DSpace at VNU: Weak expansiveness for actions of sofic groups

JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.1 (1-32) Journal of Functional Analysis ••• (••••) •••–••• Contents lists available at ScienceDirect Journal of Functional Analysis www.elsevier.com/locate/jfa Weak expansiveness for actions of sofic groups Nhan-Phu Chung a,∗ , Guohua Zhang b a Department 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 a r t i c l e i n f o Article history: Received October 2014 Accepted 19 December 2014 Available online xxxx Communicated by S Vaes MSC: primary 37B05 secondary 37C85, 54H15 Keywords: Actions of sofic groups Expansive Asymptotically h-expansive Measures with maximal entropy a b s t r a c t In this paper, we shall introduce h-expansiveness and asymptotical h-expansiveness for actions of sofic groups By definition, each h-expansive action of a sofic group is asymptotically h-expansive We show that each expansive action of a sofic group is h-expansive, and, for any given asymptotically h-expansive action 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 * Corresponding author E-mail addresses: cnphu@hcmus.edu.vn (N.-P Chung), chiaths.zhang@gmail.com (G Zhang) http://dx.doi.org/10.1016/j.jfa.2014.12.013 0022-1236/© 2014 Elsevier Inc All rights reserved JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.2 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Introduction Dynamical system theory is the study of qualitative properties of group actions on spaces with certain structures In this paper, by a topological dynamical system we mean a continuous action of a countable discrete sofic group on a compact metric space Sofic groups were defined implicitly by Gromov in [26] and explicitly by Weiss in [48] They include all amenable groups and residually finite groups Recently, Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable discrete sofic group admitting a generating measurable partition with finite entropy [7,9] The main idea here is to replace the important Følner sequence of a countable discrete amenable group with a sofic approximation for a countable discrete sofic group Very soon after [7], in the spirit of L Bowen’s sofic measure-theoretic entropy, Kerr and Li developed an operator-algebraic approach to sofic entropy [32,34] which applies not only to continuous actions of countable discrete sofic groups on compact metric spaces but also to all measure-preserving actions of countable discrete sofic groups on standard probability measure spaces From then on, there are many other papers, presenting different but equivalent definitions of sofic entropy [31,50], extending sofic entropy to sofic pressure [14] and to sofic mean dimension [35], and discussing combinatorial independence for actions of sofic groups [33] Let X be a compact metric space Any homeomorphism T : X → X generates naturally a topological dynamical system by considering the group {T n : n ∈ Z} Even in the case the given map T : X → X is just a continuous map (may be non-invertible), we still call it a topological dynamical system by considering the semi-group {T n : n ∈ Z+ } A self-homeomorphism of a compact metric space is expansive if, for each pair of distinct points, some iterate of the homeomorphism separates them by a definite amount Expansiveness is in fact a multifaceted dynamical condition which plays a very important role in the exploitation of hyperbolicity in smooth dynamical systems [39] In the setting of considering a continuous mapping over a compact metric space, two classes of weak expansiveness, the h-expansiveness and asymptotical h-expansiveness, were introduced by Rufus Bowen [6] and Misiurewicz [40], respectively By definition, each h-expansive system is asymptotically h-expansive Both of h-expansiveness and asymptotical h-expansiveness turn out to be important in the study of smooth dynamical systems [12,18,21,22,37] It is direct to define expansiveness for actions of groups That is, let G be a discrete group acting on a compact metric space X (with the metric ρ), then we say that (X, G) is expansive if there is δ > 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 systems are standard examples for expansive actions This obvious extension of the notion of expansiveness has been investigated extensively in algebraic actions for Zd [24,38,44,45] and for more general groups [8,15,17]; and a general framework of dynamics of d ∈ N commuting homeomorphisms over a compact metric space, in terms of expansive behavior along lower dimensional subspaces of Rd , was first proposed by Boyle and Lind [11] JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.3 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Now, a natural question rises: how to define weak expansiveness using entropy techniques when considering actions of countable sofic groups? The problem is addressed in this paper When considering a continuous mapping over a compact metric space, R Bowen introduced h-expansiveness using separated and spanning subsets by considering topological entropy of special subsets [6], and then Misiurewicz introduced asymptotical h-expansiveness, which is weaker than h-expansiveness, using open covers by introducing tail entropy [40] Observe that the notion of tail entropy was first introduced by Misiurewicz in [40] where he called it topological conditional entropy, and then Buzzi called this quantity local entropy in [13] Here, we follow Downarowicz and Serafin [23] and Downarowicz [20] It was Li who first used open covers for actions of sofic groups to consider sofic mean dimension [35], and then this idea was used in [50] to consider equivalently the entropy for actions of sofic groups To fix the problem of defining weak expansiveness naturally for actions of sofic groups, we shall use open covers again to introduce the properties of h-expansiveness and asymptotical h-expansiveness in the spirit of Misiurewicz [40] The idea turns out to be successful From definition each h-expansive action of a sofic group is asymptotically h-expansive; we shall prove that each expansive action of a sofic group is h-expansive (Theorem 3.1), and hence, h-expansiveness and asymptotical h-expansiveness are indeed