofX withthewholespaceasitsunion.Ifelements of
acoverarepairwisedisjoint,thenitiscalledapartition.DenotebyCX ,Co
Throughout the whole paper, we shall fix the conventionlog 0=−∞.
Nowwerecallthesoficmeasure-theoreticentropyintroducedin[7,§2]
Let α = {A1, · · · , A k } ∈ P X , k ∈ N and σ : G → Sym(d), d ∈ N, and let ζ be
the uniform probability measure on {1, · · · , d } and β = {B1, · · · , B k } a partition of
{1, · · · , d }. Assumeμ ∈ M(X, G). For F ∈ F G, we denote by Map(F, k) the set of allfunctionsφ : F → N suchthatφ(f ) ≤ k forallf ∈ F ,andweset
d F (α, β) =
φ∈Map(F,k)
μ(A φ)− ζ(B φ),where
A φ=
f∈F
f −1 A φ(f ) and B φ=
f∈F σ(f ) −1 B φ(f ) for each φ ∈ Map(F, k).
Now for each ε > 0, let AP μ (σ, α : F, ε) (or just AP (σ, α : F, ε) if there is no anyambiguity)bethesetofallpartitionsβ = {B1, · · · , B k } of {1, · · · , d } with d F (α, β) ≤ ε.
Inparticular,|AP (σ, α : F, ε) | ≤ k d.Wedefine
ObservethatAP (σ i , α : F, ε) maybe empty,and inthe casethatAP (σ i , α : F, ε)=∅
foralllargeenoughi ∈ N wehaveH μ,Σ (α : F, ε)=−∞ bytheconventionlog 0=−∞.
Hence,h μ,Σ (α) maytakeavaluein[0,log|α|] ∪ {−∞}.
The main result of [7] tells us that, if there exists an α ∈ P X generating the
σ-algebraBX(inthesenseofμ)thenthequantityh μ,Σ (α) isindependentoftheselection
of such a partition, and this quantity, denoted by h μ,Σ (X, G), is called the theoretic μ-entropy of (X, G). Indeed,L Bowen defined themeasure-theoretic entropy
measure-inamoregeneralcasewhen theaction admits agenerating partitionβ (notnecessaryfinite) with finite Shannon entropy [7] We say that the partition β ⊂ B X generates the σ-algebra BX (in the sense of μ) if foreach B ∈ B X there exists A ∈ A suchthat
μ(AΔB) = 0,whereA isthesmallestG-invariant sub-σ-algebra ofBX containingβ.
Trang 6Observethat,foranexpansiveaction(X, G) ofasoficgroupwithanexpansivestant δ > 0, if ξ ∈ P X satisfies diam(ξ) < δ, where diam(ξ) denotes the maximaldiameterofsubsetsinξ, then,foreachμ ∈ M(X, G) (if M (X, G) = ∅), ξ generates BX
con-[46, Theorem 5.25],andso thequantityh μ,Σ (X, G) iswelldefined
For technical reasons for r1, r2∈ [−∞, ∞] we set r1+ r2=−∞ by convention in the case that either r1=−∞ or r2=−∞, and for r1, r2∈ (−∞, ∞] we set r1+ r2=∞ by convention in the case that either r1=∞ or r2=∞.
We say that a function f : Y → [−∞, ∞) defined over a compact metric space Y
is upper semi-continuous if lim supy →y f (y ) ≤ f(y) for each y ∈ Y The followingresultshowsthateachexpansiveactionofasoficgroupadmitsameasurewithmaximalentropy
Theorem 2.1 Let (X, G) be an expansive action of a sofic group with M (X, G) = ∅ Then h •,Σ (X, G) : M (X, G) → [0, ∞) ∪ {−∞} is an upper semi-continuous function.
Proof The proofisinspiredby[46,Theorem 8.2]
Letδ > 0 beanexpansiveconstantfor(X, G) and ξ ∈ P X withdiam(ξ) < δ.Thenξ
generatesBX andso h μ,Σ (X, G) ∈ [0,log|ξ|] ∪ {−∞} foreachμ ∈ M(X, G).
