Communications in Mathematics and Statistics

, Volume 1, Issue 2, pp 213–257

Combinatorial Independence and Sofic Entropy

Article

Abstract

We undertake a local analysis of combinatorial independence as it connects to topological entropy within the framework of actions of sofic groups.

Keywords

Independence Sofic entropy Loeb space Algebraic action Fuglede-Kadison determinant Li-Yorke chaos 

1 Introduction

Among the various phenomena in dynamics associated with randomness, weak mixing and entropy stand out for the depth of their theory and the breadth of their applications (see for example [11, 13]). In the setting of discrete acting groups, weak mixing makes sense in general while entropy, as classically formulated, requires the group to be amenable. One can view these two concepts in a unified way across both measurable and topological dynamics by means of the combinatorial notion of independence. In close parallel with the 1 theorems of Rosenthal and Elton-Pajor in Banach space theory, weak mixing and positive entropy reflect two of the basic regimes in which combinatorial independence can occur across the orbit of a tuple of sets in a dynamical system. The first of these asks for independence over a subset of the group having infinite cardinality, while the other requires this subset to satisfy a positive density condition. Inspired by work in the local theory of entropy [16], the authors studied this connection between independence, weak mixing, and entropy in [21, 22] as part of a program to develop a general theory of combinatorial independence in dynamics.

Combinatorial independence is the basic set-theoretic expression of randomness in which we are concerned not with the size of intersections, as in the probabilistic context, but merely with their nonemptiness. A collection {(Ai,1,…,Ai,k)}iJ of k-tuples of subsets of a set X is independent if for every nonempty finite set FJ and function ω:F→{1,…,k} the intersection ⋂iFAi,ω(i) is nonempty. If a group G is acting on X then given a tuple (A1,…,Ak) of subsets of X we say that a set JG is an independence set for (A1,…,Ak) if the collection {(s−1A1,…,s−1Ak)}sJ is independent. In the case of an action of a countable amenable group G on a compact Hausdorff space X, the independence densityI(A) of a tuple A=(A1,…,Ak) of subsets of X is defined as the limit of φA(F)/|F| as the nonempty finite set FG becomes more and more left invariant, where φA(F) denotes the maximum of the cardinalities of the independence sets for A which are contained in F [21, Proposition 3.23]. We then say that a tuple (x1,…,xk)∈Xk is an IE-tuple if for every product neighborhood U1×⋯×Uk of (x1,…,xk) the tuple U=(U1,…,Uk) satisfies I(U)>0. This condition on U is equivalent to the existence of an independence set JG for U which has positive density with respect to a given tempered Følner sequence {Fi}i∈ℕ in the sense that limi→∞|FiJ|/|Fi|>0. It turns out that a nondiagonal tuple is an IE-tuple if and only if it is an entropy tuple [21, Sect. 3]. For general acting G we define IT-tuples in the same way as IE-tuples except that we ask instead for the independence set to have infinite cardinality. Versions of IE-tuples and IT-tuples can also be defined for probability-measure-preserving actions by requiring the independence condition on the set J to hold whenever we remove small parts of the sets s−1U1,…,s−1Uk for each sG. The following are some of the results relating independence, weak mixing, and entropy that were established in [21, 22].
  1. (1)

    A continuous action GX of an Abelian group on a compact Hausdorff space is (topologically) weakly mixing if and only if every tuple of points in X is an IT-tuple (in which case the action is said to be uniformly untame of all orders).

     
  2. (2)

    A probability-measure-preserving action G↷(X,μ) of an arbitrary group is weakly mixing if and only if its universal topological model is uniformly untame of all orders.

     
  3. (3)

    A continuous action GX of a discrete amenable group on a compact Hausdorff space has positive topological entropy if and only if it has a nondiagonal IE-pair (for G=ℤ this was first proved by Huang and Ye in [18] using measure-theoretic techniques). Moreover, the action has uniformly positive entropy if and only if every pair of points in X is an IE-pair, and uniformly positive entropy of all orders if and only if every tuple of points in X is an IE-tuple.

     
  4. (4)

    A probability-measure-preserving action G↷(X,μ) of a discrete amenable group has positive measure entropy if and only if there is a nondiagonal measure IE-pair in some topological model. Moreover, the action has complete positive entropy if and only if every tuple of points is an IE-tuple in the universal topological model (for G=ℤ this was proved by Glasner and Weiss in [15]).

     
Chung and the second author applied IE-tuples in [7] as part of a new approach for studying the relation between homoclinicity and entropy in expansive algebraic actions that enabled them to break the commutativity barrier and establish some duality-type equivalences for polycyclic-by-finite acting groups. In this case, and more generally for actions of a countable amenable group on a compact group X by automorphisms, the analysis of IE-tuples is governed by a single closed invariant normal subgroup of X called the IE group [7, Sect. 7].
Recent seminal work of Bowen in [5] has expanded the scope of the classical theory of entropy for actions of discrete amenable groups to the much broader realm of sofic acting groups. For a countable group G, soficity can be expressed as the existence of a sequence Σ={σi:G→Sym(di)} of maps from G into finite permutation groups which is asymptotically multiplicative and free in the sense that
$$\begin{array}{r@{\quad}l} \mbox{(i)} & \displaystyle\lim_{i\to\infty} \frac{1}{d_i} \big| \bigl\{ k\in\{ 1, \dots,d_i \} : \sigma_{i,st} (k) = \sigma_{i,s} \sigma_{i,t} (k) \bigr\} \big| = 1\quad \mbox{for all}\ s,t\in G,\quad \mbox{and} \\[12pt] \mbox{(ii)} &\displaystyle\lim_{i\to\infty} \frac{1}{d_i} \big| \bigl\{ k\in\{ 1, \dots,d_i \} : \sigma_{i,s} (k) \neq\sigma_{i,t} (k) \bigr\} \big| = 1\quad \mbox{for all distinct}\ s,t\in G. \end{array} $$
Such a sequence for which limi→∞di=∞ we call a sofic approximation sequence. By measuring the asymptotic exponential growth of dynamical models which are compatible with a fixed sofic approximation sequence, Bowen defined in [5] a collection of invariants for probability-measure-preserving actions of a countable sofic group admitting a generating partition with finite Shannon entropy. A remarkable application of this sofic measure entropy was a far-reaching extension of the Ornstein-Weiss classification of Bernoulli actions of amenable groups.

The authors developed in [23] a more general operator-algebraic approach to sofic entropy that enables one to remove the generator hypothesis (see also [20] for a formulation in terms of finite partitions). This led to a sofic version of topological entropy as well as a variational principle relating it to sofic measure entropy. We used this variational principle to compute the sofic topological entropy of a principal algebraic action \(G\curvearrowright\widehat {{\mathbb{Z}}G/{\mathbb{Z}}G f}\) of a countable residually finite group in the case that the sofic approximation sequence arises from finite quotients of G and f is invertible in the full group C-algebra C(G). In line with previous work on algebraic actions [6, 8, 26, 31] (see also the more recent [27]), this value turns out to be equal to the logarithm of the Fuglede-Kadison determinant of f in the group von Neumann algebra \({\mathcal{L}}G\). We also showed how topological entropy can be used to give a proof that Gottschalk’s surjunctivity conjecture holds for countable sofic groups, a result originally established by Gromov in [17], where the idea of soficity itself first appeared.

In the present work we initiate a local analysis of independence as it connects to topological entropy within this broadened framework of actions of sofic groups. Given a continuous action GX of a countable sofic group and a sofic approximation sequence Σ={σi:G→Sym(di)} for G, we define the notion of a Σ-IE-tuple by externalizing the positive independence density condition in the amenable case to the finite sets {1,…,di} appearing in the sequence Σ (Definition 4.3). We show in Sect. 4 that Σ-IE-tuples share many of the same properties as IE-tuples for actions of discrete amenable groups. In particular, the action GX has positive entropy with respect to Σ if and only if there is a nondiagonal Σ-IE-pair in X×X. On the other hand, we do not know whether the product formula holds in general for Σ-IE-tuples. However, granted that we use a free ultrafilter \({\mathfrak{F}}\) over ℕ to express the independence density condition in the definition of Σ-IE-tuples, we demonstrate in Theorem 5.2 that the product formula holds under the assumption of ergodicity on the action of the commutant of G inside the group of measure-preserving automorphisms of the Loeb space \(\prod_{\mathfrak{F}}\{ 1,\dots,d_{i} \}\) which arise from permutations of the sets {1,…,di}. We then prove that this commutant acts ergodically when G is residually finite and Σ is built from finite quotients of G (Theorem 5.7), and also when G is amenable and Σ is arbitrary (Theorem 5.8). In the case that G is nonamenable, a combination of results of Elek and Szabo [12, Theorem 2] and Paunescu [35] shows that there exist Σ for which the ergodicity condition fails.

The definition of IE-tuples for amenable G, as given in [21], involves an asymptotic density condition over finite subsets of G which become more and more invariant. Although density in this sense loses its meaning in the nonamenable case, we might nevertheless ask what the external independence density in the definition of Σ-IE-tuples implies about the degree of independent behavior across orbits in X. We observe in Proposition 4.6 that every Σ-IE-tuple (and more generally every sofic IE-tuple as defined in Definition 4.3) is an orbit IE-tuple, by which we mean that for every product neighborhood U1×⋯×Uk of the given tuple (x1,…,xk) in Xk, the tuple (U1,…,Uk) has positive independence density over G in the sense that there is a q>0 such that every finite set FG has a subset of cardinality at least q|F| which is an independence set for (U1,…,Uk) (note that this definition makes sense for any acting group G). We show moreover in Theorem 4.8 that, for amenable G, Σ-IE-tuples, IE-tuples, and orbit IE-tuples are all the same thing. This puts us in the pleasant and somewhat surprising situation that IE-tuples can be identified by a density condition that does not structurally depend on amenability for its formulation, and raises the question about the relation between entropy and orbit IE-tuples for nonamenable sofic G. In another direction, Theorem 7.1 asserts that if a tuple of subsets of X has positive independence density over G then it has an infinite independence set in G, which implies that every orbit IE-tuple is an IT-tuple.

By a theorem of Chung and the second author in [7], an algebraic action of a countable group G is expansive if and only if it is either the dual action \(G\curvearrowright X_{A} := \widehat{({\mathbb{Z}}G)^{n} /({\mathbb{Z}}G)^{n} A}\) for some n∈ℕ and matrix AMn(ℤG) which is invertible in Mn(1(G)), or the restriction of such an action to a closed G-invariant subgroup of XA. In the same paper it is shown that an expansive algebraic action GX of a polycyclic-by-finite group has completely positive entropy with respect to the Haar measure precisely when the IE group is equal to X, which is equivalent to every tuple of points in X being an IE-tuple. It is also shown that, when G is amenable, every action of the form GXA with A invertible in Mn(1(G)) has the property that every tuple of points in X is an IE-tuple (see Lemma 5.4 and Theorems 7.3 and 7.8 in [7]). We prove in Theorem 6.7 that if G is a countable sofic group, n∈ℕ, and A is a matrix in Mn(ℤG) which is invertible in Mn(1(G)), then the algebraic action GXA has the property that every tuple of points in XA is a Σ-IE-tuple for every sofic approximation sequence Σ. We use this to answer a question of Deninger in the case that G is residually finite by combining it with an argument from [7] and the entropy computation for principal algebraic actions from [23] mentioned above to deduce that if f is an element of ℤG which is invertible in 1(G) and has no left inverse in ℤG then the Fuglede-Kadison determinant of f satisfies \(\det_{{\mathcal{L}}G} f > 1\) (Corollary 6.8). Deninger asked whether this holds for all countable groups [9, Question 26], and affirmative answers were given in [10] for residually finite amenable G and more generally in [7] for amenable G.

For a continuous action GX of a countably infinite group on a compact metrizable space with compatible metric ρ, we say that a pair (x,y)∈X×X is a Li-Yorke pair if
$$\limsup_{G \ni s\to\infty}\rho(sx, sy)>0 \quad \text{and}\quad \liminf_{G \ni s\to\infty}\rho(sx, sy)=0 , $$
where the limit supremum and limit infimum mean the limits of supsGFρ(sx,sy) and infsGFρ(sx,sy), respectively, over the net of finite subsets F of G. Note that the definition of Li-Yorke pair does not depend on the choice of the metric ρ. The action GX is said to be Li-Yorke chaotic if there is an uncountable subset Z of X such that every nondiagonal pair (x,y) in Z×Z is a Li-Yorke pair. The notion of Li-Yorke chaos stems from [28]. In the case of a continuous map T:XX, a theorem of Blanchard, Glasner, Kolyada, and Maass in [3] states that positive entropy implies Li-Yorke chaos. In [21] the authors strengthened this by showing that for every k≥2 and product neighborhood U1×⋯×Uk of a nondiagonal IE-tuple (x1,…,xk)∈Xk there are Cantor sets ZiUi for i=1,…,k such that
  1. (1)

    every nonempty tuple of points in ⋃iZi is an IE-tuple, and

     
  2. (2)
    for all m∈ℕ, distinct y1,…,ym∈⋃iZi, and \(y_{1}' ,\dots, y_{m}'\in\bigcup_{i} Z_{i}\) one has
    $$\liminf_{n\to\infty} \max_{1\leq i\leq m} \rho\bigl(T^n y_i,y_i' \bigr) = 0 . $$
     
In Theorem 8.1 we show that a similar result holds when G is sofic and IE-tuples are replaced by Σ-IE-tuples as defined with respect to a free ultrafilter \({\mathfrak{F}}\) on ℕ, where Σ is any sofic approximation sequence for G. Using \({\mathfrak{F}}\) in the definition of entropy, we deduce that if the action has positive entropy for some Σ then it is Li-Yorke chaotic. As a corollary, if the action GX is distal then hΣ(X,G)=0 or −∞. In particular, when G is amenable every distal action GX has zero entropy, which is well known in the case G=ℤ [34].
The following diagram illustrates how some of the main results of the paper relate various properties of actions of a countable discrete group G on a compact metrizable space X, which we assume to have more than one point. In the left column we assume that G is sofic and that Σ is a fixed but arbitrary sofic approximation sequence. The unlabeled implications are trivial. By pair we mean an element of X×X. See [21] for terminology related to tameness and nullness.