two classes of weak expansiveness Additionally, similar to the setting of considering a continuous mapping over a compact metric space, for any given asymptotically h-expansive action of a sofic group, the entropy function (with respect to measures) is upper semi-continuous (Theorem 3.5) and hence the system admits a measure with maximal entropy Observe that the asymptotically h-expansive property was first introduced and studied by Misiurewicz for Z-actions using the language of tail entropy We can define tail entropy for actions of amenable groups in the same spirit, and so it is quite natural to ask if we could define asymptotical h-expansiveness for actions of amenable groups along the line of tail entropy The answer turns out to be true, that is, our definitions of weak expansiveness for actions of sofic groups are equivalent to the definitions given in the same 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-expansive system, that is, any continuous endomorphism of a compact metric group with finite entropy is asymptotically h-expansive We shall show that in fact this holds in a more general setting with the help of Theorem 6.1, precisely, any action of a countable discrete amenable group acting on a compact metric group by continuous automorphisms is asymptotically h-expansive if and only if the action has finite entropy (Theorem 7.1) The paper is organized as follows In Section we prove that each expansive action of a sofic group admits a measure with maximal entropy based on the sofic measure-theoretic entropy introduced in [7] In Section we introduce h-expansive and asymptotically h-expansive actions of sofic groups in the spirit of Misiurewicz [40] Each h-expansive action of a sofic group is asymptotically h-expansive by the definitions We show that each expansive action of a sofic group is h-expansive, and each asymptotically h-expansive action of a sofic group admits a measure with maximal entropy (in fact, its entropy JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.4 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• function is upper semi-continuous with respect to measures) In Section we present our first interesting non-trivial example of an h-expansive action of a sofic group which is in fact the profinite action of a countable group In order to understand further our introduced weak expansiveness for actions of sofic groups in the setting of amenable groups, in Section we define tail entropy for actions of amenable groups in the same spirit of Misiurewicz And then in Section we compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in Section when the group is amenable It turns out that these two definitions are equivalent in this setting And then in Section we show that any action of a countable discrete amenable group acting on a compact metric group by continuous automorphisms is asymptotically h-expansive if and only if the action has finite entropy Expansive actions of sofic groups Let G be a countable discrete group For each d ∈ N, denote by Sym(d) the permutation group of {1, · · · , d} We say that G is sofic if there is a sequence Σ = {σi : G → Sym(di ), g → σi,g , di ∈ N}i∈N such that i→∞ di lim a ∈ {1, · · · , di } : σi,s σi,t (a) = σi,st (a) = for all s, t ∈ G and lim i→∞ di a ∈ {1, · · · , di } : σi,s (a) = σi,t (a) =1 for all distinct s, t ∈ G Here, by | • | we mean the cardinality of a set • Such a sequence Σ with limi→∞ di = ∞ is referred as a sofic approximation of G Observe that the condition limi→∞ di = ∞ is essential for the variational principle concerning entropy of actions of sofic groups (see [32] and [50] for the global and local variational principles, respectively), and it is automatic if G is infinite 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], we mainly prove that, for an expansive action of a sofic group, the entropy function is upper semi-continuous with respect to measures, and hence the action admits a measure with maximal entropy Additionally, we show that in general the entropy function of a finite open cover is also upper semi-continuous with respect to measures Denote by M (X) the set of all Borel probability measures on X, which is a compact metric space if endowed with the well-known weak star topology; and by M (X, G) the set of all G-invariant elements μ in M (X), i.e., μ(A) = μ(g −1 A) for each g ∈ G and all A ∈ BX , where BX is the Borel σ-algebra of X Note that if M (X, G) = ∅ then it is a compact metric space JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.5 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• For a set Y , we denote by FY the set of all non-empty finite subsets of Y By a cover of X we mean a family of subsets of X with the whole space as its union If elements of a cover are pairwise disjoint, then it is called a partition Denote by CX , CoX , CcX and PX the set of all finite Borel covers, finite open covers, finite closed covers and finite Borel partitions of X, respectively For V ∈ CX and ∅ = K ⊂ X we set V = {V : V ∈ V}, and set N (V, K) to be the minimal cardinality of sub-families of V covering K (with N (V, ∅) = by convention) Throughout the whole paper, we shall fix the convention log = −∞ Now we recall the sofic measure-theoretic entropy introduced in [7, §2] Let α = {A1 , · · · , Ak } ∈ PX , k ∈ N and σ : G → Sym(d), d ∈ N, and let ζ be the uniform probability measure on {1, · · · , d} and β = {B1 , · · · , Bk } a partition of {1, · · · , d} Assume μ ∈ M (X, G) For F ∈ FG , we denote by Map(F, k) the set of all functions φ : F → N such that φ(f ) ≤ k for all f ∈ F , and we set μ(Aφ ) − ζ(Bφ ) , dF (α, β) = φ∈Map(F,k) where f −1 Aφ(f ) Aφ = σ(f )−1 Bφ(f ) and Bφ = f ∈F for each φ ∈ Map(F, k) f ∈F Now for each ε > 0, let APμ (σ, α : F, ε) (or just AP (σ, α : F, ε) if there is no any ambiguity) be the set of all partitions β = {B1 , · · · , Bk } of {1, · · · , d} with dF (α, β) ≤ ε In particular, |AP (σ, α : F, ε)| ≤ kd We define Hμ,Σ (α : F, ε) = lim sup i→∞ log AP (σi , α : F, ε) ≤ log |α|, di Hμ,Σ (α : F ) = lim Hμ,Σ (α : F, ε) = inf Hμ,Σ (α : F, ε) ≤ log |α|, ε→0 ε>0 hμ,Σ (α) = inf Hμ,Σ (α : F ) ≤ log |α| F ∈FG Observe that AP (σi , α : F, ε) may be empty, and in the case