Now fix η > 0 and μ ∈ M(X, G). It suffices to find an open set U ⊂ M(X, G)
containingμ suchthath ν,Σ (X, G) ≤ h μ,Σ (X, G) + η foreachν ∈ U.
We chooseF ∈ F G and ε > 0 such thatH μ,Σ (ξ : F, 2ε) ≤ h μ,Σ (X, G) + η.Say ξ = {A1, · · · , A k } and let0< ε1 < 2M ε2 with M = | Map(F, k) | = k |F | Letφ ∈ Map(F, k).
Since μ isregular,thereexists acompactset K φ ⊂ A φ withμ(A φ \ K φ)< ε1,andthenforeachi = 1, · · · , k we define
by the construction of ξ , and so using Urysohn’s Lemma we can chooseu φ ∈ C(X),
where C(X) denotes the set of all real-valued continuous functions over X, with 0 ≤
u φ ≤ 1 whichequals 1onK φ andvanishesonX \ int(A
φ).Set
U =
ν ∈ M(X, G) :ν(u )− μ(u )< ε for all φ ∈ Map(F, k)
Trang 7which is an open set of M (X, G) containing μ. Let ν ∈ U. Then ν(A φ) ≥ ν(u φ) > μ(u φ)− ε1 ≥ μ(K φ)− ε1 and hence μ(A φ)− ν(A
φ) < 2ε1 for each φ ∈ Map(F, k).
Observe {A φ : φ ∈ Map(F, k) } ∈ P X and {A
φ : φ ∈ Map(F, k) } ∈ P X Note that if
p1, · · · , p m , q1, · · · , q m , c arenonnegative realnumberswith m ∈ N suchthat m
i=1 p i = m
i=1 q i= 1 andp j − q j < c foreachj = 1, · · · , m then
As diam(ξ ) < δ, ξ ∈ P X generates BX by the construction of δ, and so we get
h ν,Σ (X, G) ≤ h μ,Σ (X, G) + η foreachν ∈ U as desired.Thisfinishestheproof 2
InthespiritofL.Bowen’sentropyasabove,KerrandLiintroducedalternativelythesoficmeasure-theoreticentropy[32,34]as follows
Let(Y, ρ) beametricspaceandε > 0. A set∅ = A ⊂ Y issaidtobe(ρ, ε)-separated
if ρ(x, y) ≥ ε for all distinct x, y ∈ A. We write N ε (Y, ρ) for the maximal cardinality
offinite non-empty (ρ, ε)-separated subsets ofY (and set N ε(∅, ρ)= 0 by convention)
A basic factisthatif ∅ = A ⊂ Y is amaximal finite (ρ, ε)-separated subsetof Y then
foreachy ∈ Y thereexists x ∈ A such thatρ(x, y) < ε.
Foreachd ∈ N and (x1, · · · , x d ), (x 1, · · · , x d)∈ X d,weset
Trang 8By [34, Proposition 3.4] the measure-theoretic μ-entropy of (X, G) can be defined as(recallingtheconventionlog 0=−∞)
.
The sofic measure-theoretic entropy can be defined equivalently using finite opencoversasfollows[50,§2].WeremarkthatitwasLiwhofirstusedopencoversforactions
of soficgroupsto considersofic meandimensionin[35]
For U ∈ C X and d ∈ N,we denote by Ud thefinite Borel cover of X d consisting of
U1× U2× · · · × U d,whereU1, · · · , U d ∈ U.LetU∈ C X,weset (recallingtheconventionlog 0=−∞)
In particular, h F,δ,μ,L (G,U) takes the value −∞ if X F,δ,σi,μ,L di = ∅ for all i ∈ N large
enough.Now wedefinethemeasure-theoretic μ-entropy of U as
h μ (G,U) = inf
L ∈FC(X) Finf∈FG δ>0inf h F,δ,μ,L (G, U) ≤ log N(U, X).