The organization of the paper is as follows. In Sect. 2 we set up some basic notation and review sofic topological entropy. In Sect. 3 we introduce orbit IE-tuples and prove a product formula for them. Section 4 introduces Σ-IE-tuples and includes our results relating them to orbit IE-tuples. In Sect. 5 we focus on the product formula for Σ-IE-tuples and the question of ergodicity for the action of G′ on the Loeb space. Section 6 contains the material on algebraic actions. In Sect. 7 we prove that positive independence density for a tuple of subsets implies the existence of an infinite independence set, showing that orbit IE-tuples are IT-tuples. Finally in Sect. 8 we establish the theorem connecting independence and entropy to Li-Yorke chaos in the sofic framework.

2 Sofic Topological Entropy

We review here the definition of sofic topological entropy [23, 24] and in the process introduce some of the basic notation and terminology appearing throughout the paper. Our approach will bypass the operator algebra technology that appears in [23, 24].

Let Y be a set equipped with a pseudometric ρ and let ε≥0. A set AY is said to be (ρ,ε)-separated if ρ(x,y)≥ε for all distinct x,yA. Write Nε(Y,ρ) for the maximum cardinality of a (ρ,ε)-separated subset of Y.

Let GX be a continuous action of a countable sofic group on a compact metrizable space. Let Σ={σi:G→Sym(di)} be a sofic approximation sequence for G, meaning that
$$\begin{array}{r@{\quad}l} \mbox{(i)} & \displaystyle\lim_{i\to\infty} \frac{1}{d_i} \big| \bigl\{ k\in\{ 1,\dots,d_i \} : \sigma_{i,st} (k) = \sigma_{i,s} \sigma_{i,t} (k) \bigr\} \big| = 1\quad \mbox{for all}\ s,t\in G,\\[12pt] \mbox{(ii)} & \displaystyle\lim_{i\to\infty} \frac{1}{d_i} \big| \bigl\{ k\in\{ 1,\dots,d_i \} : \sigma_{i,s} (k) \neq\sigma_{i,t} (k) \bigr\} \big| = 1\quad \mbox{for all distinct}\ s,t\in G, \end{array} $$
and di→∞ as i→∞. Depending on the situation, for a∈{1,…,di} we may write σi,s(a), σi(s)a, or sa to denote the image of a under the evaluation of σi at s.
Let ρ be a continuous pseudometric on X. For a given d∈ℕ, we define on the set of all maps from {1,…,d} to X the pseudometrics

Definition 2.1

Let F be a nonempty finite subset of G and δ>0. Let σ be a map from G to Sym(d) for some d∈ℕ. Define Map(ρ,F,δ,σ) to be the set of all maps φ:{1,…,d}→X such that ρ2(φσs,αsφ)<δ for all sF, where αs is the transformation xsx of X.

Definition 2.2

Let F be a nonempty finite subset of G and δ>0. For ε>0 define where F in the third line ranges over the nonempty finite subsets of G. In the case that Map(ρ,F,δ,σi) is empty for all sufficiently large i, we set \(h_{\varSigma,2}^{\varepsilon}(\rho,F, \delta) = -\infty\). We similarly define \(h_{\varSigma,\infty}^{\varepsilon}(\rho,F, \delta)\), \(h_{\varSigma,\infty}^{\varepsilon}(\rho,F)\), \(h_{\varSigma,\infty}^{\varepsilon}(\rho)\) and hΣ,∞(ρ) using Nε(⋅,ρ) in place of Nε(⋅,ρ2).

Instead of the limit supremum above we could have taken a limit over a fixed free ultrafilter on ℕ, whose utility is apparent for example if we wish to have a product formula (see Sect. 5). We will also use this variant in Sect. 8.

The pseudometric ρ is said to be dynamically generating if for every pair of distinct points x,yX there is an sG such that ρ(sx,sy)>0.

Lemma 2.3

Suppose thatρandρare continuous pseudometrics onXand thatρis dynamically generating. LetFbe a nonempty finite subset ofGandδ>0. Then there exist a nonempty finite subsetFofGandδ′>0 such that for anyd∈ℕ and sufficiently good sofic approximationσ:G→Sym(d) one has Map(ρ′,F′,δ′,σ)⊆Map(ρ,F,δ,σ).

Proof

List the elements of G as s1,s2,…. Since ρ′ is dynamically generating, we have the compatible metric ρ″ on X defined by
$$\rho''(x, y)=\sum_{k=1}^{\infty} \frac{1}{2^k}\rho'(s_kx, s_ky). $$
It follows that there are a nonempty finite subset F″ of G and a δ″>0 such that, for all x,yX, if maxsFρ′(sx,sy)<δ″ then ρ(x,y)<δ/2. Set F′=F″∪(FF). Let δ′>0 and let σ be a map from G to Sym(d) for some d∈ℕ. Let φ∈Map(ρ′,F′,δ′,σ). Then Suppose that \(2\sqrt{\delta'}<\delta''\) and σ is a good enough sofic approximation for G so that
$$\big|\bigl\{a\in\{1, \dots, d\}: (s_1s_2)a=s_1(s_2a) \text{ for all } s_1\in F'', s_2\in F\bigr\}\big|\ge\bigl(1-\delta'\bigr)d. $$
Then It follows that when δ′ is small enough independently of d and σ, one has φ∈Map(ρ,F,δ,σ). □

The following proposition is contained in Proposition 2.4 of [24], whose statement and proof use the operator-algebraic formulation of sofic topological entropy from [23].

Proposition 2.4

Letρandρbe continuous pseudometrics onXwhich are dynamically generating. Then
$$h_{\varSigma,2} (\rho) = h_{\varSigma,2} \bigl(\rho'\bigr) = h_{\varSigma,\infty} (\rho) = h_{\varSigma,\infty} \bigl(\rho'\bigr) . $$

Proof

Since the pseudometric ρ dominates the pseudometric ρ2, we have hΣ,2(ρ)≤hΣ,∞(ρ).

Next we argue that hΣ,∞(ρ)≤hΣ,2(ρ). Let F be a finite subset of G, δ>0, and σ a map from G to Sym(d) for some d∈ℕ. Let 1/2>ε>0. Let η>0 be the minimum of ε2 and the reciprocal of the minimum cardinality of an (ε/2)-spanning subset of X with respect to ρ. Given a φ∈Map(ρ,F,δ,σ), every element in the open (ρ2,η)-ball in Map(ρ,F,δ,σ) around φ agrees with φ to within \(\sqrt{\eta}\), and hence to within ε, on a subset of {1,…,d} of cardinality at least (1−η)d. Thus the maximum cardinality of a (ρ,ε)-separated subset of the open (ρ2,η)-ball around φ is at most \(\sum_{j=0}^{\lfloor\eta d\rfloor} \binom{d}{j} \eta^{-j}\), which by Stirling’s approximation is bounded above, for all d sufficiently large, by eβdηηd for some β>0 not depending on d with β→0 as ε→0. Hence
$$N_\varepsilon\bigl(\mathrm{Map}(\rho,F,\delta,\sigma),\rho_\infty \bigr) \leq e^{\beta d} \eta^{-\eta d} N_\eta\bigl( \mathrm{Map}(\rho,F,\delta ,\sigma ),\rho_2 \bigr) . $$
It follows that
$$h_{\varSigma,\infty}^\varepsilon(\rho) \leq h_{\varSigma,2}^\eta( \rho) + \beta- \eta\log\eta, $$
and since βηlogη→0 as ε→0 we conclude that hΣ,∞(ρ)≤hΣ,2(ρ).

Finally we show that hΣ,2(ρ)≤hΣ,2(ρ′), which will establish the proposition as we can interchange the roles of ρ and ρ′.

Let ε>0. Since ρ is dynamically generating, we can find a finite set KG and an ε′>0 such that, for all x,yX, if \(\rho (sx,sy) < \sqrt{3\varepsilon'}\) for all sK then \(\rho' (x,y) < \varepsilon/\sqrt{2}\). By shrinking ε′ if necessary we may assume that 3ε′|K|<ε2/2. Take a finite set FG containing K and a δ>0 with δε′ such that \(h_{\varSigma,2}^{\varepsilon'} (\rho,F,\delta) \leq h_{\varSigma ,2}^{\varepsilon'} (\rho) + \varepsilon\). Since ρ′ is dynamically generating, by Lemma 2.3 there are a nonempty finite set F′⊆G and a δ′>0 such that for any d∈ℕ and sufficiently good sofic approximation σ:G→Sym(d) we have Map(ρ′,F′,δ′,σ)⊆Map(ρ,F,δ,σ).

Given φ,ψ∈Map(ρ′,F′,δ′,σ) such that ρ2(φ,ψ)<ε′, for each sK we have, writing αs for the transformation xsx of X, This implies that there is a set W⊆{1,…,d} of cardinality at least (1−3ε′|K|)d such that for all aW we have \(\rho(s\varphi(a),s\psi(a)) < \sqrt{3\varepsilon'}\) for every sK and hence \(\rho' (\varphi(a),\psi(a)) < \varepsilon/\sqrt{2}\). As a consequence, assuming (as we may by normalizing) that X has ρ′-diameter at most one,
$$\rho'_2 (\varphi, \psi) \leq\sqrt{(\varepsilon/ \sqrt{2})^2 + 3\varepsilon' |K|} < \varepsilon. $$
It follows that
$$N_\varepsilon\bigl(\mathrm{Map}\bigl(\rho' ,F', \delta' ,\sigma\bigr),\rho'_2 \bigr) \leq N_{\varepsilon'} \bigl(\mathrm{Map}(\rho,F,\delta,\sigma),\rho_2 \bigr) $$
and hence \(h_{\varSigma,2}^{\varepsilon}(\rho' ,F',\delta' ) \leq h_{\varSigma ,2}^{\varepsilon'} (\rho,F,\delta)\), so that Since ε was an arbitrary positive number we conclude that hΣ,2(ρ′)≤hΣ,2(ρ). □

Definition 2.5

The topological entropyhΣ(X,G) of the action GX with respect to Σ is defined to be the common value in Proposition 2.4 over all dynamically generating continuous pseudometrics on X.

Note that the approximate multiplicativity of a sofic approximation was only needed in the proof of Lemma 2.3 to handle the situation in which one of ρ and ρ′ is not an actual metric. Indeed we could have defined topological entropy more easily by using the obvious fact that hΣ,∞(ρ) takes a common value over all compatible metrics on X, with Proposition 2.4 then being regarded as a Kolmogorov-Sinai theorem. As with the (n,ε)-separated set definition of topological entropy for single transformations, it is by considering pseudometrics that we can compute the entropy for a nontrivial example like the shift action G↷{1,…,k}G. In this case one can see that the value is logk independently of Σ by considering the pseudometric ρ on {1,…,k}G given by ρ(x,y)=0 or 1 depending on whether or not the coordinates of x and y at e agree. Indeed logk is easily seen to be an upper bound, and given a nonempty finite set FG, a δ>0, and a good enough sofic approximation σ:G→Sym(d) we can construct a (ρ,1/2)-separated subset of Map(ρ,F,δ,σ) of cardinality kd by associating to every ω∈{1,…,k}d some φω∈Map(ρ,F,δ,σ) defined by φω(a)(s−1)=ω(σs(a)) for all a∈{1,…,d} and sG.

For actions of amenable G, the entropy hΣ(X,G) coincides with the classical topological entropy for every Σ [23]. Such an action always has a largest zero-entropy factor (i.e., a zero-entropy factor such that every zero-entropy factor factors through it), called the topological Pinsker factor [4]. More generally for sofic G, with respect to a fixed Σ there exists a largest factor of the action GX which has entropy either 0 or −∞ (note that the value −∞ does not occur for actions of amenable G). This follows from the fact that the property of having entropy 0 or −∞ is preserved under taking countable products and restricting to closed invariant sets. (The property of having entropy −∞ is also preserved under taking countable products, though we do not know what happens to the property of having entropy 0.) We say that the action has completely positive entropy with respect toΣ if each of its nontrivial factors has positive entropy with respect to Σ.

Unlike in the amenable case, the largest factor with entropy 0 or −∞ might have factors with positive entropy. In fact for every nonamenable G there exist zero-entropy actions of G which have factors with positive entropy: Take an action GX with hΣ(X,G)>0 and an action GY which has no G-invariant Borel probability measure, and consider the action of G on K:=(X×Y)∐{z} where z is a point on which G acts trivially. Then hΣ(K,G)=0 but the quotient action on X∐{z} satisfies hΣ(X∐{z},G)>0.