that AP (σi , α : F, ε) = ∅ for all large enough i ∈ N we have Hμ,Σ (α : F, ε) = −∞ by the convention log = −∞ Hence, hμ,Σ (α) may take a value in [0, log |α|] ∪ {−∞} The main result of [7] tells us that, if there exists an α ∈ PX generating the σ-algebra BX (in the sense of μ) then the quantity hμ,Σ (α) is independent of the selection of such a partition, and this quantity, denoted by hμ,Σ (X, G), is called the measuretheoretic μ-entropy of (X, G) Indeed, L Bowen defined the measure-theoretic entropy in a more general case when the action admits a generating partition β (not necessary finite) with finite Shannon entropy [7] We say that the partition β ⊂ BX generates the σ-algebra BX (in the sense of μ) if for each B ∈ BX there exists A ∈ A such that μ(AΔB) = 0, where A is the smallest G-invariant sub-σ-algebra of BX containing β JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.6 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Observe that, for an expansive action (X, G) of a sofic group with an expansive constant δ > 0, if ξ ∈ PX satisfies diam(ξ) < δ, where diam(ξ) denotes the maximal diameter of subsets in ξ, then, for each μ ∈ M (X, G) (if M (X, G) = ∅), ξ generates BX [46, Theorem 5.25], and so the quantity hμ,Σ (X, G) is well defined 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 following result shows that each expansive action of a sofic group admits a measure with maximal entropy 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 proof is inspired by [46, Theorem 8.2] Let δ > be an expansive constant for (X, G) and ξ ∈ PX with diam(ξ) < δ Then ξ generates BX and so hμ,Σ (X, G) ∈ [0, log |ξ|] ∪ {−∞} for each μ ∈ M (X, G) Now fix η > and μ ∈ M (X, G) It suffices to find an open set U ⊂ M (X, G) containing μ such that hν,Σ (X, G) ≤ hμ,Σ (X, G) + η for each ν ∈ U We choose F ∈ FG and ε > such that Hμ,Σ (ξ : F, 2ε) ≤ hμ,Σ (X, G) + η Say ξ = ε |F | Let φ ∈ Map(F, k) {A1 , · · · , Ak } and let < ε1 < 2M with M = | Map(F, k)| = k Since μ is regular, there exists a compact set Kφ ⊂ Aφ with μ(Aφ \ Kφ ) < ε1 , and then for each i = 1, · · · , k we define f Kφ : φ(f ) = i ⊂ Ai Li = f ∈F Then L1 , · · · , Lk are pairwise disjoint compact subsets of X, and so there exists ξ = {A1 , · · · , Ak } ∈ PX such that diam(ξ ) < δ and, for each j = 1, · · · , k, Lj ⊂ int(Aj ) where int(Aj ) denotes the interior of Aj Observe that f −1 Aφ(f ) Kφ ⊂ int f ∈F = int Aφ f −1 Aφ(f ) here Aφ = f ∈F by the construction of ξ , and so using Urysohn’s Lemma we can choose uφ ∈ C(X), where C(X) denotes the set of all real-valued continuous functions over X, with ≤ uφ ≤ which equals on Kφ and vanishes on X \ int(Aφ ) Set U = ν ∈ M (X, G) : ν(uφ ) − μ(uφ ) < ε1 for all φ ∈ Map(F, k) JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.7 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• which is an open set of M (X, G) containing μ Let ν ∈ U Then ν(Aφ ) ≥ ν(uφ ) > μ(uφ ) − ε1 ≥ μ(Kφ ) − ε1 and hence μ(Aφ ) − ν(Aφ ) < 2ε1 for each φ ∈ Map(F, k) Observe {Aφ : φ ∈ Map(F, k)} ∈ PX and {Aφ : φ ∈ Map(F, k)} ∈ PX Note that if m p1 , · · · , pm , q1 , · · · , qm , c are nonnegative real numbers with m ∈ N such that i=1 pi = m i=1 qi = and pj − qj < c for each j = 1, · · · , m then qi − p i = (pj − qj ) < mc j=i and hence |pi − qi | < mc for any i = 1, · · · , m This implies |ν(Aφ ) − μ(Aφ )| < 2ε1 M for each φ ∈ Map(F, k), and so ν Aφ − μ(Aφ ) ≤ 2ε1 M ≤ ε φ∈Map(F,k) Thus APν (σi , ξ : F, ε) ⊂ APμ (σi , ξ : F, 2ε) for each i ∈ N, and then Hν,Σ ξ : F ≤ Hν,Σ ξ : F, ε ≤ Hμ,Σ (ξ : F, 2ε) ≤ hμ,Σ (X, G) + η As diam(ξ ) < δ, ξ ∈ PX generates BX by the construction of δ, and so we get hν,Σ (X, G) ≤ hμ,Σ (X, G) + η for each ν ∈ U as desired This finishes the proof ✷ In the spirit of L Bowen’s entropy as above, Kerr and Li introduced alternatively the sofic measure-theoretic entropy [32,34] as follows Let (Y, ρ) be a metric space and ε > A set ∅ = A ⊂ Y is said to be (ρ, ε)-separated if ρ(x, y) ≥ ε for all distinct x, y ∈ A We write Nε (Y, ρ) for the maximal cardinality of finite non-empty (ρ, ε)-separated subsets of Y (and set Nε (∅, ρ) = by convention) A basic fact is that if ∅ = A ⊂ Y is a maximal finite (ρ, ε)-separated subset of Y then for each y ∈ Y there exists x ∈ A such that ρ(x, y) < ε For each d ∈ N and (x1 , · · · , xd ), (x1 , · · · , xd ) ∈ X d , we set ρd (x1 , · · · , xd ), x1 , · · · , xd d = max ρ xi , xi i=1 For F ∈ FG , δ > and σ : G → Sym(d), g → σg with d ∈ N, put d d XF,δ,σ = (x1 , · · · , xd ) ∈ X d : max s∈F i=1 ρ (sxi , xσs (i) ) < δ ; d and then for μ ∈ M (X) and L ∈ FC(X) , we set d XF,δ,σ,μ,L = (x1 , · · · , xd ) ∈ d XF,δ,σ : max f ∈L d d f (xi ) − μ(f ) < δ i=1 (2.1) JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.8 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• By [34, Proposition 3.4] the measure-theoretic μ-entropy of (X, G) can be defined as (recalling the convention log = −∞) hμ (G, X) = sup inf inf inf lim sup ε>0 L∈FC(X) F ∈FG δ>0 i→∞ di log Nε XF,δ,σ , ρdi i ,μ,L di The sofic measure-theoretic entropy can be defined equivalently using finite open covers as follows [50, §2] We remark that it was Li who first used open covers for actions of sofic groups to consider sofic mean dimension in [35] For U ∈ CX and d ∈ N, we denote by Ud the finite Borel cover of X d consisting of U1 × U2 × · · · × Ud , where U1 , · · · , Ud ∈ U Let U ∈ CX , we set (recalling the convention log = −∞) hF,δ,μ,L (G, U) = lim sup i→∞ di log N Udi , XF,δ,σ i ,μ,L di di In particular, hF,δ,μ,L (G, U) takes the value −∞ if XF,δ,σ = ∅ for all i ∈ N large i ,μ,L enough Now we define the measure-theoretic μ-entropy of U as hμ (G, U) = inf inf inf hF,δ,μ,L (G, U) ≤ log N (U, X) L∈FC(X) F ∈FG δ>0 It is not hard to check that hμ (G, X) = sup hμ (G, U) U∈Co X Moreover, by the proof of [32, Theorem 6.1], it was proved implicitly hμ (G, X) = −∞ (and hence hμ (G, U) = −∞ for all U ∈ CoX ) for each μ ∈ M (X) \ M (X, G) Observe that both of hμ (G, U) and hμ (G, X) may take the value of −∞, and by [32, 34] if μ ∈ M (X, G) and BX admits a generating partition (in the sense of μ) with finite Shannon entropy then hμ (G, X) is just the quantity hμ,Σ (X, G) introduced before The following result is easy to obtain: Proposition 