It isnothard tocheckthat
h μ (G, X) = sup
U∈C o X
h μ (G, U).
Moreover, by theproof of [32, Theorem 6.1], itwas provedimplicitly h μ (G, X) =−∞
(and henceh μ (G,U)=−∞ forallU∈ C o
X)foreachμ ∈ M(X) \ M(X, G).
Observethatbothofh μ (G,U) andh μ (G, X) maytakethevalueof−∞,andby[32,34] ifμ ∈ M(X, G) andBX admitsageneratingpartition(inthesense ofμ)withfiniteShannonentropythenh μ (G, X) isjustthequantityh μ,Σ (X, G) introducedbefore.Thefollowing resultiseasyto obtain:
Proposition 2.2 LetU∈ C o
X Then h • (G,U): M (X) → [0, log N ( U, X)] ∪ {−∞} is an upper semi-continuous function.
Proof Letμ ∈ M(X).Foranyε > 0 wemaychooseL ∈ F C(X) , F ∈ F G andδ > 0 such
thath F,2δ,μ,L (G,U)≤ h μ (G,U)+ ε. Nowweconsiderthenon-emptyopenset
μ ∈ V =ν ∈ M(X) :ν(f ) − μ(f)< δ for all f ∈ L.
Then foreach ν ∈ V wehaveX d
F,δ,σ,ν,L ⊂ X d
F,2δ,σ,μ,Lfor eachσ : G → Sym(d), d ∈ N,
which impliesh ν (G,U)≤ h F,δ,ν,L (G,U)≤ h F,2δ,μ,L (G,U)≤ h μ (G,U)+ ε. Thisimpliesthattheconsideredfunctionisuppersemi-continuous 2
Trang 93 Weakexpansivenessforactionsofsofic groups
Notethat,whenconsideringacontinuousmappingoveracompactmetricspace,sincetheintroductionofh-expansivenessandasymptoticalh-expansiveness,bothofthemturnouttobe veryimportantclassesintheresearch areaofdynamical systems.Itisshown
by R Bowen [6] thatpositively expansivesystems, expansive homeomorphisms, morphisms ofa compactLie group and Axiom A diffeomorphisms areall h-expansive,
endo-byMisiurewicz [40] thateverycontinuousendomorphism ofacompact metricgroupisasymptoticallyh-expansive ifithas finite entropy,and byBuzzi [13] thatany C ∞ dif-feomorphismonacompactmanifoldisasymptoticallyh-expansive.Moreover,therearemorenice characterizations of asymptotical h-expansiveness obtained recently for thissetting.Forexample,atopologicaldynamicalsystemisasymptoticallyh-expansiveifandonlyifitadmitsaprincipalextension toasymbolicsystembyBoyleandDownarowicz
[10], i.e., a symbolic extension which preserves entropy for each invariant measure; ifandonlyifitishereditarilyuniformlylowerablebyHuang,Yeandthesecondauthorofthepresent paper [28] (for adetaileddefinitionof the hereditarilyuniformly lowerablepropertyanditsstorysee [28])
Inthissectionweexploresimilarweakexpansivenessforactionsofsoficgroups
By[34, Proposition 2.4]thetopological entropy of (X, G) canbedefinedas
,
which is introduced and discussed in [32,34] Before proceeding, we need to recall thetopological entropy for actions of sofic groups introduced in [50, §2] using finite opencovers.LetU∈ C X.ForF ∈ F G and δ > 0 weset
Observeagainthath F,δ (G,U) takesthevalueof−∞ whenever X F,δ,σi di =∅ foralli ∈ N
largeenough.Now wedefinethetopological entropy of U as
h(G,U) = inf
F ∈FG δ>0inf h F,δ (G, U) ≤ log N(U, X).
Itisnothardtocheck that
h(G, X) = sup
U∈C o X h(G, U).
Observethatbothof h(G,U) andh(G, X) maytakethevalueof−∞.