3 Orbit IE-tuples

Let GX be a continuous action of a discrete group on a compact Hausdorff space. Recall from the introduction that if A=(A1,…,Ak) is a tuple of subsets of X then we say that a subset F of G is an independence set forA if for every finite subset J of F and every function ω:J→{1,…,k} we have ⋂sJs−1Aω(s)≠∅.

Definition 3.1

Let A=(A1,…,Ak) be a tuple of subsets of X. We define the independence density ofA (overG) to be the largest q≥0 such that every finite set FG has a subset of cardinality at least q|F| which is an independence set for A.

Definition 3.2

We say that a tuple x=(x1,…,xk)∈Xk is an orbit IE-tuple (or orbit IE-pair in the case k=2) if for every product neighborhood U1×…×Uk of x the tuple (U1,…,Uk) has positive independence density. Write IEk(X,G) for the set of all orbit IE-tuples of length k.

As Theorem 4.8 below demonstrates, the notation IEk(X,G) is consistent with its use in [21] when G is amenable.

The equality in the next theorem statement is understood with respect to the identification of ((x1,…,xk),(y1,…,yk))∈Xk×Yk and ((x1,y1),…,(xk,yk))∈(X×Y)k.

Theorem 3.3

LetGXandGYbe continuous actions on compact Hausdorff spaces. Letk∈ℕ. Then
$$\mathrm{IE}_k (X\times Y,G)= \mathrm{IE}_k (X,G) \times \mathrm{IE}_k (Y,G). $$

Proof

The inclusion IEk(X×Y,G)⊆IEk(X,G)×IEk(Y,G) is trivial. To prove the other direction, it suffices to show that if A=(A1,…,Ak) is a tuple of subsets of X with independence density q and B=(B1,…,Bk) is a tuple of subsets of Y with independence density r, then A×B:=(A1×B1,…,Ak×Bk) has independence density at least qr. Let F be a nonempty finite subset of G. Then we can find a JF with |J|≥q|F| which is an independence set for A. We can then find a J1J with |J1|≥r|J| which as an independence set for B. Then J1 is an independence set for A×B and |J1|≥qr|F|. □

In [21] we defined a tuple x=(x1,…,xk)∈Xk to be an IN-tuple if for every product neighborhood U1×…×Uk of x the tuple (U1,…,Uk) has arbitrarily large finite independence sets. The following fact is obvious.

Proposition 3.4

Suppose thatGis infinite. Then every orbit IE-tuple is an IN-tuple.

We will strengthen this assertion in Theorem 7.1.

4 Σ-IE-tuples

Unless otherwise stated, throughout this section G is a countable sofic discrete group, subject to further hypotheses as appropriate. We suppose G to be acting continuously on a compact metrizable space X, and ρ denotes a dynamically generating continuous pseudometric on X unless otherwise stated.

In order to be able to define the notion of a sofic IE-tuple as appears in Proposition 4.6, we will set up our definitions for a general sofic approximation net Σ={σi:G→Sym(di)}, which is formally defined in the same way as the sequential version.

Definition 4.1

Let A=(A1,…,Ak) be a tuple of subsets of X. Let F be a nonempty finite subset of G and δ>0. Let σ be a map from G to Sym(d) for some d∈ℕ. We say that a set \({\mathcal{J}}\subseteq\{ 1,\dots,d\}\) is a (ρ,F,δ,σ)-independence set forA if for every function \(\omega: {\mathcal{J}}\to\{ 1,\dots,k \}\) there exists a φ∈Map(ρ,F,δ,σ) such that φ(a)∈Aω(a) for every \(a\in{\mathcal{J}}\).

Definition 4.2

Let A=(A1,…,Ak) be a tuple of subsets of X. Let Σ={σi:G→Sym(di)} be a sofic approximation net for G. We say that A has positive upper independence density overΣ if there exists a q>0 such that for every nonempty finite set FG and δ>0 there is a cofinal set of i for which A has a (ρ,F,δ,σi)-independence set of cardinality at least qdi. By Lemma 2.3 this definition does not depend on the choice of ρ.

For the purposes of Sects. 5 and 8, we will consider a variation of the above definition in which cofinality is replaced by the stronger requirement of membership in a fixed free ultrafilter \({\mathfrak{F}}\) on ℕ. The resulting notion of positive upper independence density over Σ with respect to \({\mathfrak{F}}\) will then be used when interpreting the following definition of Σ-IE-tuples.

By the universal sofic approximation net for G we mean the net (σ,F)↦σ indexed by the directed set of pairs (σ,F) where σ is a map from G to Sym(d) for some d∈ℕ and F is a finite subset of G, and (σ′:G→Sym(d′),F′)≻(σ:G→Sym(d),F) means that d′≥d and |V(σ′,F)|/d′≥|V(σ,F)|/d, where V(ω,F) for a map ω:G→Sym(c) denotes the set of all a∈{1,…,c} such that σ(s)σ(t)a=σ(st)a for all s,tF and σ(s)aσ(t)a for all distinct s,tF.

Definition 4.3

Let Σ={σi:G→Sym(di)} be a sofic approximation net for G. We say that a tuple x=(x1,…,xk)∈Xk is a Σ-IE-tuple (or Σ-IE-pair in the case k=2) if for every product neighborhood U1×…×Uk of x the tuple (U1,…,Uk) has positive upper independence density over Σ. We say that x is a sofic IE-tuple (or sofic IE-pair in the case k=2) if it is a Σ-IE-tuple for the universal sofic approximation net Σ. We denote the Σ-IE-tuples of length k by \(\mathrm{IE}^{\varSigma}_{k} (X,G)\) and the sofic IE-tuples of length k by \(\mathrm{IE}_{k}^{\rm sof} (X,G)\).

Note that \(\mathrm{IE}^{\varSigma}_{k} (X,G) \subseteq\mathrm{IE}_{k}^{\rm sof} (X,G)\) for every sofic approximation net Σ.

We define Σ-IE-tuples and sofic IE-tuples of sets in the same way as for points above.

Remark 4.4

It follows from Lemma 3.3 of [21] that a nondiagonal tuple of points in X is a Σ-IE-tuple if and only if it is a Σ-entropy tuple in the sense of Sect. 5 in [40]. In particular, if in analogy with the amenable case we define the action to have uniformly positive entropy with respect toΣ when every nondiagonal pair in X×X is a Σ-entropy pair, then the action has this property precisely when every pair in X×X is a Σ-IE-pair.

We will need the following consequence of Karpovsky and Milman’s generalization of the Sauer-Shelah lemma [19, 36, 38].

Lemma 4.5

([19])

Givenk≥2 andλ>1 there is a constantc>0 such that, for alln∈ℕ, ifS⊆{1,2,…,k}{1,2,…,n}satisfies |S|≥((k−1)λ)nthen there is anI⊆{1,2,…,n} with |I|≥cnandS|I={1,2,…,k}I.

Proposition 4.6

A sofic IE-tuple is an orbit IE-tuple.

Proof

Fix a compatible metric ρ on X. Let x=(x1,…,xk) be an IE-tuple and U1×…×Uk a product neighborhood of x. We will show that the tuple (U1,…,Uk) has positive independence density over G.

Suppose first that k>1. Take \(1<\lambda<\frac{k}{k-1}\). Then we have the constant c>0 in Lemma 4.5.

Let V1×…×Vk be a product neighborhood of x such that for some κ>0 the κ-neighborhood of Vj is contained in Uj for all 1≤jk. Then there exists a q>0 such that for every nonempty finite subset F of G and δ>0 there is a cofinal set of i for which the tuple (V1,…,Vk) has a (ρ,F,δ,σi)-independence set \({\mathcal{J}}_{i}\) of cardinality at least qdi.

Let F be a nonempty finite subset of G. We will show F has a subset of cardinality at least (cq/2)|F| which is an independence set for the tuple (U1,…,Uk). Let δ be a small positive number to be determined in a moment.

Take an i in the above cofinal set with \(|{\mathcal{W}}_{i}|\ge (1-\delta)d\) for \({\mathcal{W}}_{i}:=\{a\in\{1, \dots, d_{i}\}: F\overset{\sigma_{i}(\cdot )a}{\rightarrow} \sigma_{i}(F)a \text{ is injective}\}\). For each \(\omega: {\mathcal{J}}_{i}\rightarrow\{1, \dots, k\}\), take a φω∈Map(ρ,F,δ,σ) such that φω(a)∈Vω(a) for all \(a\in{\mathcal{J}}_{i}\). Then |{a∈{1,…,di}:ρ(ω(a),φω(sa))≤δ1/2}|≥(1−δ)di for each sF and hence |Λω|≥(1−|F|δ)di for
$$\varLambda_\omega:=\bigl\{a\in\{1, \dots, d_i\}: \rho \bigl(s\varphi_\omega (a), \varphi_\omega(sa)\bigr)\le \delta^{1/2} \text{ for all } s\in F\bigr\}. $$
Set n=|F|. When <1/2, the number of subsets of {1,…,d} of cardinality no greater than nδd is equal to \(\sum_{j=0}^{\lfloor n\delta d \rfloor} \binom{d}{j}\), which is at most \(n\delta d \binom{d}{n\delta d}\), which by Stirling’s approximation is less than exp(βd) for some β>0 depending on δ and n but not on d when d is sufficiently large with β→0 as δ→0 for a fixed n. Thus when δ is small enough and i is large enough, there is a subset Ωi of \(\{1, \dots, k\}^{{\mathcal{J}}_{i}}\) with \((\frac{k}{(k-1)\lambda} )^{q d_{i}}|\varOmega_{i}|\ge k^{|{\mathcal{J}}_{i}|}\) such that the set Λω is the same, say Θi, for every ωΩi, and |Θi|/di>1−|F|δ. Then
$$|\varOmega_i|\ge k^{|{\mathcal{J}}_i|} \biggl(\frac{(k-1)\lambda}{k} \biggr)^{q d_i} \ge k^{|{\mathcal{J}}_i|} \biggl(\frac{(k-1)\lambda}{k} \biggr)^{|{\mathcal{J}} _i|}=\bigl((k-1)\lambda\bigr)^{|{\mathcal{J}}_i|}. $$
By our choice of c, we can find a subset \({\mathcal{J}}'_{i}\) of \({\mathcal{J}}_{i}\) with \(|{\mathcal{J}}'_{i}|\ge c|{\mathcal{J}}_{i}|\ge cq d_{i}\) such that every \(\xi: {\mathcal{J}}'_{i}\rightarrow\{1, \dots, k\}\) extends to some ωΩi.
Writing ζ for the uniform probability measure on {1,…,d}, we have and hence \(\sum_{s\in F}1_{{\mathcal{J}}'_{i}}(sa_{i})\ge(cq-(|F|+1)\delta)|F|\) for some \(a_{i}\in{\mathcal{W}}_{i}\cap\varTheta_{i}\). Then |Ji|≥(cq−(|F|+1)δ)|F| for \(J_{i}:=\{s\in F: sa_{i}\in {\mathcal{J}}'_{i}\}\).

We claim that Ji is an independence set for the tuple (U1,…,Uk) when δ<κ2. Let \(f\in\{1, \dots, k\}^{J_{i}}\). Since \(a_{i}\in{\mathcal{W}}_{i}\), the map \(J_{i}\overset{\sigma_{i}(\cdot )a_{i}}{\rightarrow} \sigma_{i}(J_{i})a_{i}\) is bijective. Thus we can define \(\xi'\in\{1, \dots, k\}^{\sigma(J_{i})a_{i}}\) by ξ′(sai)=f(s) for sJi. Extend ξ′ to some \(\xi\in\{1, \dots, k\}^{{\mathcal{J}}'_{i}}\). Then we can extend ξ to some ωΩi. For every sJi, since \(sa_{i}\in{\mathcal{J}}_{i}\) and aiΘi=Λω, we have \(\varphi_{\omega}(s a_{i})\in V_{\omega(s a_{i})}=V_{f(s)}\) and ρ(ω(ai),φω(sai))≤δ1/2<κ. By the choice of κ we have ω(ai)∈Uf(s). This proves our claim.

Taking δ to be small enough, we have |Ji|≥(cq−(|F|+1)δ)|F|≥(cq/2)|F| as desired.

The case k=1 can be established by a simpler version of the above argument that considers only a single map of the form φω and does not require the invocation of the constant c from Lemma 4.5 and the associated use of Stirling’s approximation. □

For the remainder of this subsection, Σ={σi:G→Sym(di)} is a fixed but arbitrary sofic approximation sequence.

In the case that G is amenable, the independence density I(A) of a tuple A of subsets of X was defined on page 887 of [21] as the limit of φA(F)/|F| as the nonempty finite set FG becomes more and more left invariant, where φA(F) denotes the maximum of the cardinalities of the independence sets for A which are contained in F. A tuple (x1,…,xk)∈Xk is an IE-tuple if for every product neighborhood U1×⋯×Uk of (x1,…,xk) the independence density I(U) of the tuple U=(U1,…,Uk) is positive.

To establish Theorem 4.8 we need the following version of the Rokhlin lemma for sofic approximations, which appears as Lemma 4.6 in [24]. For λ≥0, a collection of subsets of {1,…,d} is said to λ-cover {1,…,d} if its union has cardinality at least λd.