2.2 Let U ∈ CoX Then h• (G, U) : M (X) → [0, log N (U, X)] ∪ {−∞} is an upper semi-continuous function Proof Let μ ∈ M (X) For any ε > we may choose L ∈ FC(X) , F ∈ FG and δ > such that hF,2δ,μ,L (G, U) ≤ hμ (G, U) + ε Now we consider the non-empty open set μ ∈ V = ν ∈ M (X) : ν(f ) − μ(f ) < δ for all f ∈ L d d Then for each ν ∈ V we have XF,δ,σ,ν,L ⊂ XF,2δ,σ,μ,L for each σ : G → Sym(d), d ∈ N, which implies hν (G, U) ≤ hF,δ,ν,L (G, U) ≤ hF,2δ,μ,L (G, U) ≤ hμ (G, U) + ε This implies that the considered function is upper semi-continuous ✷ JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.9 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Weak expansiveness for actions of sofic groups Note that, when considering a continuous mapping over a compact metric space, since the introduction of h-expansiveness and asymptotical h-expansiveness, both of them turn out to be very important classes in the research area of dynamical systems It is shown by R Bowen [6] that positively expansive systems, expansive homeomorphisms, endomorphisms of a compact Lie group and Axiom A diffeomorphisms are all h-expansive, by Misiurewicz [40] that every continuous endomorphism of a compact metric group is asymptotically h-expansive if it has finite entropy, and by Buzzi [13] that any C ∞ diffeomorphism on a compact manifold is asymptotically h-expansive Moreover, there are more nice characterizations of asymptotical h-expansiveness obtained recently for this setting For example, a topological dynamical system is asymptotically h-expansive if and only if it admits a principal extension to a symbolic system by Boyle and Downarowicz [10], i.e., a symbolic extension which preserves entropy for each invariant measure; if and only if it is hereditarily uniformly lowerable by Huang, Ye and the second author of the present paper [28] (for a detailed definition of the hereditarily uniformly lowerable property and its story see [28]) In this section we explore similar weak expansiveness for actions of sofic groups By [34, Proposition 2.4] the topological entropy of (X, G) can be defined as h(G, X) = sup inf inf lim sup ε>0 F ∈FG δ>0 i→∞ di log Nε XF,δ,σ , ρdi , i di which is introduced and discussed in [32,34] Before proceeding, we need to recall the topological entropy for actions of sofic groups introduced in [50, §2] using finite open covers Let U ∈ CX For F ∈ FG and δ > we set hF,δ (G, U) = lim sup i→∞ di log N Udi , XF,δ,σ i di di Observe again that hF,δ (G, U) takes the value of −∞ whenever XF,δ,σ = ∅ for all i ∈ N i large enough Now we define the topological entropy of U as h(G, U) = inf inf hF,δ (G, U) ≤ log N (U, X) F ∈FG δ>0 It is not hard to check that h(G, X) = sup h(G, U) U∈Co X Observe that both of h(G, U) and h(G, X) may take the value of −∞ The sofic topological entropy and sofic measure-theoretic entropy are related to each other [32, Theorem 6.1] and [50, Theorem 4.1]: for U ∈ CoX , JID:YJFAN 10 AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.10 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• h(G, X) = sup hμ (G, X) and h(G, U) = μ∈M (X,G) max μ∈M (X,G) hμ (G, U), (3.1) where in the right-hand sides as above we set it to −∞ by convention if M (X, G) = ∅ In the spirit of Misiurewicz [40], the above idea can be used to introduce h-expansiveness and asymptotical h-expansiveness for actions of sofic groups Let U1 , U2 ∈ CX For F ∈ FG and δ > we set hF,δ (G, U1 |U2 ) = lim sup i→∞ di log max N Ud1i , XF,δ,σ ∩ V ≤ hF,δ (G, U1 ), i d di V ∈U2 i h(G, U1 |U2 ) = inf inf hF,δ (G, U1 |U2 ) ≤ h(G, U1 ), F ∈FG δ>0 h(G, X|U2 ) = sup h(G, U1 |U2 ) ≤ h(G, X), U1 ∈Co X h∗ (G, X) = inf h(G, X|U2 ) ≤ h(G, X) U2 ∈Co X Then h(G, X|{X}) = h(G, X) by the definitions We say that (X, G) is h-expansive if h(G, X|U) ≤ for some U ∈ CoX , and asymptotically h-expansive if h∗ (G, X) ≤ Each h-expansive action of a sofic group is asymptotically h-expansive by definition The next result shows that each expansive action of a sofic group is h-expansive, and thus these two kinds of expansiveness are indeed weak expansiveness Theorem 3.1 Let (X, G) be an expansive action of a sofic group with κ > an expansive constant and U ∈ CX Assume that diam(U) ≤ cκ for some c < Then h(G, X|U) ≤ Proof Let V ∈ CoX and ε > It suffices to prove that h(G, V|U) ≤ ε Let τ > be a Lebesgue number of V As (X, G) is an expansive action of a sofic group with κ > an expansive constant, it is not hard to choose F ∈ FG such that maxs∈F ρ(sx, sx ) < κ implies ρ(x, x ) < τ2 (for example see [46, Chapter 5, §5.6]) Now let δ > be small enough such that 8δ |F | |Θd | · |V| (1−c)2 κ2 ·d < eεd for all d ∈ N large enough, where Θd is the set of all subsets θ of {1, · · · , d} with |θ| < 8δ |F | · d (1 − c)2 κ2 For any map σ : G → Sym(d), g → σg with d ∈ N, recall d d XF,δ,σ = (x1 , · · · , xd ) ∈ X d : max s∈F i=1 ρ (sxi , xσs (i) ) < δ , d (3.2) JID:YJFAN 18 AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.18 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Note that V is an open subset of X F and KF ⊂ X F is a closed subset So there exists δ > such that KF,δ ⊂ V This finishes the proof ✷ Then, following the ideas of [34, Lemma 5.1] we have: Proposition 6.4 Let U, U2 ∈ CoX and U1 ∈ CcX with U1 h(G, U|U1 ) ≤ (G, U|U2 ) and thus U2 Then h(G, X|U1 ) ≤ (G, X|U2 ) Proof Let ε > We choose > η > small enough such that (G, U|U2 ) + ε + 2η log |U| ≤ (G, U|U2 ) + 2ε 1−η (6.1) and K ∈ FG , δ > such that, once F ∈ FG satisfies |KF ΔF | ≤ δ |F | then max log N (UF , K) ≤ (G, U|U2 ) + ε |F | K∈(U2 )F (6.2) Observing that U2 ∈ CoX and U1 ∈ CcX satisfy U1 U2 , we can choose δ > such that, for each U1 ∈ U1 there exists U2 ∈ U2 containing the open δ -neighborhood of U1 Now let l ∈ N and η > be as given by Lemma 6.2 with respect to τ = η and η −1 In FG we take eG ∈ F1 ⊂ · · · ⊂ Fl such that |Fk−1 Fk \ Fk | ≤ η |Fk | for each k = 2, · · · , l and |KFk ΔFk | ≤ δ |Fk | for all k = 1, · · · , l As the group G is amenable, such subsets F1 , · · · , Fl must exist Thus, by (6.2), l max k=1 max log N (UFk , K) ≤ (G, U|U2 ) + ε |Fk | K∈(U2 )Fk (6.3) For each k = 1, · · · , l and any K ∈ (U2 )Fk , let δ(k, K) > be as given by Lemma 6.3 with respect to K, Fk and U, and then set δk = min{δ(k, K) : K ∈ (U2 )Fk } We take δ > such that δ ≤ min{(δ )2 , δ12 , · · · , δl2 , |Fηl | } and [|Fl |δd] j=0 d j < (1 + ε)d for all large enough d ∈ N (6.4) Now let σ : G → Sym(d) be a good enough sofic approximation for G with some d ∈ N (and hence d ∈ N is large enough) If (x1 , · · · , xd ) ∈ XFdl ,δ,σ then d max s∈Fl i=1 ρ (sxi , xσs (i) ) < δ, d which implies that |J(x1 , · · · , xd , Fl )| ≥ (1 − |Fl |δ)d, where JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.19 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• J(x1 , · · · , xd , Fl ) = i ∈ {1, · · · , d} : max ρ(sxi , xσs (i) ) < s∈Fl √ 19 δ Denote by Θ the set of all subsets of {1, · · · , d} with at least (1 − |Fl |δ)d many elements, and for each θ ∈ Θ denote by XFdl ,δ,σ,θ the set of all (x1 , · · · , xd ) ∈ XFdl ,δ,σ with J(x1 , · · · , xd , Fl ) = θ Then [|Fl |δd] |Θ| = j=0 d j < (1 + ε)d using (6.4) XFdl ,δ,σ,θ = XFdl ,δ,σ and (6.5) θ∈Θ Let θ ∈ Θ As σ is good enough, by Lemma 6.2 there exist C1 , · · · , Cl ⊂ θ with (1) (2) (3) (4) the sets σ(Fk )Ck , k ∈ {1, · · · , l} are pairwise disjoint; {σ(Fk )c : c ∈ Ck } is η-disjoint for each k = 1, · · · , l; {σ(Fk )Ck : k ∈ {1, · · · , l}} (1 − 2η)-covers {1, · · · , d}; and for every k ∈ {1, · · · , l} and c ∈ Ck , Fk s → σs (c) is bijective Set Jθ = {1, · · · , d} \ {σ(Fk )Ck : k ∈ {1, · · · , l}} Then l |Jθ | ≤ 2ηd |Fk | · |Ck | ≤ and k=1 d 1−η l σ(Fk )Ck ≤ k=1 d 1−η (6.6) (i) Now let W ∈ (U1 )d , say W = and for each i = 1, · · · , d the open i=1 U1 (i) (i) δ -neighborhood of U1 is contained in U2 ∈ U2 For each k = 1, · · · , l and any ck ∈ Ck , as Ck ⊂ θ and Fk ⊂ Fl , if (x1 , · · · , xd ) ∈ XFdl ,δ,σ,θ ∩ W , then max ρ(xσs (ck ) , sxck ) < s∈Fk (σs (ck )) and so sxck ∈ U2 (σs (ck )) (as xσs (ck ) ∈ U1 s−1 U2 (σs (ck )) xck ∈ √ δ≤δ , ) for all s ∈ Fk by the selection of δ , thus (denoted by Q) ∈ (U2 )Fk , s∈Fk which implies by applying Lemma 6.3 and (6.3) that we can cover (xi )i∈σ(Fk )ck : (x1 , · · · , xd ) ∈ XFdl ,δ,σ,θ ∩ W ⊂ (xi )i∈σ(Fk )ck : max ρ(xσs (ck ) , sx) < δk for some x ∈ Q s∈Fk by at most (observing the selection of δk ) N (UFk , Q) ≤ e|Fk |·[h a (G,U|U2 )+ε] JID:YJFAN 20 AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.20 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• elements of Uσ(Fk )ck , and so it is not hard to cover (xi )i∈σ(Fk )Ck : (x1 , · · · , xd ) ∈ XFdl ,δ,σ,θ ∩ W using at most e|Ck |·|Fk |·[h a (G,U|U2 )+ε] elements of Uσ(Fk )Ck Thus l log N Ud , XFdl ,δ,σ,θ ∩ W ≤ |Ck | · |Fk | · (G, U|U2 ) + ε + |Jθ | log |U| k=1 ≤d (G, U|U2 ) + ε + 2η log |U| 1−η ≤ d (G, U|U2 ) + 2ε using (6.6) using (6.1) (6.7) Combining (6.5) with (6.7) we obtain log N Ud , XFdl ,δ,σ ∩ W ≤ d (G, U|U2 ) + 2ε + log(1 + ε) By the arbitrariness of ε > and W ∈ (U1 )d we obtain the conclusion ✷ We also have [34, Lemma 4.6], an improved version of Lemma 6.2 for an amenable group Recall that the group G is amenable throughout the whole section Lemma 6.5 Let ≤ τ < 1, < η < and K ∈ FG , δ > Then there are an l ∈ N and F1 , · · · , Fl ∈ FG with |KFk \ Fk | < δ|Fk | and |Fk K \ Fk | < δ|Fk | for all k = 1, · · · , l, such that for every good enough sofic approximation σ : G → Sym(d) for G with some d ∈ N and any set V ⊂ {1, · · · , d} with |V | ≥ (1 − τ )d, there exist C1 , · · · , Cl ⊂ V such that (1) the sets σ(Fk )Ck , k ∈ {1, · · · , l} are pairwise disjoint; (2) {σ(Fk )Ck : k ∈ {1, · · · , l}} (1 − τ − η)-covers {1, · · · , d}; and (3) for every k ∈ {1, · · · , l}, the map Fk × Ck (s, c) → σs (c) is bijective Let U ∈ CX , ε > and F ∈ FG , δ > We set hF,δ (G, ε|U) = lim sup i→∞ di log max Nε XF,δ,σ ∩ V, ρdi i di V ∈Udi recalling (2.1) , h(G, ε|U) = inf inf hF,δ (G, ε|U) F ∈FG δ>0 Now let V ∈ CX and ε1 , ε2 > Assume that diam(V) < ε1 and any open ball with radius ε2 is contained in some element of V It is easy to obtain JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.21 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• 21 hF,δ (G, ε1 |U) ≤ hF,δ (G, V|U) ≤ hF,δ (G, ε2 |U), h(G, ε1 |U) ≤ h(G, V|U) ≤ h(G, ε2 |U) Thus we have h(G, X|U) = lim h(G, ε|U) = sup h(G, ε|U) ε→0 (6.8) ε>0 Following the ideas of [34, Lemma 5.2] we have: Proposition 6.6 Let U ∈ CX Then h(G, X|U) ≥ (G, X|U) Proof Let U1 ∈ CoX We only need to prove h(G, X|U) ≥ (G, U1 |U) We choose ε > such that any open ball with ρ-radius ε is contained in some element of U1 , and let θ > 0, F ∈ FG , δ > We are to finish the proof by showing d log max Nε XF,δ,σ ∩ V, ρd ≥ (G, U1 |U) − 3θ d V ∈Ud (6.9) once σ : G → Sym(d) is a good enough sofic approximation for G with some d ∈ N Let M > be large enough and δ > small enough such that the diameter of the space X is at most M and √ δM< δ , δ log |U| < θ and − δ (G, U1 |U) ≥ (G, U1 |U) − θ (6.10) Applying Lemma 6.5, there are an l ∈ N and F1 , · · · , Fl ∈ FG so that l min k=1 s∈F |s−1 Fk ∩ Fk | ≥1−δ |Fk | (6.11) and l k=1 max log N (U1 )Fk , K ≥ max 0, (G, U1 |U) − θ , |Fk | K∈UFk (6.12) such that once σ : G → Sym(d) is a good enough sofic approximation for G with some d ∈ N then there exist C1 , · · · , Cl ⊂ {1, · · · , d} satisfying (1) (2) (3) (4) the sets σ(Fk )Ck , k ∈ {1, · · · , l} are pairwise disjoint; {σ(Fk )Ck : k ∈ {1, · · · , l}} (1 − δ )-covers {1, · · · , d}; for every k ∈ {1, · · · , l}, the map Fk × Ck (s, c) → σs (c) is bijective; and for all k ∈ {1, · · · , l} and s ∈ F, sk ∈ Fk , ck ∈ Ck , σssk (ck ) = σs σsk (ck ) Remark again that since the group G is amenable, such subsets F1 , · · · , Fl must exist JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.22 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• 22 Now assume that σ : G → Sym(d) is a good enough sofic approximation for G with some d ∈ N and let C1 , · · · , Cl ⊂ {1, · · · , d} be constructed as above For each k ∈ {1, · · · , l} and any K ∈ UFk , we take a maximal (ρFk , ε)-separated subset Ek,K of K which is obviously finite, where ρFk x, x = max ρ gx, gx g∈Fk for x, x ∈ X Then for each y ∈ K there exists an x ∈ Ek,K such that ρFk (x, y) < ε Observe that any open ball with ρ-radius ε is contained in some element of U1 , and hence any open ball with ρFk -radius ε is contained in some element of (U1 )Fk , thus |Ek,K | ≥ N (U1 )Fk , K (6.13) Now let (y1 , · · · , yl ) be any l-tuple with