Thesofictopologicalentropyandsofic measure-theoreticentropyarerelatedtoeachother[32,Theorem 6.1]and[50, Theorem 4.1]:forU∈ C o,
Trang 10h(G, X) = sup
μ∈M(X,G) h μ (G, X) and h(G,U) = max
μ∈M(X,G) h μ (G, U), (3.1)
where intheright-handsidesasabovewesetitto −∞ byconventionifM (X, G)=∅.
In thespirit of Misiurewicz [40], the aboveideacan be used to introduce
h-expans-iveness and asymptotical h-expansiveness for actionsof sofic groups.Let U1,U2 ∈ C X.ForF ∈ F G andδ > 0 weset
h ∗ (G, X) = inf
U 2∈C o X h(G, X |U2)≤ h(G, X).
Then h(G, X |{X}) = h(G, X) by the definitions Wesay that(X, G) is h-expansive if h(G, X |U) ≤ 0 forsomeU∈ C o
X,and asymptotically h-expansive if h ∗ (G, X) ≤ 0.
Each h-expansiveactionofasofic groupis asymptoticallyh-expansivebydefinition.The next result shows that each expansive action of asofic group is h-expansive,andthus thesetwokindsofexpansivenessareindeedweakexpansiveness
Theorem3.1.Let (X, G) be an expansive action of a sofic group with κ > 0 an expansive constant andU∈ C X Assume that diam(U)≤ cκ for some c < 1 Then h(G, X |U) ≤ 0.
Proof Let V∈ C o
X and ε > 0.It sufficestoprovethath(G, V|U) ≤ ε.
Let τ > 0 be a Lebesgue number of V As (X, G) is an expansive action of a soficgroup with κ > 0 an expansive constant, it is not hard to choose F ∈ F G such thatmaxs ∈F ρ(sx, sx )< κ implies ρ(x, x )< τ2 (forexamplesee[46,Chapter 5, §5.6]).Nowletδ > 0 besmall enoughsuchthat
|Θ d | · |V|(18δ2 −c)2κ2 |F | ·d < e εd (3.2)foralld ∈ N largeenough,whereΘ d isthesetofallsubsetsθ of {1, · · · , d } with
Trang 11F,δ,σ ∩ V (if it is not empty) For any
(x 1, · · · , x d)∈ X d
F,δ,σ ∩ V ,applying (3.3)to(x1, · · · , x d) and (x 1, · · · , x d) and observing
ρ(sx i , sx i)≤ ρ(sx i , x σs (i))+ ρ(x σs (i) , x σs (i))+ ρ(sx i , x σs (i)),itiseasytosee
Inotherwords,ifweassociate eachθ ∈ Θ d withX d
F,δ,σ,V,θ, thesetofall(x 1, · · · , x d) in
andhenceh(G, V|U) ≤ ε bythedefinition,finishingtheproof 2
Observingthat,forV1,V2∈ C X andK ⊂ X,
N (V1, K) ≤ N(V2, K) · max
V ∈V2
N (V1, K ∩ V ), (3.5)
itisdirectto obtainthefollowing easywhileusefulobservation
Lemma3.2 LetU1,U2∈ C X and μ ∈ M(X) Then
h μ (G,U1)≤ h μ (G,U2) + h(G,U1|U2) and h μ (G, X) ≤ h μ (G,U2) + h(G, X |U2),
h(G,U )≤ h(G, U ) + h(G,U |U ) and h(G, X) ≤ h(G, U ) + h(G, X |U ).
Trang 12Proof Let {F n : n ∈ N} ⊂ F G increase to thewhole group G and {δ n > 0 : n ∈ N}
decrease to0.Bythedefinitionsitisdirectto see
h μ (G,U) = inf
L ∈FC(X) nlim→∞ h Fn,δn,μ,L (G,U)foreachU∈ C X and
par-h(G, X |U2)=∞ (recallingourtechnicalconventionof−∞+∞=−∞).Theremainingitems canbeprovedsimilarly 2
As directcorollaries,wehave:
Corollary 3.3 h ∗ (G, X) < ∞ if and only if h(G, X) < ∞, and h ∗ (G, X)=−∞ if and only if h(G, X)=−∞.