Lemma 4.7

LetGbe a countable amenable discrete group. Let 0≤τ<1 and 0<η<1. LetKbe a nonempty finite subset ofGandδ>0. Then there are an∈ℕ, nonempty finite subsetsF1,…,FofGwith |KFkFk|<δ|Fk| and |FkKFk|<δ|Fk| for allk=1,…,, a finite setFGcontaininge, and anη′>0 such that, for everyd∈ℕ, every mapσ:G→Sym(d) for which there is a setB⊆{1,…,d} satisfying |B|≥(1−η′)dand
$$\sigma_{st}(a)=\sigma_s\sigma_t(a),\qquad \sigma_s(a)\neq\sigma_{s'}(a),\qquad \sigma_e(a)=a $$
for allaBands,t,s′∈Fwithss′, and every setV⊆{1,…,d} with |V|≥(1−τ)d, there existC1,…,CVsuch that
  1. (1)

    for everyk=1,…,, the map (s,c)↦σs(c) fromFk×Cktoσ(Fk)Ckis bijective,

     
  2. (2)

    the family {σ(F1)C1,…,σ(F)C} is disjoint and (1−τη)-covers {1,…,d}.

     

Theorem 4.8

Suppose thatGis amenable. Then IE-tuples, orbit IE-tuples, Σ-IE-tuples, and sofic IE-tuples are all the same thing.

Proof

By Proposition 4.6, sofic IE-tuples and Σ-IE-tuples are orbit IE-tuples. That orbit IE-tuples are IE-tuples is clear in view of the definition of the independence density I(A) of a tuple A of subsets of X. It thus remains to show that IE-tuples are both sofic IE-tuples and Σ-IE-tuples.

To prove that IE-tuples are Σ-IE-tuples, it suffices to demonstrate that, given a tuple U=(U1,…,Uk) of subsets of X with I(U)>0, the tuple U has positive upper independence density over Σ. Set λ=I(U)>0. Let F be a nonempty finite subset of G and δ>0.

Let η>0, to be determined. By Lemma 4.7 we can find an ∈ℕ and nonempty finite sets F1,…,FG such that (i) the sets F1,…,F are sufficiently left invariant so that for each i=1,…, there is a set JiFi which is an independence set for U and has cardinality at least λ|Fi|/2, and (ii) for every good enough sofic approximation σ:G→Sym(d) there exist C1,…,C⊆{1,…,d} satisfying the following:
  1. (1)

    for every i=1,…, and cCi, the map sσs(c) from Fi to σ(Fi)c is bijective,

     
  2. (2)

    the family of sets σ(Fi)c for i=1,…, and cCi is disjoint and (1−η)-covers {1,…,d}.

     
Let σ:G→Sym(d) be a sufficiently good sofic approximation for G for some d∈ℕ. For every \(h = (h_{1},\dots,h_{\ell})\in\prod_{i=1}^{\ell}X^{C_{i}}\) take a map φh:{1,…,d}→X such that
$$\varphi_h (sc) = s\bigl(h_i (c)\bigr) $$
for all i∈{1,…,}, cCi, and sFi. We may assume in our invocation of Lemma 4.7 above that the sets F1,…,F are sufficiently left invariant so that, assuming that η is sufficiently small and σ is a sufficiently good sofic approximation, we have φh∈Map(ρ,F,δ,σ) for every \(h\in\prod_{i=1}^{\ell}X^{C_{i}}\). Write \({\mathcal{J}}\) for the subset \(\bigcup_{i=1}^{\ell}\bigcup_{s\in J_{i}} \bigcup_{c\in C_{i}} \sigma_{s} (c)\) of {1,…,d}. From (1) and (2) we obtain
$$|{\mathcal{J}}|= \sum_{i=1}^\ell|J_i| |C_i|\geq\sum_{i=1}^\ell \frac {\lambda}{2} |F_i| |C_i| \geq\frac{\lambda}{2} (1-\eta)d \geq\frac{\lambda}{4} d $$
assuming that η≤1/2. Now whenever we are given \(\omega_{i} \in\{ 1,\dots,k\}^{J_{i}}\) for i=1,…, we can find, since each Ji is an independence set for U, an \(h = (h_{1},\dots,h_{\ell})\in\prod_{i=1}^{\ell}X^{C_{i}}\) such that \(sh_{i} (c) \in U_{\omega_{i} (s)}\) for all i=1,…, and sJi. The maps φh for such h then witness the fact that \({\mathcal{J}}\) is a (ρ,F,δ,σ)-independence set for U. It follows that U has positive upper independence density over Σ. Hence IE-tuples are Σ-IE-tuples.

The above argument also shows that IE-tuples are sofic IE-tuples, and so we are done.

One can also give the following direct proof that IE-tuples are orbit IE-tuples. It suffices to show that if A=(A1,…,Ak) is a tuple of subsets of X and q>0 is such that for every nonempty finite subset K of G and ε>0 there exist a nonempty finite subset F of G with |KFF|≤ε|F| and a JF with |J|≥q|F| which is an independence set for A, then the independence density of A over G is at least q. Let F1 be a nonempty finite subset of G. Let 1>δ>0. Take ε>0 be a small number which we shall determine in a moment. Then there exist a nonempty finite subset F of G with \(|F_{1}^{-1}F\setminus F|\le\varepsilon|F|\) and a JF with |J|≥q|F| which is an independence set for A. Set \(F'=\{s\in F: F_{1}^{-1}s\subseteq F\}\). Taking ε to be small enough, we have |F′|≥(1−δ)|F|. Note that the function \(\sum_{s\in F}1_{F_{1}s}\) has value |F1| at every point of F′. Thus We also have Therefore we can find an sF with Since F1∩(JF′)s−1 is an independence set for A, we deduce that F1 has a subset of cardinality at least (qδ)|F1| which is an independence set for A. Letting δ→0, we get that F1 has a subset of cardinality at least q|F1| which is an independence set for A. Therefore the independence density I(A) of A is at least q, and so we conclude that IE-tuples are orbit IE-tuples. □

The surprising fact above is that IE-tuples are orbit IE-tuples in the amenable case. It is clear however for a Bernoulli action that all tuples are orbit IE-tuples. Notice also that the argument above works equally well if in the definition of Σ-IE-tuples we use positive upper independence density over Σ with respect to a fixed free ultrafilter \({\mathfrak{F}}\).

Remark 4.9

The product formula for IE-tuples as defined in the amenable framework was established in Theorem 3.15 of [21] using a measure-theoretic argument. We can now combine Theorems 4.8 and 3.3 to obtain a combinatorial proof.

Remark 4.10

The proof of Theorem 4.8 shows that the independence density I(A), as defined on page 887 of [21] and recalled before the theorem statement, coincides with the independence density defined in Definition 3.1. We may thus use the notation I(A) without ambiguity to denote the more general independence density of Definition 3.1.

Remark 4.11

When G is amenable, it is clear from the classical (n,ε)-separated set formulation of topological entropy that the entropy of an action GX is bounded below by the supremum of I(A)logk over all pairs (k,A) where k∈ℕ and A is a k-tuple of pairwise disjoint closed subsets of X. For Bernoulli actions the two quantities are equal. In the nonamenable case, the entropy fails in general to be bounded below by sup(k,A)I(A)logk, where I(A) is as defined in Remark 4.10. Indeed an example of Ornstein and Weiss [33, Appendix C] shows that the Bernoulli action \(F_{2}\curvearrowright\{ 0,1 \}^{F_{2}}\) over the free group on two generators has Bernoulli factors over arbitrarily large finite sets of symbols, in which case the supremum is infinite.

We next aim to establish some basic properties of Σ-IE-tuples in Proposition 4.16.

From Lemma 3.6 of [21] we obtain:

Lemma 4.12

Letk∈ℕ. Then there is a constantc>0 depending only onkwith the following property. LetA=(A1,…,Ak) be ak-tuple of subsets ofXand supposeA1=A1,1A1,2. LetFbe a nonempty finite subset ofGandδ>0. Letσbe a map fromGto Sym(d) for somed∈ℕ. If a setJ⊆{1,…,d} is a (ρ,F,δ,σ)-independence set forA, then there exists anIJsuch that |I|≥c|J| andIis a (ρ,F,δ,σ)-independence set for (A1,1,…,Ak) or (A1,2,…,Ak).

From Lemma 4.12 we get:

Lemma 4.13

LetA=(A1,…,Ak) be ak-tuple of subsets ofXwhich has positive upper independence density overΣ. Suppose thatA1=A1,1A1,2. Then at least one of the tuples (A1,1,…,Ak) and (A1,2,…,Ak) has positive upper independence density overΣ.

Lemma 4.14

hΣ(X,G)>0 if and only if there are disjoint closed subsetsA0andA1ofXsuch that (A0,A1) has positive upper independence density overΣ.

Proof

Let ρ be a compatible metric on X with diamρ(X)≤1. Then hΣ,∞(ρ)=hΣ(X,G). The “if” part is obvious. So assume hΣ,∞(ρ)>0. Then \(h^{6\varepsilon}_{\varSigma, \infty}(\rho)>0\) for some ε>0. Set \(c=h^{6\varepsilon}_{\varSigma, \infty}(\rho)/2\).

Take a finite (ρ,2ε)-dense subset Z of X. Consider on X the continuous pseudometrics ρz, for zZ, and ρ′ given by
$$\rho^z(x, y)=\big|\rho(x, z)-\rho(y, z)\big|, \qquad \rho'(x, y)=\max_{z\in Z}\rho^z(x, y). $$
Note that if ρ(x,y)≥6ε for some x,yX, then ρ′(x,y)≥2ε. It follows that if d∈ℕ and φ and ψ are maps from {1,…,d} to X with ρ(φ,ψ)≥6ε, then \(\rho'_{\infty}(\varphi, \psi)\ge2\varepsilon\).
Take an increasing sequence {Fn}n∈ℕ of nonempty finite subsets of G with union G and a decreasing sequence {δn}n∈ℕ of positive numbers converging to 0. For each n∈ℕ, there is a cofinal set In of i for which one has N6ε(Map(ρ,Fn,δn,σi),ρ)≥exp(cdi). Then \(N_{2\varepsilon}(\mathrm{Map}(\rho, F_{n}, \delta_{n}, \sigma_{i}), \rho'_{\infty})\ge\exp(c d_{i})\) for all iIn. For each iIn and zZ take a \((\rho^{z}_{\infty}, \varepsilon)\)-separated subset Wi,z of Map(ρ,Fn,δn,σi) of maximum cardinality. Then
$$N_{2\varepsilon}\bigl(\mathrm{Map}(\rho, F_n, \delta_n, \sigma_i), \rho'_\infty\bigr) \le\prod _{z\in Z}|W_{i, z}| = \prod _{z\in Z}N_\varepsilon\bigl(\mathrm{Map}(\rho, F_n, \delta_n, \sigma_i), \rho^z_\infty\bigr). $$
Thus \(N_{\varepsilon}(\mathrm{Map}(\rho, F_{n}, \delta_{n}, \sigma_{i}), \rho^{z_{n, i}}_{\infty})\ge\exp(c d_{i}/|Z|)\) for some zn,iZ. Replacing In by a confinal subset if necessary, we may assume that zn,i is the same, say zn, for all iIn. Passing to a subsequence of {(Fn,δn)}n∈ℕ if necessary, we may assume that zn is the same, say \(\mathfrak{z}\), for all n∈ℕ.

Note that if W is a \((\rho^{\mathfrak{z}}_{\infty}, \varepsilon )\)-separated subset of Map(ρ,Fn,δn,σi), then the set \(\{ \rho(\mathfrak{z}, \cdot)\circ\varphi: \varphi\in W\} \) in \(\ell_{\infty}^{d_{i}}\) is (∥⋅∥,ε)-separated. By [15, Lemma 2.3], there are constants c′ and δ>0 depending only on c/|Z| and ε such that for every n∈ℕ and large enough iIn there are a tn,i∈[0,1] and a subset Jn,i of {1,…,di} with |Jn,i|≥cdi so that for every ω:Jn,i→{0,1} there are a φω∈Map(ρ,Fn,δn,σi) such that for all aJn,i we have \(\rho(\mathfrak{z}, \varphi_{\omega}(a))\ge t_{n, i}+\delta\) or \(\rho(\mathfrak{z}, \varphi_{\omega}(a))\le t_{n, i}-\delta\) depending on whether ω(a)=0 or ω(a)=1. Replacing In by a confinal subset if necessary, we may assume that there is a tn∈[0,1] such that |tn,itn|<δ/4 for all iIn. Replacing {Fn,δn}n∈ℕ by a subsequence if necessary, we may assume that there is a t∈[0,1] such that |tnt|<δ/4 for all n∈ℕ. Set \(A_{0}=\{x\in X: \rho(\mathfrak{z}, x)\ge t+\delta/2\}\) and \(A_{1}=\{x\in X: \rho(\mathfrak{z}, z)\le t-\delta/2\}\). Then for every n∈ℕ and iIn, the set Jn,i is a (ρ,Fn,δn,σi)-independence set for (A0,A1). Thus (A0,A1) has positive upper independence density over Σ. □

The following is obvious.

Lemma 4.15

hΣ(X,G)≥0 if and only ifXas a 1-tuple has positive upper independence density overΣ.

Proposition 4.16

The following are true:
  1. (1)

    Let (A1,…,Ak) be a tuple of closed subsets ofXwhich has positive upper independence density overΣ. Then there exists aΣ-IE-tuple (x1,…,xk) withxjAjfor all 1≤jk.

     
  2. (2)

    \(\mathrm{IE}_{1}^{\varSigma}(X, G)\)is nonempty if and only ifhΣ(X,G)≥0.