yk ∈ Ek,Kc , where k ∈ {1, · · · , l} and Kc ∈ UFk for each c ∈ Ck c∈Ck From the construction of C1 , · · · , Cl , it is not hard to see that there exists at least one point (x1 , · · · , xd ) ∈ X d such that once i ∈ σ(Fk )Ck for some k ∈ {1, · · · , l}, say i = σsk (ck ) with sk ∈ Fk and ck ∈ Ck , then xi = sk yk (ck ) Let (x1 , · · · , xd ) ∈ X d be such a point (corresponding to the l-tuple (y1 , · · · , yl )) Let s ∈ F and i ∈ {1, · · · , d} Once i = σsk (ck ) for some sk ∈ Fk and ck ∈ Ck , k ∈ {1, · · · , l}, if ssk ∈ Fk , then sxi = ssk yk (ck ) = xσssk (ck ) = xσs σsk (ck ) = xσs (i) Which implies that d d ρ2 (sxi , xσs (i) ) = i=1 d ρ2 (sxi , xσs (i) ) ≤ i∈{1,···,d}\E M2 {1, · · · , d} \ E , d (6.14) where l σ s−1 Fk ∩ Fk Ck E= k=1 Using the construction of C1 , · · · , Cl again, by (6.11) one has l l s−1 Fk ∩ Fk · |Ck | ≥ − δ |E| = k=1 |Fk | · |Ck | ≥ d − δ ≥ d − 2δ k=1 (6.15) Combining (6.14) with (6.15), we obtain d d ρ2 (sxi , xσs (i) ) ≤ 2δ M < i=1 δ2 d by the selection (6.10) of M and δ In particular, (x1 , · · · , xd ) ∈ XF,δ,σ JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.23 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• 23 On one hand, from the above constructions, it is easy to check that, given Kc ∈ UFk for each k ∈ {1, · · · , l} and any c ∈ Ck , there exists W ∈ W with l W= Uσ(Fk )Ck × {X}{1,···,d}\ l k=1 σ(Fk )Ck , k=1 l such that if a point (x1 , · · · , xd ) ∈ X d corresponds to an l-tuple from k=1 c∈Ck Ek,Kc then (x1 , · · · , xd ) ∈ W On the other hand, assume that (x1 , · · · , xd ) ∈ X d and (x1 , · · · , xd ) ∈ X d correspond to distinct l-tuples (y1 , · · · , yl ) and (y1 , · · · , yl ) from l c∈Ck Ek,Kc , where Kc ∈ UFk for each k ∈ {1, · · · , l} and any c ∈ Ck , respeck=1 tively Thus, for some k ∈ {1, · · · , l} and c ∈ Ck , yk (c) and yk (c) are distinct elements of Ek,Kc , in particular, ρFk (yk (c), yk (c)) ≥ ε, and then ρd (x1 , · · · , xd ), x1 , · · · , xd ≥ max ρ xi , xi i∈σ(Fk )c = max ρ syk (c), syk (c) = ρFk yk (c), yk (c) ≥ ε s∈Fk This implies that l l d max Nε XF,δ,σ ∩ W, ρd ≥ W ∈W |Ek,Kc | Ek,Kc = k=1 c∈Ck k=1 c∈Ck l ≥ N (U1 )Fk , Kc using (6.13) k=1 c∈Ck Combining the above estimation with the fact that {σ(Fk )Ck : k ∈ {1, · · · , l}} (1 − δ )covers {1, · · · , d}, we obtain that d log |U|δ d max Nε XF,δ,σ ∩ V, ρd V ∈Ud d ≥ log max Nε XF,δ,σ ∩ W, ρd W ∈W l |Ck | max log N (U1 )Fk , K ≥ k=1 K∈UFk l ≥ |Ck | · |Fk | · max 0, (G, U1 |U) − θ using (6.12) k=1 ≥ d − δ max 0, (G, U1 |U) − θ ≥ d (G, U1 |U) − 2θ using (6.10) Then (6.9) follows from (6.10) and (6.16) ✷ (6.16) JID:YJFAN 24 AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.24 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Now let us prove our main result of this section Proof of Theorem 6.1 Assume that (X, G) is asymptotically h-expansive Let ε > Then there exists U1 ∈ CoX such that h(G, X|U1 ) < ε Thus ha,∗ (G, X) ≤ (G, X|U1 ) < ε by Proposition 6.6, and finally ha,∗ (G, X) = Now assume that ha,∗ (G, X) = and let ε > By the definition, there exists U3 ∈ CoX with (G, X|U3 ) < ε As X is a compact metric space, we can take U4 ∈ CoX with U4 U3 , then h(G, X|U4 ) < ε by Proposition 6.4, and so h∗ (G, X) ≤ h(G, X|U4 ) < ε, thus h∗ (G, X) ≤ That is, (X, G) is asymptotically h-expansive We could prove similarly the remaining part of the theorem ✷ By the same proof we have: Theorem 6.7 h∗ (G, X) = ha,∗ (G, X) Setting U1 = U2 = {X} in Proposition 6.4 and U = {X} in Proposition 6.6, we obtain directly the following observation [34, Theorem 5.3] Corollary 6.8 h(G, X) = (G, X) Actions over compact metric groups In this section we shall provide more interesting asymptotically h-expansive examples when we consider actions of countable discrete amenable groups Recall that any C ∞ diffeomorphism on a compact manifold is asymptotically h-expansive by Buzzi [13] Moreover, if we consider a differentiable action (X, G) in the sense that the homeomorphism of X given by each g ∈ G is a C (1) map, where X is a compact smooth manifold (here we allow a smooth manifold to have different dimensions for different connected components, even including zero dimension) and G is a countable discrete amenable group containing Z as a subgroup of infinite index, then the action (X, G) has zero topological entropy (see for example [36, Lemma 5.7] by Li and Thom), and so it is h-expansive Moreover, inspired by Misiurewicz’s work [40, §7], in the following we prove: Theorem 7.1 Let G be a countable discrete amenable group acting on a compact metric group X by continuous automorphisms Then the action (X, G) is asymptotically h-expansive if and only if (G, X) < ∞ Remark 7.2 Let G be a countable discrete amenable group acting on a compact metrizable group X by continuous automorphisms If (X, G) has finite topological entropy then, by Corollary 3.7, Corollary 6.8 and Theorem 7.1, it admits an invariant measure with maximal entropy If the amenable group G is infinite and (X, G) has infinite topological entropy, then, by the affinity property of measure-theoretic entropy (for actions of a JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.25 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• 25 countable discrete infinite amenable group), it will also admit an invariant measure with maximal entropy Indeed, if we let μ be the normalized Haar measure of the compact metric group X (and hence μ is automatically G-invariant), then the measure-theoretic μ-entropy coincides with the topological entropy of the system [16, Theorem 2.2] (remark that X was assumed to be abelian in [16, Theorem 2.2], which is in fact not needed) Note that this fact was first pointed out by Berg [4] in the case of G = Z Remark 7.3 The conclusion of Theorem 7.1 does not hold if we remove the group structure from X even for the Z-actions There are many such actions The first one may be Gurevic’s example (for the detailed construction see for example [46, p 192]), which is a Z-action having finite entropy and without any invariant measure attaining maximal entropy Another example is [40, Example 6.4], which is a Z-action with finite entropy such that it is not asymptotically h-expansive, while each of its invariant measures has maximal entropy