Corollary 3.4.Assume that (X, G) is h-expansive Then there existsU∈ C o
X with h(G, X) = h(G, U) and h μ (G, X) = h μ (G, U) for each μ ∈ M(X).
Moreover, wecanprovethefollowingresult
Theorem 3.5.Let μ ∈ M(X) Then
Trang 13Now we assume that (X, G) is asymptotically h-expansive, that is, h ∗ (G, X) ≤ 0.
Thenh(G, X) < ∞ byCorollary 3.3,andhenceh η (G, X) ∈ [0, ∞) ∪ {−∞} foreachη ∈
M (X) by(3.1)(recallingthefactthath η (G, X)=−∞ foreachη ∈ M(X) \ M(X, G)).
Moreover,using(3.6)weobtain
lim sup
ν→μ h ν (G, X) ≤ h μ (G, X) + h ∗ (G, X) ≤ h μ (G, X)
fromourtechnicalconvention.Thisfinishestheproof 2
Remark 3.6 As a direct corollary of Theorem 3.5, we have: if the action (X, G) is
asymptotically h-expansive then both of its sofic topological mean dimension and itssoficmetricmeandimensionwithrespecttoanycompatiblemetricareatmostzero[35,Proposition 4.3andTheorem 6.1].Forthedefinitionofsofictopologicalmeandimensionandsoficmetricmeandimensionsee[35, §2and §4],respectively
CombiningTheorem 3.5with(3.1)onehas:
Corollary3.7.Each asymptotically h-expansive action of a sofic group admits a measure with maximal entropy.
Ingeneral,theconversedoesnothold,forexample,anyZ-actionwithinfiniteentropyadmits a measure with infiniteentropy, whereas, it is not asymptotically h-expansive.
Furthermore,[40,Example6.4]showsusaZ-actionwithfiniteentropysuchthatitisnotasymptoticallyh-expansive,whileeachinvariantmeasurehasmaximalentropy.Observethat,asweshallshowlater,thedefinitionsofweakexpansivenessgivenhereforactions
ofsoficgroupsareequivalenttodefinitionsgiveninthesamespiritofMisiurewicz’sideaswhenthegroupisamenable
Remark3.8 Theorem 2.1isaconsequenceof Theorem 3.1andTheorem 3.5
4 Profiniteactions
Inthissectionweprovideourfirstinterestingnon-trivialh-expansiveactionofasoficgroupusing thelanguageoftheprofiniteaction
Recallthattheaction(X, G) is distal ifinfg ∈G ρ(gx, gy) > 0 foralldistinctx, y ∈ X,
and equicontinuous if for each δ > 0 there exists ε > 0 such that ρ(x, y) ≤ ε implies ρ(gx, gy) ≤ δ forallg ∈ G.
Thefollowingresultshouldbe known,we providehereaproofforcompleteness.Lemma 4.1 Assume that the action (X, G) is equicontinuous Then it is distal And if additionally X is infinite then it is not expansive.