     
  3. (3)

    \(\mathrm{IE}_{2}^{\varSigma}(X, G)\setminus\Delta_{2}(X)\)is nonempty if and only ifhΣ(X,G)>0, where Δ2(X) denotes the diagonal inX2.

     
  4. (4)

    \(\mathrm{IE}_{k}^{\varSigma}(X, G)\)is a closed subset ofXkwhich is invariant under the product action.

     
  5. (5)

    Letπ:(X,G)→(Y,G) be a factor map. Then\((\pi\times\cdots\times\pi)(\mathrm{IE}_{k}^{\varSigma}(X, G))\subseteq \mathrm{IE}_{k}^{\varSigma}(Y, G)\).

     
  6. (6)

    Suppose thatZis a closedG-invariant subset ofX. Then\(\mathrm{IE}_{k}^{\varSigma}(Z, G )\subseteq\mathrm{IE}_{k}^{\varSigma}(X, G)\).

     

Proof

Assertion (1) follows from Lemma 4.13 and a simple compactness argument. Assertion (2) follows from assertion (1) and Lemma 4.15. Assertion (3) follows directly from assertion (1) and Lemma 4.14. Assertion (4) follows from the observation that, given a compatible metric ρ of X, for any sG, nonempty finite subset F of G, and δ>0 there is a δ′>0 such that, for every d∈ℕ and map σ:G→Sym(d) which is a good enough sofic approximation for G, if φ∈Map(ρ,{s−1}∪(s−1F),δ′,σ), then \(\alpha_{s}\circ\varphi\circ\sigma_{s^{-1}}\in\mathrm{Map}(\rho, F, \delta, \sigma)\), where αs is the transformation xsx of X. Assertions (5) and (6) are trivial. □

Remark 4.17

The inclusion in (5) above is an equality when G is amenable, since Σ-IE-tuples are the same as IE-tuples by Theorem 4.8. Equality can fail however if G is nonamenable: Take an action GX with hΣ(X,G)=−∞ and an action GY with hΣ(Y,G)>0. Then GY has a nondiagonal Σ-IE-pair, while the product action GX×Y, which factors onto GY via the second coordinate projection, satisfies hΣ(X×Y,G)=−∞ and hence has no nondiagonal Σ-IE-pairs.

Remark 4.18

The analogue for orbit IE-tuples of the localization in Proposition 4.16(1) does not hold in the nonamenable case. Indeed for any action GX of a discrete group the 1-tuple X has positive independence density, while the boundary action F2∂F2 of the free group on two generators (where ∂F2 consists of infinite reduced words in the standard generators and their inverses, with the action by left concatenation and reduction) is easily seen not to admit any orbit IE-1-tuples.

From Proposition 4.16(5) we get the following. As in Theorem 3.3, the inclusion below is understood with respect to the identification of ((x1,…,xk),(y1,…,yk))∈Xk×Yk and ((x1,y1),…,(xk,yk))∈(X×Y)k.

Proposition 4.19

\(\mathrm{IE}^{\varSigma}_{k} (X\times Y,G)\subseteq\mathrm{IE}^{\varSigma}_{k} (X,G) \times\mathrm{IE}^{\varSigma}_{k} (Y,G)\).

The problem of the reverse inclusion will be taken up in the next section.

For the remainder of this section X is the unit ball of p(G) for some 1≤p<∞ equipped with the pointwise convergence topology, and the action GX is by left shifts. We will use some of the above results to compute the sofic topological entropy of this action to be zero when G is infinite.

Recall from the end of Sect. 3 that a tuple x=(x1,…,xk)∈Xk is an IN-tuple if for every product neighborhood U1×…×Uk of x the tuple (U1,…,Uk) has arbitrarily large finite independence sets. We write INk(X,G) for the set of IN-tuples of length k.

Lemma 4.20

For everyk∈ℕ the set INk(X,G) consists of the single element (0,…,0).

Proof

Clearly (0,…,0)∈INk(X) for every k∈ℕ. Also note that if x=(x1,…,xk)∈INk(X) then x1,…,xk∈IN1(X). Thus it suffices to show IN1(X)⊆{0}.

Let xX with x≠0. Then (tx)e≠0 for some tG. Set r=|(tx)e|/2>0 and U={yX:|ye|≥r}. Then U is a neighborhood of tx in X. Let FG be a finite independence set for U. Then ⋂sFs−1U is nonempty. Take y∈⋂sFs−1U. Then syU and hence \(|y_{s^{-1}}|=|(sy)_{e}|\ge r\) for every sF. It follows that
$$|F|r^p\le\sum_{s\in F}|y_{s^{-1}}|^p \le\|y\|_p^p\le1, $$
and hence |F|≤rp. Therefore \(tx\not\in\mathrm{IN}_{1}(X)\). Since IN1(X) is G-invariant, \(x\not\in\mathrm{IN}_{1}(X)\). □

Proposition 4.21

Suppose thatGis infinite. ThenhΣ(X,G)=0.

Proof

By Lemma 4.20, Propositions 4.16(2), and Propositions 4.6 and 3.4, we have hΣ(X,G)≤0. Since X has the fixed point 0, we have hΣ(X,G)≥0. Therefore hΣ(X,G)=0. □

5 Product Formula for IE-tuples

In order to hope for a product formula for Σ-IE-tuples beyond the amenable case, we must be able to witness independence density in some uniform way, in analogy with the definition of orbit IE-tuples in Sect. 3 (see Theorem 3.3). This can be achieved by taking a free ultrafilter \({\mathfrak{F}}\) on ℕ and requiring that the independence sets in Definition 4.2 exist for a set of i belonging to \({\mathfrak{F}}\) instead a cofinal set of i. Thus for the purposes of this section we fix a free ultrafilter \({\mathfrak{F}}\) on ℕ and switch to definition of Σ-IE-tuples based on this interpretation of positive density. We will similarly understand sofic topological entropy to be defined by using an ultralimit over \({\mathfrak{F}}\) in Definition 2.5 instead of the limit supremum. We do not know whether our product formula results, Proposition 5.1 and Theorem 5.2, hold for the original definitions.

For the first part of our discussion, up to and including Lemma 5.6, G is a countable sofic group and \(\varSigma=\{\sigma_{i}: G\rightarrow\mathrm{Sym} (d_{i})\}_{i=1}^{\infty}\) a fixed but arbitrary sofic approximation sequence for G.

Proposition 5.1

LetGact continuously on compact metrizable spacesXandY. Then
$$h_\varSigma(X\times Y, G)=h_\varSigma(X, G)+h_\varSigma(Y, G). $$

Proof

Fix compatible metrics ρX and ρY on X and Y respectively. Define a compatible metric ρX×Y on X×Y by
$$\rho^{X\times Y}\bigl((x_1, y_1), (x_2, y_2)\bigr)=\rho^X(x_1, x_2)+ \rho^Y(y_1, y_2) $$
for (x1,y1),(x2,y2)∈X×Y.
Let d∈ℕ. Identify (X×Y){1,…,d} with X{1,…,d}×Y{1,…,d} naturally. Note that for all φ,φ′∈X{1,…,d} and ψ,ψ′∈Y{1,…,d} one has
$$\max \bigl(\rho^X_2\bigl(\varphi, \varphi' \bigr), \rho^Y_2\bigl(\psi, \psi' \bigr) \bigr) \le\rho^{X\times Y}_2\bigl((\varphi, \psi), \bigl( \varphi', \psi'\bigr)\bigr) \le\rho^X_2 \bigl(\varphi, \varphi'\bigr)+\rho^Y_2 \bigl(\psi, \psi'\bigr). $$
Let F be a nonempty finite subset of G, δ>0, and ε>0, and let σ be a map from G to Sym(d). Then Map(ρX,F,δ,σ)×Map(ρY,F,δ,σ)⊆Map(ρX×Y,F,2δ,σ). Furthermore, for every \((\rho^{X}_{2}, \varepsilon)\)-separated subset Open image in new window of Map(ρX,F,δ,σ) and every \((\rho^{Y}_{2}, \varepsilon)\)-separated subset Open image in new window of Map(ρX,F,δ,σ), the set Open image in new window is \((\rho^{X\times Y}_{2}, \varepsilon )\)-separated. It follows that \(h^{\varepsilon}_{\varSigma, 2}(\rho^{X\times Y}, F, 2\delta )\ge h^{\varepsilon}_{\varSigma, 2}(\rho^{X}, F, \delta)+h^{\varepsilon}_{\varSigma, 2}(\rho^{Y}, F, \delta)\), and hence hΣ(X×Y,G)≥hΣ(X,G)+hΣ(Y,G).

Note that for any \((\rho^{X}_{2}, \varepsilon)\)-spanning subset Open image in new window of Map(ρX,F,δ,σ) and any \((\rho^{Y}_{2}, \varepsilon)\)-spanning subset Open image in new window of Map(ρX,F,δ,σ), the set Open image in new window is \((\rho^{X\times Y}_{2}, 2\varepsilon )\)-spanning for (though not necessarily contained in) Map(ρX×Y,F,δ,σ). It follows that N4ε(ρX×Y,F,δ,σ)≤Nε(ρX,F,δ,σNε(ρY,F,δ,σ), and hence \(h^{4\varepsilon}_{\varSigma, 2}(\rho^{X\times Y}, F, \delta )\le h^{\varepsilon}_{\varSigma, 2}(\rho^{X}, F, \delta)+h^{\varepsilon}_{\varSigma , 2}(\rho^{Y}, F, \delta)\). Consequently, hΣ(X×Y,G)≤hΣ(X,G)+hΣ(Y,G). □

The Loeb space and the Loeb measure were introduced by Loeb in [32]. An exposition can be found in [1]. The Loeb space is the ultraproduct space \(\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}\). A subset Y of \(\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}\) is called internal if it is of the form \(\prod_{\mathfrak{F}}Y_{i}\) for a sequence {Yi}i∈ℕ with Yi⊆{1,…,di} for all i∈ℕ. The collection ℑ of inner subsets is an algebra. The Loeb measure is the unique probability measure μ on the σ-algebra \({\mathfrak{B}}\) generated by ℑ such that \(\mu(Y)=\lim_{i\to{\mathfrak{F}}} |{\mathcal{Y}}_{i}|/d_{i}\) for every internal set \(Y=\prod_{\mathfrak{F}}{\mathcal{Y}}_{i}\). For every \(Z\in{\mathfrak{B}}\) there exists a Y∈ℑ such that μ(YΔZ)=0.

For each d∈ℕ, denote by ρHamm the normalized Hamming distance on Sym(d) defined by
$$\rho_{\mathrm{Hamm}}\bigl(\tau, \tau'\bigr)=\frac{1}{d} \big|\bigl\{a\in\{1, \dots, d\}: \tau(a)\neq\tau'(a)\bigr\}\big|. $$

The ultraproduct group \(\prod_{\mathfrak{F}}\mathrm{Sym}(d_{i})\) has a natural action on \(\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}\) preserving μ. One has a bi-invariant pseudometric ρL on \(\prod_{\mathfrak{F}}\mathrm{Sym}(d_{i})\) defined by \(\rho_{L}(\tau, \tau')=\mu(\{y\in\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}: \tau y\neq\tau'y\})\). For any \(\tau=(\tau_{i})_{i}, \tau'=(\tau'_{i})_{i}\in\prod_{\mathfrak{F}}\mathrm{Sym} (d_{i})\) with \(\tau_{i}, \tau'_{i}\in\mathrm{Sym}(d_{i})\) for all i∈ℕ, one has \(\rho_{L}(\tau, \tau')= \lim_{i\to{\mathfrak{F}}}\rho_{\mathrm{Hamm}}(\tau_{i}, \tau'_{i})\). Denote by \({\mathfrak{G}}\) the quotient group of \(\prod_{\mathfrak{F}}\mathrm{Sym}(d_{i})\) by ρL. Then we may think of \({\mathfrak{G}}\) as acting on \(\prod_{\mathfrak{F}}\{1, \dots, d_{i}\} \) by μ-preserving transformations.

The sofic approximation sequence Σ gives rise to a natural group embedding of G into \({\mathfrak{G}}\). Thus we may think of G as a subgroup of \({\mathfrak{G}}\). Denote by G′ the subgroup of \({\mathfrak{G}}\) consisting of elements commuting with G.

As before, the equality below is understood with respect to the identification of ((x1,…,xk),(y1,…,yk))∈Xk×Yk and ((x1,y1),…,(xk,yk))∈(X×Y)k.

Theorem 5.2

Suppose that the action ofGon\((\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}, {\mathfrak{B}} , \mu)\)is ergodic. LetGact continuously on compact metrizable spacesXandY. Letk∈ℕ. Then
$$\mathrm{IE}^\varSigma_k (X\times Y,G)=\mathrm{IE}^\varSigma_k (X,G) \times\mathrm{IE}^\varSigma_k (Y,G). $$

We prove the theorem by way of the following results.

Definition 5.3

Let G act continuously on a compact metrizable space X. Let ρ be a dynamically generating continuous pseudometric on X. Let A=(A1,…,Ak) be a tuple of subsets of X. We say that an internal set \(Y=\prod_{\mathfrak{F}}Y_{i}\) with Yi⊆{1,…,di} for all i∈ℕ is an independence set for A if for every nonempty finite subset F of G and every δ>0 the set of all i∈ℕ for which Yi is a (ρ,F,δ,σi)-independence set for A belongs to \({\mathfrak{F}}\).