The “only if” part of Theorem 7.1 comes from Corollary 3.3 and Corollary 6.8 The proof of its “if” part relies on the concept of homogeneous measures And so first we recall the definition of homogeneous measures following [40, §7], which was first introduced by R Bowen in [5] Let G be a group acting on a compact metric space X, and denote by α the action Let μ ∈ M (X) For each U ∈ CoX , we set U ∈ U : x ∈ U and μ(U ) = P (U) = x∈X max V ∈U,x∈V μ(V ) ∈ CoX The measure μ is called α-homogeneous if there exist mappings D : CoX → CoX and c : CoX → (0, ∞) such that for any U ∈ CoX and each F ∈ FG we have μ(V ) ≤ c(U)μ(U ) for all U ∈ P (UF ) and V ∈ D(U)F In general, it is not easy to check if a measure is homogeneous While for G = Z Misiurewicz gave a sufficient condition for the existence of such a measure [40, Theorem 7.2], which can be generalized to a general group G as follows Following the spirit of Misiurewicz [40], let H be a group acting on a compact metric space (X, ρ) with the action Φ, that is, Φ is a homomorphism of H into the group of all homeomorphisms of X Recall that Φ is transitive if for any x, y ∈ X there exists g ∈ H with gx = y, and equicontinuous if for each ε > there exists δ > such that ρ(x1 , x2 ) < δ implies ρ(gx1 , gx2 ) < ε for all x1 , x2 ∈ X and g ∈ H Recall that μ ∈ M (X) is invariant with respect to Φ if gμ = μ for each g ∈ H While the transitivity here is different from the usual one in topological dynamics, the definitions of equicontinuity and invariance of a measure are just as usual JID:YJFAN 26 AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.26 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Lemma 7.4 Let G be a group acting on a compact metric space X, and denote by α the action Let H be a group and T an action of G on H by homomorphisms g → Tg for each g ∈ G, equivalently, each Tg : H → H is a homomorphism Now assume that Φ is a transitive equicontinuous action of H on X by homeomorphisms such that μ ∈ M (X) is invariant with respect to Φ and g(hx) = (Tg h)(gx) for all x ∈ X, g ∈ G and h ∈ H Then μ is α-homogeneous Proof Let U ∈ CoX We aim to construct some D(U) ∈ CoX with c(U) = such that both of mappings D and c satisfy the condition of μ being α-homogeneous Let ε > be a Lebesgue number of U By the equicontinuity of Φ there exists δ > such that ρ(x1 , x2 ) ≤ δ implies ρ(hx1 , hx2 ) < 3ε for all x1 , x2 ∈ X and h ∈ H Now let W be a non-empty open set with diameter at most δ For each x ∈ X, denote by Wx the set of all y ∈ X such that both x and y are contained in hW for some h ∈ H, then ρ(x, y) < 3ε for each y ∈ Wx , and so Wx ⊂ Ux for some Ux ∈ U Observe that the family of non-empty open subsets {hW : h ∈ H} covers X by the transitivity of the action Φ, and so we could choose D(U) ∈ CoX with D(U) ⊂ {hW : h ∈ H} Now we show that D(U) and c(U) = satisfy the required properties Let F ∈ FG Set WF,x = γ∈F γ −1 Wγx ⊂ γ∈F γ −1 Uγx ∈ UF for each x ∈ X Let U ∈ D(U)F , and say U = g∈F g −1 (hg W ) with hg ∈ H for any g ∈ F Then gx ∈ hg W and hence hg W ⊂ Wgx for all x ∈ U and g ∈ F Thus, for any x ∈ U , g −1 (hg W ) ⊂ U= g∈F and hence (observing WF,x ⊂ g −1 Wgx = WF,x x g∈F γ∈F γ −1 Uγx ∈ UF ) μ(U ) ≤ μ(WF,x ) ≤ max μ(V ) : x ∈ V and V ∈ UF (7.1) Let x ∈ X and k ∈ H For y ∈ X, y ∈ kWx , equivalently k−1 y ∈ Wx , if and only if both x and k−1 y are contained in hW for some h ∈ H, equivalently both kx and y are contained in khW for some h ∈ H, if and only if y ∈ Wkx Thus kWx = Wkx Now let h ∈ H, Y ⊂ X and g ∈ G By assumptions, g(h(g −1 Y )) = (Tg h)(g(g −1 Y )) = (Tg h)Y , equivalently, h(g −1 Y ) = g −1 ((Tg h)Y ) And hence h g −1 Wgx = g −1 (Tg h)Wgx = g −1 (W(Tg h)(gx) ) = g −1 (Wg(hx) ) Which implies that, for all x ∈ X and any h ∈ H, F ∈ FG , γ −1 Wγx hWF,x = h γ∈F h γ −1 Wγx = = γ∈F γ −1 Wγ(hx) = WF,hx γ∈F As the action Φ is transitive and μ is invariant with respect to Φ, we obtain JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.27 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• μ(WF,x ) = μ(WF,y ) for all x, y ∈ X and any F ∈ FG 27 (7.2) Summing up, for each F ∈ FG and any U ∈ D(U)F and Q ∈ P (UF ), let x ∈ U , and by the construction of P (UF ) we could choose y ∈ Q with μ(Q) = max{μ(V ) : y ∈ V and V ∈ UF }, and then μ(U ) ≤ μ(WF,x ) using (7.1) = μ(WF,y ) ≤ max μ(V ) : y ∈ V and V ∈ UF using (7.2) using (7.1) again = μ(Q) Recalling c(U) = 1, we see that the measure μ is α-homogeneous ✷ As a direct corollary, we have: Proposition 7.5 Let G be a group acting on a compact metric group X by continuous automorphisms, and denote by α the action Let μ ∈ M (X) be the normalized Haar measure of the group X Then μ is α-homogeneous Proof Set H = X and let T be the action α of G on H by automorphisms Tg : H → H, h → gh Now let Φ be the action of H on X by homeomorphisms h : X → X, x → hx for all h ∈ X Obviously, the action Φ is transitive and the measure μ is invariant with respect to the action Φ and g(hx) = (gh)(gx) = (Tg h)(gx) for all x ∈ X, g ∈ G and h ∈ H, recalling that G acts on H = X by automorphisms Observe that by the well-known Birkhoff–Kakutani Theorem the compact metric group X admits a left-invariant compatible metric (for example see [25, p 430, Lemma C.2]), which implies that the action Φ is equicontinuous Thus the measure μ is α-homogeneous by Lemma 7.4 ✷ For a Z-action acting on a compact metric space X, if the action admits a homogeneous measure and has finite topological entropy, then it is asymptotically h-expansive [40, Theorem 7.1] In the following, we shall see that the proof there also works for a general countable discrete infinite amenable group G Let (X, G) be an action of a countable group G on a compact metric space X with μ ∈ M (X), F ∈ FG , U ∈ CoX We put MF (U) = max μ(U ) > and mF (U) = U ∈UF For U, V ∈ CoX , MF (V) ≤ MF (U) whenever V MF (V) ≤ MF D(U) ≤ c(U)mF (U) U ∈P (UF ) μ(U ) U, and if μ is α-homogeneous then and hence mF (U) ≥ MF (V) >0 c(U) (7.3) JID:YJFAN 28 AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.28 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• whenever V D(U) Let U ∈ CoX and Y ⊂ X The star of Y with respect to U and the star of U are defined, respectively, as st(Y, U) = {U ∈ U : U ∩ Y = ∅} and St U = st(U, U) : U ∈ U ∈ CoX Following [40, Lemma 7.1], it is easy to obtain: Lemma 7.6 Let (X, G) be an action of a countable group G on a compact metric space X with μ ∈ M (X) and U, V ∈ CoX with V U Then max N (St V)F , K · mF (V) ≤ MF (St U) K∈UF Proof Observe St(VF ) (St V)F as done in [40, Lemma 3.3] Let U ∈ UF with max N (St V)F , K ≤ max N St(VF ), K = N St(VF ), U =: p K∈UF K∈UF (7.4) Using [40, Lemma 3.1], we could select a disjoint family C ⊂ P (VF ) such that C ∩ U = ∅ for each C ∈ C and U ⊂ {st(C, P (VF )) : C ∈ C} Observing P (VF ) ⊂ VF by the definition, C ⊂ VF (and hence st(C, VF ) ∈ St(VF ) for each C ∈ C) and U ⊂ {st(C, VF ) : C ∈ C}, which implies p ≤ |C| Now say U = γ∈F γ −1 Uγ with Uγ ∈ U for each γ ∈ F , then st(U, VF ) = γ −1 Uγ = ∅ W ∈ VF : W ∩ γ∈F st γ −1 Uγ , γ −1 V = ⊂ γ∈F ⊂ γ −1 st(Uγ , V) γ∈F γ −1 st(Uγ , U) (as V U) ∈ (St U)F (as Uγ ∈ U for all γ ∈ F ) γ∈F (7.5) As C ⊂ VF and C ∩ U = ∅ for each C ∈ C, C ⊂ st(U, VF ) by the definition Now recalling that C ⊂ P (VF ) is a disjoint family and p ≤ |C|, we have p · mF (V) ≤ μ C ≤ μ st(U, VF ) ≤ MF (St U) by (7.5) Then the conclusion follows from (7.4) and the above estimation ✷ The following result generalizes [40, Theorem 7.1] to the case of G being a general countable discrete infinite amenable group Proposition 7.7 Let α be the action (X, G) with G a countable discrete infinite amenable group and μ ∈ M (X, G) Assume that μ is α-homogeneous and (G, X) < ∞ Then the action (X, G) is asymptotically h-expansive JID:YJFAN AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.29 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• 29 Proof As μ is α-homogeneous, let D : CoX → CoX and c : CoX → (0, ∞) correspond to μ Setting U = {X} (and hence St U = {X}) in Lemma 7.6 we obtain N (St V)F , X · mF (V) ≤ 1, equivalently, mF (V) ≤ for each V ∈ CoX And then once V, W ∈ CoX satisfy W and hence combining (7.3) with (7.6) we obtain mF (V) ≥ MF (W) ≥ c(V) N (WF , X)c(V) N ((St V)F , X) (7.6) D(V), N (WF , X)MF (W) ≥ and MF (W) ≤ c(V)mF (V) ≤ c(V) N ((St V)F , X) (7.7) Now fix any C, D ∈ CoX We choose V ∈ CoX with St V C, A ∈ CoX with St A D(V) and E ∈ CoX with St E D by [40, Proposition 3.5]; and then find G ∈ CoX with G A, B ∈ CoX with B A and B E (and hence St B D) and W ∈ CoX with W D(B) F (B ) Choose B ∈ CoX such that B D(B), then by (7.3), mF (B) ≥ Mc(B) > for any F ∈ FG And hence we have max N (St B)F , K ≤ max N (St B)F , K K∈AF K∈(G)F ≤ MF (St A) mF (B) ≤ c(V) · c(B) · as St A D(V) and W (as G A) (using Lemma 7.6, as B N (WF , X) N ((St V)F , X) A) using (7.7) D(B), which implies (G, St B|G) ≤ (G, W) − (G, St V) from the definitions given at the end of Section 5; and then (recalling that G is a countable discrete infinite amenable group and observing St V C and St B D) (G, D|G) ≤ (G, St B|G) ≤ (G, W) − (G, St V) ≤ (G, X) − (G, C) By the arbitrariness of D ∈ CoX , one has ha,∗ (G, X) ≤ (G, X|G) ≤ (G, X) − (G, C), and then by the arbitrariness of C ∈ CoX and observing (G, X) < ∞, we obtain ha,∗ (G, X) = That is, (X, G) is asymptotically h-expansive by Theorem 6.1 ✷ Now we are ready to prove Theorem 7.1 JID:YJFAN 30 AID:7165 /FLA [m1L; v1.143-dev; Prn:6/01/2015; 15:49] P.30 (1-32) N.-P Chung, G Zhang / Journal of Functional Analysis ••• (••••) •••–••• Proof of Theorem 7.1 The “only if” part comes from Corollary 3.3 Now we prove the “if” part If G is finite, it is trivial to see from the definition that (G, X) < ∞ implies the finiteness of the group X, then the action (X, G) is expansive and hence asymptotically h-expansive Now assume that G is infinite Let μ ∈ M (X) be the normalized Haar measure of the compact metric group X By Proposition 7.5 and Proposition 7.7, we only need to show μ ∈ M (X, G), which is obvious by the assumption, as it is well known that each continuous surjective homomorphism of the compact metric group X preserves μ ✷ It is natural to ask the following question which we haven’t solved currently Question 7.8 Does Theorem 7.1 hold when the acting group is sofic? At the end of this section, we present further discussions about actions of countable groups over a compact metrizable abelian group by continuous automorphisms Let α be an action of a countable group G acting on a compact metrizable abelian group X by continuous automorphisms On one hand, by [15, Theorem 3.1], the action (X, G) is expansive if and only if there exist some k ∈ N, some left ZG-submodule J of (ZG)k , and some A ∈ Mk (ZG) being invertible in Mk ( (G)) such that the left ZG-module X is isomorphic to (ZG)k /J and the rows of A are contained in J On the other hand, suppose that G is amenable and X is a finitely presented left ZG-module, and write X as (ZG)k /(ZG)n A for some k, n ∈ N and A ∈ Mn×k (ZG), then (G, X) < ∞ if and only if the additive map (ZG)k → (ZG)n sending a to aA∗ is injective [15, Theorem 4.11], which implies that (X, G) is asymptotically h-expansive by Theorem 7.1 As a consequence, if G is amenable, then for any non-zero divisor f in ZG which is not invertible in (G), the principal algebraic action αf , the canonical action of G over ZG/ZGf , is asymptotically h-expansive but not expansive See [15, §3 and §4] for more details Acknowledgments N.-P Chung would like to thank Lewis Bowen for asking and discussing the question “When the actions of sofic groups are expansive, they admit measures of maximal entropy?” which started this project The joint work began when N.-P Chung visited School of Mathematical Sciences, Fudan University on July, 2012 Part of the work was carried out while N.-P Chung attended Arbeitsgemeinschaft “Limits of Structures” at Oberwolfach on April, 2013 Most of this work was done while he was doing postdoc at Max Planck Institute, Mathematics in the Sciences, Leipzig Both authors thank Hanfeng Li for helpful suggestions which resulted in substantial improvements to this paper The authors are also grateful to Tomasz Downarowicz, Ben Hayes and Wen Huang for helpful comments, and to Thomas Ward for bringing the reference [4] into our attention which is related to Remark 7.2 We thank the unknown referee for useful comments JID:YJFAN AID:7165 /FLA [m1L; 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