Trang 14Proof First weprovethat(X, G) isdistal Else,there exist distinct points x1, x2 ∈ X
with infg ∈G ρ(gx1, gx2)= 0 As (X, G) is equicontinuous, there exists ε > 0 such that
ρ(x, y) < ε implies ρ(gx, gy) ≤ 1
2ρ(x1, x2) for each g ∈ G. Let g ∈ G be such that
ρ(g x1, g x2) < ε. Then 0 < ρ(x1, x2) = ρ((g )−1 g x1, (g )−1 g x2) ≤ 1
2ρ(x1, x2) by theselectionofε, acontradiction
NowadditionallyweassumethatX isinfinite.If(X, G) isexpansive,thenthereexists
δ > 0 withsupg ∈G ρ(gx, gy) > δ foralldistinctx, y ∈ X.Usingagaintheequicontinuity
of (X, G), we could choose ε > 0 such that ρ(x, y) < ε implies ρ(gx, gy) ≤ δ for
each g ∈ G. As X isinfinite, bythecompactness ofX we couldchoose distinctpoints
y1, y2 ∈ X with ρ(y1, y2) < ε , acontradiction to theselection of δ. That is, (X, G) is
notexpansive,finishingtheproof 2
See[3]foramoredetailedstoryofdistalactionsandequicontinuousactions
Observethat,bydefinitioneachactionof asofic groupish-expansiveifithaslogical entropyat most zero,and Kerr and Liprovedthat eachdistal actionof asoficgroup has topological entropy at most zero [33, Corollary 8.5] Note that [33, Corol-lary 8.5]isstatedforthesofictopologicalentropydefinedusingultrafilter.However,thelimsup versionof itfollows directlyfromthe ultrafilterversion, sinceonecanpassto asubsequence where thequantity converges and thenone canuseany free ultrafilteronthis subsequence.Thuseachdistalactionofasoficgroupish-expansive.Withthehelp
topo-of thisobservation,wecanprovide ourfirstinterestingh-expansiveexample
Let G be a countable group A chain of G is a sequence G = G0 ≥ G1 ≥ · · · of
subgroupswithfiniteindicesinG.Forachain(G n),wehaveatreestructureT (G, (G n))defined naturallyas follows Thevertices are {gG n : n ∈ N, g ∈ G} and (g1G n , g2G m)
is an edge ifm = n+ 1 and g2G m ⊂ g1G n The boundary∂T (G, (G n)) of T (G, (G n))consistsofallsequences(x0, x1, · · ·) ofverticeswithx nadjacenttox n+1foreachn ∈ Z+.Then ∂T (G, (G n)) isacompactmetrizable spaceendowedwiththetopologygenerated
by theopenbasis consistingof allsubsetsO x={(x0, x1, · · ·) ∈ ∂T (G, (G n)): x N = x }
withx ∈ G/G NandN ∈ Z+.ThenaturalleftactionsofG on G/G ninducetheprofinite action (∂T (G, (G n )), G),anactionofG on ∂T (G, (G n)) byhomeomorphisms.Inthiscasethe profiniteaction(∂T (G, (G n )), G) isequicontinuous, sinceforanyx = gG n ∈ G/G n
and h ∈ G we havehO x = O y with y = hgG n ∈ G/G n
Combining withtheabovediscussions, weobtain:
Proposition 4.2 Let G be a countable sofic group and (G n ) a chain of G Then the profinite action of G on ∂T (G, (G n )) is h-expansive Furthermore if ∂T (G, (G n )) of
T (G, (G n )) is infinite then the action is not expansive.
Dynamicalpropertiesofprofiniteactionshavebeenstudiedextensivelyforresiduallyfinite groups in[1,2] They were also used to investigate orbit equivalence rigidityforKazhdan property(T) groups[30,42]
Trang 155 Tailentropyforactionsofcountablediscreteamenablegroups
In this section we shall introduce tail entropy for actions of countable discreteamenablegroupsinthesamespiritofMisiurewicz
RecallthatG isacountable discretegroup.Denote bye G theunitof G G iscalled
amenable,ifthere exists asequence {F n : n ∈ N} ⊂ F G,called aFølner sequence of G,
of G,thenF n = G once n islargeenough
Throughout this section and the next, we will assume that G is always a countable discrete amenable group.
Thewell-known Ornstein–Weiss Lemmaplays acrucial role inthe study ofentropytheoryforactionsofamenablegroups[41](seealso[19,29,43,49,47]).Thefollowingver-sionofitistakenfrom[27, 1.3.1]
Proposition5.1 Let f : F G → R be a nonnegative function such that f (Eg) = f (E) and
f (E ∪ F ) ≤ f(E) + f (F ) for all E, F ∈ F G and g ∈ G Then for any Følner sequence {F n : n ∈ N} of G the sequence { f (Fn)
|Fn| : n ∈ N} converges and the value of the limit is independent of the selection of the Følner sequence {F n : n ∈ N}.