From Lemma 2.3 it is easy to see that Definition 5.3 does not depend on the choice of ρ.

Consistent with our interpretation of the equality in Theorem 5.2, in Proposition 5.4 and Lemma 5.6 we understand A×B to mean (A1×B1,…,Ak×Bk) where A=(A1,…,Ak) and B=(B1,…,Bk).

Proposition 5.4

LetGact continuously on compact metrizable spacesXandY. LetAandBbek-tuples of subsets ofXandYrespectively for somek∈ℕ. Then the following hold:
  1. (1)

    Ahas positive upper independence density overΣif and only ifAhas an internal independence setZwithμ(Z)>0.

     
  2. (2)

    The set of internal independence sets forAisG′-invariant.

     
  3. (3)

    An internal set is an independence set forA×Bif and only if it is an independence set for bothAandB.

     

Proof

Fix a compatible metric ρ on X which gives X diameter at most 1.

(1) The “if” part is obvious. Suppose that A has positive upper independence density over Σ. Let q>0 be as in Definition 4.2. Let {Fn}n∈ℕ be an increasing sequence of finite subsets of G with ⋃n∈ℕFn=G. For each n∈ℕ, denote by \(W'_{n}\) the set of all i∈ℕ for which there is a (ρX,Fn,1/n,σi)-independence set Zi for A with ζ(Zi)≥q. Also set \(W_{n}=W'_{n}\setminus\{1, \dots, n-1\}\). Then \(W'_{n}\in{\mathfrak{F}}\) by our assumption and hence \(W_{n}\in {\mathfrak{F}}\) for each n∈ℕ. Note that the sequence {Wn}n∈ℕ is decreasing, and ⋂n∈ℕWn=∅.

We define an internal set \(Z=\prod_{\mathfrak{F}}{\mathcal{Z}}_{i}\) as follows. If i∈ℕ∖W1, we take any \({\mathcal{Z}}_{i} \subseteq\{1, \dots, d_{i}\}\). If iWnWn+1 for some n∈ℕ, we take \({\mathcal{Z}}_{i}\) to be a (ρ,Fn,1/n,σi)-independence set for A with \(|{\mathcal{Z}}_{i}|/d_{i}\ge q\). Then \({\mathcal{Z}}_{i}\) is a (ρ,Fn,1/n,σi)-independence set for A for all n∈ℕ and iWn. Thus Z is an internal independence set for A. As \(|{\mathcal{Z}}_{i}|/d_{i}\ge q\) for all iW1, we have \(\mu (Z)=\lim_{i\to {\mathfrak{F}}}\zeta(Z_{i})\ge q\). This proves the “only if” part.

(2) Let \(Z=\prod_{\mathfrak{F}}{\mathcal{Z}}_{i}\) be an internal independence set for A, and let τ=(τi)iG′. Then \(\tau Z=\prod_{\mathfrak{F}}\tau_{i}{\mathcal{Z}}_{i}\). Let F be a nonempty finite subset of G and 1>δ>0. Then the set W of all i∈ℕ for which \({\mathcal{Z}}_{i}\) is a (ρ,F,δ,σi)-independence set for A is in \({\mathfrak{F}}\). Since τG′, the set V of all i∈ℕ for which \(\max_{s\in F}\rho_{\mathrm{Hamm}}(\tau^{-1}_{i} \sigma_{i,s}, \sigma_{i,s}\tau^{-1}_{i}) \le\delta^{2}\) is also in \({\mathfrak{F}}\). Then VW is in \({\mathfrak{F}}\). For every iV, φ∈Map(ρ,F,δ,σi), and sF one has where αs is the transformation xsx of X, and hence \(\varphi\circ\tau^{-1}_{i}\in\mathrm{Map}(\rho, F, 2\delta, \sigma_{i})\). It follows that for every iVW the set \(\tau_{i} {\mathcal{Z}}_{i}\) is a (ρ,F,2δ,σi)-independence set for A. Therefore τZ is an internal independence set for A.

(3) This can be proved using arguments similar to the proof of Proposition 5.1. □

Lemma 5.5

Suppose thatΓis a subgroup of\({\mathfrak{G}}\)and the action ofΓon\((\prod_{\mathfrak{F}}\{1, \dots,\allowbreak d_{i}\}, {\mathfrak{B}}, \mu)\)is ergodic. Let\(Y, Z\in{\mathfrak{B}}\)be such thatμ(Y),μ(Z)>0. Thenμ(YτZ)>0 for someτΓ.

Proof

Set r=supFμ(⋃τFτZ) with F ranging over the nonempty countable subsets of Γ. Then we can find nonempty finite subsets F1,F2,… of Γ such that \(r=\lim_{n\to\infty}\mu (\bigcup_{\tau\in F_{n}} \tau Z)\). Set W=⋃n∈ℕFn and Z′=⋃τWτZ. Then W is a countable subset of Γ and r=μ(Z′). For every τ′∈Γ we have μ(ZτZ)=μ(⋃τWτWτZ)≤r and hence μ(τZZ)=0. Since the action of Γ on \((\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}, {\mathfrak{B}}, \mu)\) is ergodic, we conclude that r=1. Thus μ(Y)=μ(YZ′)≤∑τWμ(YτZ), and hence μ(YτZ)>0 for some τW. □

Lemma 5.6

Suppose that the action ofGon\((\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}, {\mathfrak{B}} , \mu)\)is ergodic. LetGact continuously on compact metrizable spacesXandY. Letk∈ℕ and letAandBbek-tuples of subsets ofXandY, respectively. Suppose that bothAandBhave positive upper independence density overΣ. ThenA×Balso has positive upper independence density overΣ.

Proof

This follows from Proposition 5.4 and Lemma 5.5. □

Theorem 5.2 now follows from Proposition 4.19 and Lemma 5.6.

The remainder of this section is devoted to the problem of when the ergodicity hypothesis in Theorem 5.2 is satisfied. We prove that this happens when G is residually finite and Σ arises from finite quotients of G, and also when G is amenable and Σ is arbitrary. A combination of results of Elek and Szabo [12, Theorem 2] and Paunescu [35] shows on the other hand that if G is nonamenable then there is always a sofic approximation sequence Σ for which the commutant G′ does not act ergodically.

Let G be an infinite residually finite group, and let {Gi}i∈ℕ be a sequence of finite-index normal subgroups of G such that ⋂n∈ℕinGi={e}. Then we have the sofic approximation sequence Σ={σi:G→Sym(|G/Gi|)} by identifying {1,…,|G/Gi|} with G/Gi and setting σi(s)(tGi)=stGi for s,tG.

Theorem 5.7

Under the above hypotheses, the action ofGon\((\prod_{\mathfrak{F}}\{1, \dots, |G/G_{i}|\},\allowbreak {\mathfrak{B}}, \mu)\)is ergodic.

Proof

Consider the right multiplication action σ′ of G/Gi on itself given by \(\sigma'_{i}(sG_{i})(tG_{i})=ts^{-1}G_{i}\) for s,tG. Since this commutes with σi, it suffices to show that the action of \(\prod_{\mathfrak{F}}\sigma'_{i}(G/G_{i})\subseteq G'\) on \((\prod_{\mathfrak{F}}\{1, \dots, |G/G_{i}|\}, {\mathfrak{B}}, \mu)\) is ergodic.

Let \({\mathcal{Y}}_{i}\subseteq G/G_{i}\). Then, using the 1-norm with respect to the uniform probability measure on G/Gi, Thus there is some siGiG/Gi with \(\frac{1}{|G/G_{i}|}|\sigma'_{i}(s_{i}G_{i}){\mathcal{Y}}_{i}\Delta{\mathcal{Y}}_{i}|\ge2\frac {|{\mathcal{Y}}_{i}|}{|G/G_{i}|} (1-\frac{|{\mathcal{Y}}_{i}|}{|G/G_{i}|} )\).
Let \(Y=\prod_{\mathfrak{F}}{\mathcal{Y}}_{i}\) be an internal subset of \(\prod_{\mathfrak{F}} \{1, \dots, |G/G_{n}|\}\). Take siGiG/Gi as above for each i∈ℕ. Set s=(siGi)i. Then If μ(σ′(s)YΔY)=0, then μ(Y)=0 or 1. This finishes the proof. □

Theorem 5.8

LetGbe a countable amenable group. For every sofic approximation sequenceΣforG, the action ofGon\((\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}, {\mathfrak{B}}, \mu)\)is ergodic.

The proof of Theorem 5.8 requires several lemmas.

We will use the following terminology. Let (X,μ) be a finite measure space and let δ≥0. A family of measurable subsets of X is said to δ-coverX if its union has measure at least δμ(X). A collection {Ai}iI of positive measure sets is δ-disjoint if there exist pairwise disjoint sets \(\widehat{A}_{i} \subseteq A_{i}\) such that \(\mu(\widehat{A}_{i} ) \geq(1-\delta) \mu(A_{i} )\) for all iI, and is a δ-even covering of X if there exists a number M>0 such that \(\sum_{i\in I}\mathbf{1}_{A_{i}}\le M\) and ∑iIμ(Ai)≥(1−δ)(X).

The following is the Rokhlin lemma for sofic approximations, which is based on the quasitiling theory of Ornstein and Weiss and appears as Lemma 4.5 in [24]. The statement of the latter does not contain condition (3) below, but it is not hard to see from the proof in [24] that it can be arranged.

Lemma 5.9

LetGbe a countable discrete group. Let 0≤θ<1, and 0<η<1. Then there are an′∈ℕ andκ,η″>0 such that, whenevereF1F2⊆⋯⊆Fare finite subsets ofGwith\(|(F_{k-1}^{-1}F_{k}) \setminus F_{k}|\le\kappa|F_{k}|\)fork=2,…,′, there existλ1,…,λ∈[0,1] such that for everyδ>0, every sufficiently larged∈ℕ (depending onδ), every mapσ:G→Sym(d) with a set\({\mathcal{B}}\subseteq\{1, \dots, d\}\)satisfying\(|{\mathcal{B}}|\ge(1-\eta'')d\)and
$$\sigma_{st}(a)=\sigma_s\sigma_t(a), \qquad \sigma_s(a)\neq\sigma_{s'}(a),\qquad \sigma_e(a)=a $$
for all\(a\in{\mathcal{B}}\)and\(s, t, s'\in F_{\ell'}\cup F_{\ell '}^{-1}\)withss′, and every set\({\mathcal{V}}\subseteq\{1, \dots, d\} \)with\(|{\mathcal{V}}|\ge(1-\theta)d\), there exist\({\mathcal{C}}_{1}, \dots, {\mathcal{C}}_{\ell'}\subseteq {\mathcal{V}}\)such that
  1. (1)

    for everyk=1,…,and\(c\in{\mathcal{C}}_{k}\), the mapsσs(c) fromFktoσ(Fk)cis bijective,

     
  2. (2)

    the sets\(\sigma(F_{1}){\mathcal{C}}_{1}, \dots, \sigma(F_{\ell '}){\mathcal{C}}_{\ell'}\)are pairwise disjoint and the family\(\bigcup_{k=1}^{\ell'}\{\sigma(F_{k})c: c\in{\mathcal{C}}_{k}\}\)isη-disjoint and (1−θη)-covers {1,…,d},

     
  3. (3)

    \(\sum_{k=1}^{\ell'}||\sigma(F_{k}){\mathcal{C}}_{k}|/d-\lambda_{k}|<\delta\).

     

Lemma 5.10

LetGbe a countable discrete group. LetFbe a nonempty finite subset ofG. For everyd∈ℕ, every mapσ:G→Sym(d), every set\({\mathcal{B}}\subseteq\{1, \dots, d\}\)satisfying
$$\sigma_s(a)\neq\sigma_t(a) $$
for all\(a\in{\mathcal{B}}\)and distincts,tF, every\({\mathcal{J}} \subseteq\{1, \dots, d\}\), and every 0<λ<1, there exists a\({\mathcal{V}}\subseteq{\mathcal{B}}\)such that\(|{\mathcal{V}}|\ge\frac {|{\mathcal{B}}|(1-\lambda)-d+|{\mathcal{J}}| }{1-\lambda}\)and\(|\sigma(F)a\cap{\mathcal{J}}|>\lambda|F| \)for all\(a\in{\mathcal{V}}\).

Proof

Set X={1,…,d}. Denote by ζ the uniform probability measure on X. One has Set \({\mathcal{V}}=\{a\in{\mathcal{B}}: |\sigma(F)a\cap{\mathcal{J}}|> \lambda|F|\}\). Then Thus
$$\frac{|{\mathcal{V}}|\cdot|F|}{d}+\frac{(|{\mathcal{B}}|-|{\mathcal{V}}|)\lambda |F|}{d}\ge\frac{|{\mathcal{B}}|\cdot|F|}{d}- \biggl(1- \frac {|{\mathcal{J}} |}{d} \biggr)|F|. $$
It follows that
$$|{\mathcal{V}}|\ge\frac{|{\mathcal{B}}|(1-\lambda)-d+|{\mathcal{J}}| }{1-\lambda}. $$
 □

The proof of Lemma 4.4 in [24] shows the following.

Lemma 5.11

Let (X,μ) be a finite measure space. Letδ,η∈[0,1) and let {Ai}iIbe a finiteδ-even covering ofXby positive measure sets. Then everyη-disjoint subcollection of {Ai}iIcan be enlarged to anη-disjoint subcollection of {Ai}iIwhichη(1−δ)-coversX.

Lemma 5.12

LetGbe a countable discrete group. LetFbe a nonempty finite subset ofG, 0<τ≤1, and 0<η<1/2. Then for every large enoughd∈ℕ, every mapσ:G→Sym(d) with sets\({\mathcal{B}}_{1}, {\mathcal{B}}_{2} \subseteq\{1, \dots, d\}\)satisfying\(|{\mathcal{B}}_{i}|\ge(\frac{\tau}{2}+\frac{2-2\tau}{2-\tau})d\)and
$$\sigma_s(a)\neq\sigma_{t}(a) $$
for all\(a\in{\mathcal{B}}_{i}\)and distincts,tF, and every\({\mathcal{J}}_{1}, {\mathcal{J}}_{2}\subseteq\{1, \dots, d\}\)with\(|{\mathcal{J}}_{i}|\ge \tau d\)fori=1,2, there exist\({\mathcal{C}}_{i}\subseteq{\mathcal{B}}_{i}\)such that
  1. (1)

    for everyi=1,2, the family\(\{\sigma(F)c: c\in{\mathcal{C}}_{i}\}\)isη-disjoint and\(\eta\frac{\tau}{16}\)-covers {1,…,d},

     
  2. (2)

    there is a bijection\(\varphi: {\mathcal{C}}_{1}\rightarrow {\mathcal{C}}_{2}\)such that for any\(c\in{\mathcal{C}}_{1}\), one has\(|\{s\in F: \sigma_{s}(c)\in{\mathcal{J}}_{1}, \sigma_{s}(\varphi(c))\in {\mathcal{J}}_{2}\} |\ge(\frac{\tau}{2})^{2} |F|\).

     

Proof

Note that for all distinct a,c∈{1,…,d} and sF we have
$$\sigma_s(a)\neq\sigma_s(c). $$
Taking λ=τ/2 and \({\mathcal{J}}={\mathcal{J}}_{1}\) in Lemma 5.10, we find a \({\mathcal{V}}_{1}\subseteq{\mathcal{B}}_{1}\) such that \(|{\mathcal{V}}_{1}|/d\ge\frac{|{\mathcal{B}}_{1}|(1-\lambda )/d-1+|{\mathcal{J}}_{1}|/d}{1-\lambda}\ge\frac{(\tau/2+(2-2\tau)/(2-\tau))(1-\lambda )-1+\tau}{1-\lambda}=\frac{\tau}{2}\) and \(|\sigma(F)a\cap{\mathcal{J}}_{1}|/|F|\ge\frac{\tau}{2}\) for all \(a\in {\mathcal{V}}_{1}\). Observe that
$$\sum_{c\in{\mathcal{V}}_1}\big|\sigma(F)c\big|=|F|\cdot|{\mathcal{V}}_1|\ge |F|\cdot\frac {\tau}{2}d=|F|\cdot \biggl(1- \frac{2-\tau}{2} \biggr)d , $$
so that the family \(\{\sigma(F)c\}_{c\in{\mathcal{V}}_{1}}\) is a \(\frac {2-\tau }{2}\)-even covering of {1,…,d} with multiplicity |F|. By Lemma 5.11, we can find a set \({\mathcal{W}}_{1}\subseteq{\mathcal{V}}_{1}\) such that the family \(\{\sigma (F)c\}_{c\in {\mathcal{W}}_{1}}\) is η-disjoint and \(\eta\frac{\tau}{2}\)-covers {1,…,d}. We may assume that \(|\sigma(F){\mathcal{W}}_{1}|< \eta\tau d/2 +|F|\).

List all the subsets of F with cardinality ⌈|F|τ/2⌉ as F1,…,Fn for some n∈ℕ, where ⌈x⌉ for a real number x denotes the smallest integer no less than x. Then we can write \({\mathcal{W}}_{1}\) as the disjoint union of sets \({\mathcal{W}}_{1, j}\) for 1≤jn such that \(\sigma(F_{j})c\subseteq{\mathcal{J}}_{1}\) for all 1≤jn and \(c\in{\mathcal{W}}_{1, j}\). Throwing away those empty \({\mathcal{W}}_{1, j}\), we may assume that each \({\mathcal{W}}_{1, j}\) is nonempty.

For each 1≤jn, taking λ=τ/2 and \({\mathcal{J}}={\mathcal{J}}_{2}\) in Lemma 5.10, we find a \({\mathcal{V}}_{2, j}\subseteq{\mathcal{B}}_{2}\) such that \(|{\mathcal{V}}_{2, j}|/d\ge\frac{|{\mathcal{B}}_{2}|(1-\lambda )/d-1+|{\mathcal{J}}_{2}|/d}{1-\lambda}\ge\frac{(\tau/2+(2-2\tau)/(2-\tau))(1-\lambda )-1+\tau}{1-\lambda}=\frac{\tau}{2}\) and \(|\sigma(F_{j})a\cap{\mathcal{J}}_{2}|/|F_{j}|\ge\frac{\tau}{2}\) for all \(a\in{\mathcal{V}}_{2, j}\).

We will recursively construct pairwise disjoint sets \({\mathcal{C}}_{2, 1}, \dots, {\mathcal{C}}_{2, n}\) such that the family \(\{\sigma(F)c: c\in{\mathcal{C}}_{2, j}, 1\le j\le n\}\) is η-disjoint, and \({\mathcal{C}}_{2, j}\subseteq{\mathcal{V}}_{2, j}\) and \(|{\mathcal{C}}_{2, j}|=\lfloor|{\mathcal{W}}_{1, j}|/2\rfloor\) for every 1≤jn, where ⌊x⌋ for any real number x denotes the largest integer no bigger than x.

Note that
$$\sum_{c\in{\mathcal{V}}_{2, 1}}\big|\sigma(F)c\big|=|F|\cdot|{\mathcal{V}}_{2, 1}|\ge |F|\cdot\frac{\tau}{2}d=|F|\cdot \biggl(1- \frac{2-\tau}{2} \biggr)d , $$
so that the family \(\{\sigma(F)c\}_{c\in{\mathcal{V}}_{2, 1}}\) is a \(\frac {2-\tau}{2}\)-even covering of {1,…,d} with multiplicity |F|. By Lemma 5.11, we can find a set \({\mathcal{W}}_{2, 1}\subseteq{\mathcal{V}}_{2, 1}\) such that the family \(\{\sigma(F)c\}_{c\in{\mathcal{W}}_{2, 1}}\) is η-disjoint and \(\eta\frac{\tau}{2}\)-covers {1,…,d}. Note that
$$|{\mathcal{W}}_{2, 1}|\cdot|F|\ge\big|\sigma(F){\mathcal{W}}_{2, 1}\big| \ge \eta\frac{\tau}{2}d, $$
and since the family \(\{\sigma(F)c\}_{c\in{\mathcal{W}}_{1}}\) is η-disjoint, we have Thus
$$\frac{1}{2}|{\mathcal{W}}_1|\cdot|F|<|{\mathcal{W}}_{2, 1}|\cdot|F|+|F|, $$
and hence
$$|{\mathcal{W}}_{2, 1}|> \frac{1}{2}|{\mathcal{W}}_1|-1 \ge\frac {1}{2}|{\mathcal{W}}_{1, 1}|-1. $$
Therefore we can take a subset \({\mathcal{C}}_{2, 1}\) of \({\mathcal{W}}_{2, 1}\) with cardinality \(\lfloor\frac{1}{2}|{\mathcal{W}}_{1, 1}|\rfloor\).
Suppose that we have found pairwise disjoint sets \({\mathcal{C}}_{2, 1}, \dots, {\mathcal{C}}_{2, k}\) for some 1≤k<n such that the family \(\{\sigma(F)c: c\in{\mathcal{C}}_{2, j}, 1\le j\le k\}\) is η-disjoint, and \({\mathcal{C}}_{2, j}\subseteq{\mathcal{V}}_{2, j}\) and \(|{\mathcal{C}}_{2, j}|=\lfloor|{\mathcal{W}}_{1, j}|/2\rfloor\) for every 1≤jk. Note that so that the family \(\{\sigma(F)c\}_{c\in{\mathcal{V}}_{2, k+1}\cup \bigcup _{1\le j\le k}{\mathcal{C}}_{2, j}}\) is a \(\frac{2-\tau}{2}\)-even covering of {1,…,d} with multiplicity |F|. By Lemma 5.11, we can find a set \({\mathcal{W}}_{2, k+1}\subseteq{\mathcal{V}}_{2, k+1}\setminus\bigcup_{1\le j\le k}{\mathcal{C}}_{2, j}\) such that the family \(\{\sigma(F)c\}_{c\in{\mathcal{W}}_{2, k+1}\cup\bigcup_{1\le j\le k}{\mathcal{C}}_{2, j}}\) is η-disjoint and \(\eta\frac{\tau}{2}\)-covers {1,…,d}. Note that
$$\biggl(|{\mathcal{W}}_{2, k+1}|+\sum_{1\le j\le k}|{ \mathcal{C}}_{2, j}| \biggr)\cdot|F| \ge \bigg|\sigma(F) \biggl({\mathcal{W}}_{2, k+1}\cup\bigcup_{1\le j\le k}{\mathcal{C}}_{2, j} \biggr) \bigg|\ge\eta\frac{\tau}{2}d. $$
Thus, combining with (1), we have and hence
$$|{\mathcal{W}}_{2, k+1}|> \frac{1}{2}|{\mathcal{W}}_{1, k+1}|-1. $$
Therefore we can take a subset \({\mathcal{C}}_{2, k+1}\) of \({\mathcal{W}}_{2, k+1}\) with cardinality \(\lfloor\frac{1}{2}|{\mathcal{W}}_{1, k+1}|\rfloor\). This completes the recursive construction.
For each 1≤jn take a subset \({\mathcal{C}}_{1, j}\) of \({\mathcal{W}}_{1, j}\) with cardinality \(\lfloor|{\mathcal{W}}_{1, j}|/2\rfloor\). Set \({\mathcal{C}}_{i}=\bigcup_{1\le j\le n}{\mathcal{C}}_{i, j}\) for i=1,2. Take a bijection \(\varphi: {\mathcal{C}}_{1}\rightarrow{\mathcal{C}}_{2}\) such that \(\varphi({\mathcal{C}}_{1, j})={\mathcal{C}}_{2, j}\) for all 1≤jn. For each \(c\in{\mathcal{C}}_{1}\), considering j such that \(c\in{\mathcal{C}}_{1, j}\) one has Note that
$$|{\mathcal{W}}_1|\cdot|F|\ge\big|\sigma(F){\mathcal{W}}_1\big|\ge \eta\frac {\tau}{2}d, $$
and hence for i=1,2,
$$|{\mathcal{C}}_i|=\sum_{1\le j\le n}| {\mathcal{C}}_{i, j}|\ge\sum_{1\le j\le n} \biggl( \frac{1}{2}|{\mathcal{W}}_{1, j}|-1 \biggr) \ge\frac{1}{2}|{ \mathcal{W}}_1|-2^{|F|}\ge\frac{1}{4}|{\mathcal{W}}_1| $$
when d is sufficiently large. Since the family \(\{\sigma(F)c: c\in{\mathcal{C}}_{i}\}\) is η-disjoint, we get
$$\big|\sigma(F){\mathcal{C}}_i\big|\ge(1-\eta)|{\mathcal{C}}_i| \cdot|F|\ge (1-\eta)\eta\frac {\tau}{8}d\ge\eta\frac{\tau}{16}d. $$
 □

Lemma 5.13

LetGbe a countable discrete group. Let 0<τ≤1, and 0<η<1/2 with\(\eta\frac{\tau }{16}<\frac{1-\tau'}{24}\), where\(\tau'=\frac{\tau}{2}+\frac{2-2\tau}{2-\tau}<1\). Then there are an∈ℕ andη′>0 such that, whenevereF1F2⊆⋯⊆Fare finite subsets ofGwith\(|(F_{k-1}^{-1}F_{k}) \setminus F_{k}|\le|F_{k}|\)fork=2,…,, for every large enoughd∈ℕ, every mapσ:G→Sym(d) with a set\({\mathcal{B}}\subseteq\{1, \dots, d\}\)satisfying\(|{\mathcal{B}}|\ge(1-\eta')d\)and
$$\sigma_{st}(a)=\sigma_s\sigma_t(a),\qquad \sigma_s(a)\neq\sigma_{s'}(a),\qquad \sigma_e(a)=a $$
for all\(a\in{\mathcal{B}}\)and\(s, t, s'\in F_{\ell}\cup F_{\ell}^{-1}\)withss′, and any\({\mathcal{J}}_{1}, {\mathcal{J}}_{2}\subseteq\{1, \dots, d\}\)with\(|{\mathcal{J}}_{i}|\ge \tau d\)fori=1,2, there exist\({\mathcal{C}}_{i, 1}, \dots, {\mathcal{C}}_{i, \ell }\subseteq{\mathcal{B}}\)such that
  1. (1)

    for everyi=1,2, k=1,…,and\(c\in{\mathcal{C}}_{i, k}\), the mapsσs(c) fromFktoσ(Fk)cis bijective,

     
  2. (2)

    for everyi=1,2, the sets\(\sigma(F_{1}){\mathcal{C}}_{i, 1}, \dots, \sigma(F_{\ell}){\mathcal{C}}_{i, \ell}\)are pairwise disjoint, the family\(\bigcup_{k=1}^{\ell}\{\sigma(F_{k})c: c\in{\mathcal{C}}_{i, k}\}\)isη-disjoint, and\((1-\eta)\frac{1-\tau'}{24}d\le |\bigcup_{k=1}^{\ell}\sigma (F_{k}){\mathcal{C}}_{i, k}|\le(\frac{1}{1-\eta}\frac{1-\tau'}{24}+\eta)d\),

     
  3. (3)

    for everyk=1,…,, there is a bijection\(\varphi_{k}: {\mathcal{C}}_{1, k}\rightarrow{\mathcal{C}}_{2, k}\)such that for each\(c\in{\mathcal{C}}_{1, k}\)one has\(|\{s\in F_{k}: \sigma_{s}(c)\in{\mathcal{J}}_{1}, \sigma_{s}(\varphi_{k}(c))\in{\mathcal{J}}_{2}\}|\ge(\frac{\tau}{2})^{2} |F_{k}|\).