LetW1,W2∈ C X.IfeachelementofW1iscontainedinsomeelementofW2 thenwesaythatW1 is finer than W2 (denotedbyW1 W2 or W2 W1).ThejoinW1∨ W2
isgivenbyW1∨ W2={W1∩ W2: W1∈ W1, W2∈ W2} ∈ C X, whichextendsnaturally
toafinite collectionof covers.LetF ∈ F G,weset (W1)F =
g∈F g −1W1,andthen weconsideranonnegativefunctionmW1,W2 : FG → R givenby
mW1,W2(F ) = max
K∈(W2 )F
log N (W1)F , K
for each F ∈ F G
Itiseasytoobtainthefollowingusefulobservation
Lemma5.2 mW1,W 2(E ∪ F ) ≤ mW 1,W 2(E) + mW1,W 2(F ) for all E, F ∈ F G
Proof Let E, F ∈ F G From the definition we choose K ∈ (W2)E∪F with
mW1,W2(E ∪ F ) = log N ((W1)E∪F , K). Say K1 ∈ (W2)E and K2 ∈ (W2)F with
K = K1∩ K2 (no matter if E and F are disjoint), suchK1 and K2 must exist Now
Trang 16let V1 ⊂ (W1)E cover K1 with |V1| = N ((W1)E , K1) and let V2 ⊂ (W1)F cover K2
with |V2| = N ((W1)F , K2) Obviously we can cover K1∩ K2 (i.e K) using the ily V1 ∨ V2 Observing that |V1∨ V2| ≤ |V1| · |V2| and each element of V1∨ V2 iscontained insome element of (W1)E∪F, we have that N ((W1)E∪F , K) ≤ |V1∨ V2| ≤
fam-N ((W1)E , K1)· N((W1)F , K2),whichimpliestheconclusiondirectly 2
It is easy to check G-invariance of the nonnegative function mW1,W 2 : FG → R.
Observing Lemma 5.2,wecouldapplyProposition 5.1todefine
h a (G, U|W2),
and thendefine thetopological entropy of (X, G), h a (G, X),and tail entropy of (X, G),
h a,∗ (G, X),respectively,as:
h a (G, X) = h a G, X |{X} = sup
U∈C o X
h a (G,U)
and
h a, ∗ (G, X) = inf
V∈C o X
h a (G, X |V) = inf
V∈C o X
h a (G, X |V)≥ 0.
Recalling that in the special case of G = Z acting on a compact metric space X,
equivalently, giving a homeomorphism T : X → X, the above definition recovers thedefinitiongivenbyMisiurewiczin[40],whichwasthenusedtodiscussweakexpansivenessforZ-actions.Infact,Misiurewicz[40]introducedtailentropyinthesettingofacompactHausdorffspaceX andacontinuoustransformation ofX intoitself
6 Comparisonbetweensofic andamenablecases
Thefollowingresultisthemainresultofthissection,whichshowsthatourdefinitions
of weakexpansiveness for actions of sofic groups are equivalent to definitions given inthesamespiritofMisiurewicz’s ideaswhenthegroupisamenable
...Furthermore,[40,Example6.4]showsusaZ-actionwithfiniteentropysuchthatitisnotasymptoticallyh-expansive,whileeachinvariantmeasurehasmaximalentropy.Observethat,asweshallshowlater,thedefinitionsofweakexpansivenessgivenhereforactions
ofsoficgroupsareequivalenttodefinitionsgiveninthesamespiritofMisiurewicz’sideaswhenthegroupisamenable... data-page="15">
5 Tailentropyforactionsofcountablediscreteamenablegroups
In this section we shall introduce tail entropy for actions of countable discreteamenablegroupsinthesamespiritofMisiurewicz...
Thefollowingresultisthemainresultofthissection,whichshowsthatourdefinitions
of weakexpansiveness for actions of sofic groups are equivalent to definitions given inthesamespiritofMisiurewicz’s ideaswhenthegroupisamenable