     

Proof

Set \(\eta'=\frac{1-\tau'}{2}\). Take to be the largest integer satisfying \(\ell\eta\frac{\tau}{16}\le\frac{1-\tau'}{12}\). Then \(\ell\eta \frac{\tau}{16}\ge\frac{1-\tau'}{24}\). We will recursively construct sets \({\mathcal{C}}_{i, 1}', \dots, {\mathcal{C}}_{i, \ell }'\) in reverse order so that (i) for every i=1,2 and 1≤k, the sets \(\sigma (F_{k}){\mathcal{C}}_{i, k}', \dots, \sigma(F_{\ell}){\mathcal{C}}_{i, \ell}'\) are pairwise disjoint and the family \(\bigcup_{n=k}^{\ell}\{\sigma (F_{n})c: c\in{\mathcal{C}}'_{i, n}\}\) is η-disjoint and \((\ell-k+1)\eta\frac{\tau}{16}\)-covers {1,…,d}, and (ii) for every k=1,…,, there is a bijection \(\varphi_{k}: {\mathcal{C}}_{1, k}'\rightarrow{\mathcal{C}}_{2, k}'\) such that for every \(c\in{\mathcal{C}}_{1, k}'\) one has \(|\{s\in F_{k}: \sigma_{s}(c)\in{\mathcal{J}}_{1}, \sigma_{s}(\varphi_{k}(c))\in{\mathcal{J}}_{2}\}|\ge(\frac{\tau}{2})^{2} |F_{k}|\).

Taking \({\mathcal{B}}_{i}={\mathcal{B}}\) for i=1,2 in Lemma 5.12 we find \({\mathcal{C}}_{i, \ell}'\subseteq{\mathcal{B}}\) for i=1,2 such that the family \(\{\sigma(F_{\ell})c: c\in{\mathcal{C}}_{i, \ell }'\}\) is η-disjoint and \(\eta\frac{\tau}{16}\)-covers {1,…,d} for i=1,2 and there is a bijection \(\varphi_{\ell}: {\mathcal{C}}_{1, \ell}'\rightarrow{\mathcal{C}}_{2, \ell}'\) with \(|\{s\in F_{\ell}: \sigma_{s}(c)\in{\mathcal{J}}_{1}, \sigma_{s}(\varphi_{\ell}(c))\in{\mathcal{J}}_{2}\}|\ge(\frac{\tau}{2})^{2}|F_{\ell}|\) for all \(c\in{\mathcal{C}}_{1, \ell}'\).

Suppose that 1≤k< and we have found \({\mathcal{C}}_{i, k+1}', \dots, {\mathcal{C}}_{i, \ell}'\subseteq{\mathcal{B}}\) for i=1,2 such that the sets \(\sigma(F_{k+1}){\mathcal{C}}_{i, k+1}', \dots, \sigma(F_{\ell}){\mathcal{C}}_{i,\ell }'\) are pairwise disjoint and the family \(\bigcup_{n=k+1}^{\ell}\{\sigma(F_{n})c: c\in{\mathcal{C}}_{i, n}'\}\) is η-disjoint and \((\ell-k)\eta\frac{\tau}{16}\)-covers {1,…,d} for each i=1,2, and there is a bijection \(\varphi_{j}:{\mathcal{C}}_{1, j}'\rightarrow{\mathcal{C}}_{2, j}'\) with \(|\{s\in F_{j}: \sigma_{s}(c)\in{\mathcal{J}}_{1}, \sigma_{s}(\varphi_{j}(c))\in {\mathcal{J}}_{2}\}|\ge(\frac{\tau}{2})^{2}|F_{j}|\) for all j=k+1,…, and \(c\in{\mathcal{C}}_{1, j}'\). Set \(\theta_{i, k}=|\bigcup_{j=k+1}^{\ell}\sigma(F_{j}){\mathcal{C}}_{i, j}'|/d\) and \({\mathcal{B}}_{i, k}= \{c\in{\mathcal{B}}: \sigma(F_{k})c\cap (\bigcup_{j=k+1}^{\ell}\sigma(F_{j}){\mathcal{C}}_{i, j}' )=\emptyset \} \) for i=1,2.

If 1−τ′−η′<3θm,k for some m=1,2, then we set \({\mathcal{C}}_{i, k}'=\emptyset\) for each i=1,2. Then Since the family \(\bigcup_{j=k}^{\ell}\{\sigma(F_{j})c: c\in{\mathcal{C}}_{i, j}'\}\) is η-disjoint, one has for i=1,2.
Assume that 1−τ′−η′≥3θi,k for every i=1,2. Let i∈{1,2}. For every \(c\in{\mathcal{B}}\setminus{\mathcal{B}}_{i, k}\) we have σs(c)=σt(a) for some j∈{k+1,…,}, \(a\in{\mathcal{C}}_{i, j}'\), tFj, and sFk, and hence
$$c=\sigma_{s^{-1}}\sigma_s(c)=\sigma_{s^{-1}} \sigma_t(a)=\sigma_{s^{-1}t}(a)\in\bigcup _{j=k+1}^\ell\sigma\bigl(F_k^{-1}F_j \bigr){\mathcal{C}}_{i, j}'. $$
Therefore
$${\mathcal{B}}\setminus{\mathcal{B}}_{i, k}\subseteq\bigcup _{j=k+1}^\ell\sigma\bigl(F_k^{-1}F_j \bigr){\mathcal{C}}_{i, j}'. $$
Since the family \(\bigcup_{j=k+1}^{\ell}\{\sigma(F_{j})c: c\in{\mathcal{C}}_{i, j}'\}\) is η-disjoint we have
$$\frac{1}{2}\sum_{j=k+1}^\ell|F_j| \cdot\big|{\mathcal{C}}_{i, j}'\big|\le \sum _{j=k+1}^\ell(1-\eta)|F_j|\cdot\big|{\mathcal{C}}_{i, j}'\big| \le \Bigg|\bigcup_{j=k+1}^\ell \sigma(F_j){\mathcal{C}}_{i,j}' \Bigg|= \theta_{i, k}d. $$
Thus Therefore
Taking \({\mathcal{B}}_{i}={\mathcal{B}}_{i, k}\) in Lemma 5.12, we find \({\mathcal{C}}'_{i, k}\subseteq{\mathcal{B}}_{i, k}\) for i=1,2 such that the family \(\{\sigma(F_{k})c: c\in{\mathcal{C}}_{i, k}'\}\) is η-disjoint and \(\eta\frac{\tau}{16}\)-covers {1,…,d} for i=1,2 and there is a bijection \(\varphi_{k}: {\mathcal{C}}_{1, k}'\rightarrow{\mathcal{C}}_{2, k}'\) with \(|\{s\in F_{k}: \sigma_{s}(c)\in{\mathcal{J}}_{1}, \sigma_{s}(\varphi_{k}(c))\in{\mathcal{J}}_{2}\}|\ge(\frac{\tau}{2})^{2}|F_{k}|\) for all \(c\in{\mathcal{C}}_{1, k}'\). Then for each i=1,2, the sets \(\sigma(F_{k}){\mathcal{C}}_{i, k}', \dots, \sigma(F_{\ell}){\mathcal{C}}_{i, \ell}'\) are pairwise disjoint, the family \(\bigcup_{j=k}^{\ell}\{\sigma(F_{j})c: c\in{\mathcal{C}}_{i, j}'\}\) is η-disjoint, and completing the recursive construction.
When d is large enough, take a subset \({\mathcal{C}}_{1, k}\) of \({\mathcal{C}}_{1, k}'\) for each 1≤k such that
$$\frac{1-\tau'}{24}d\le \Bigg|\bigcup_{k=1}^\ell \sigma (F_k){\mathcal{C}}_{1, k} \Bigg|\le\frac{1-\tau'}{24}d+|F_\ell| \le\frac{1-\tau'}{24}d+\eta(1-\eta)d. $$
Set \({\mathcal{C}}_{2, k}=\varphi_{k}({\mathcal{C}}_{1, k})\) for each 1≤k. Since the families \(\bigcup_{k=1}^{\ell}\{\sigma(F_{k})c: c\in{\mathcal{C}}_{i, k}\}\) for i=1,2 are η-disjoint, we have and  □

We are ready to prove Theorem 5.8.

Proof of Theorem 5.8

It suffices to show that for any internal sets \(Y=\prod_{{\mathfrak{F}} }{\mathcal{Y}}_{n}\) and \(Z=\prod_{{\mathfrak{F}}}{\mathcal{Z}}_{n}\) with strictly positive measure, there is some sG′ with μ(ZsY)>0. In turn it is enough to show that there is some λ>0 such that for every finite subset F of G and ε>0 the set of all n∈ℕ for which there is some φ∈Sym(dn) satisfying ρHamm(φσs,σsφ)<ε for all sF and \(\frac{|\varphi({\mathcal{Y}}_{n})\cap{\mathcal{Z}}_{n}|}{d_{n}}\ge \lambda\) belongs to \({\mathfrak{F}}\).

Set τ=min(μ(Y),μ(Z))/2, \(\tau'=\frac{\tau}{2}+\frac {2-2\tau}{2-\tau}\), and \(\lambda=\frac{\tau^{2}(1-\tau')}{384}\). Take 0<η<1/2 to be a small number with \(\eta\frac{\tau }{16}<\frac{1-\tau'}{24}\), to be determined in a moment. Then the set V of all n∈ℕ satisfying \(\min(|{\mathcal{Y}}_{n}|/d_{n}, |{\mathcal{Z}}_{n}|/d_{n})\ge\tau\) belongs to \({\mathfrak{F}}\). Let and η′ be as in Lemma 5.13. We may assume that η′<ε/2. Set \(\theta=\frac{1}{(1-\eta)^{2}}\frac{1-\tau'}{24}+\frac{\eta }{1-\eta}+\eta'\). Let ′,κ, and η″ be as in Lemma 5.9. Take \(F'_{1}\subseteq F'_{2}\subseteq\dots\subseteq F'_{\ell'}\subseteq F_{1}\subseteq F_{2}\subseteq\dots\subseteq F_{\ell}\) to be finite subsets of G containing e such that
  1. (1)

    \(|((F'_{\ell'})^{-1}F_{k})\setminus F_{k}|<\eta|F_{k}|\) for all k=1,…,,

     
  2. (2)

    \(|((F'_{k-1})^{-1}F'_{k})\setminus F'_{k}|<\kappa|F'_{k}|\) for all k=2,…,′ and \(|\tilde{F}'_{k}|\ge(1-\eta)|F'_{k}|\) for all k=1,…,′, where \(\tilde{F}'_{k}=\{s\in F'_{k}: Fs\subseteq F'_{k}\}\), and

     
  3. (3)

    \(|(F_{k-1}^{-1}F_{k})\setminus F_{k}|<|F_{k}|\) for all k=2,…, and \(|\tilde{F}_{k}|\ge(1-\eta)|F_{k}|\) for all k=1,…,, where \(\tilde{F}_{k}=\{s\in F_{k}: Fs\subseteq F_{k}\}\).

     
Then we have λ1,…,λ as in Lemma 5.9. When nV is large enough, one has \(|{\mathcal{B}}|\ge(1-\min(\eta', \eta''))d_{n}\) where \({\mathcal{B}}\) denotes the set of all a∈{1,…,dn} satisfying
$$\sigma_{n,st}(a)=\sigma_{n, s}\sigma_{n, t}(a), \qquad \sigma_{n, s}(a)\neq\sigma_{n, s'}(a),\qquad \sigma_{n, e}(a)=a $$
for all s,t∈(FF)∪(FF)−1 and distinct \(s, s'\in F_{\ell}\cup F_{\ell}^{-1}\), and one has \({\mathcal{C}}_{i, 1}, \dots, {\mathcal{C}}_{i, \ell}\subseteq{\mathcal{B}}\) for i=1,2 as in Lemma 5.13 for d=dn, σ=σn, \({\mathcal{J}}_{1}={\mathcal{Y}}_{n}\), and \({\mathcal{J}}_{2}={\mathcal{Z}}_{n}\).
Let i∈{1,2}. Set \({\mathcal{V}}_{i}=\{c\in{\mathcal{B}}: \sigma_{n}(F'_{\ell'})c \cap\bigcup_{k=1}^{\ell}\sigma_{n}(F_{k}){\mathcal{C}}_{i, k}=\emptyset\}\). Then \({\mathcal{B}}\setminus{\mathcal{V}}_{i}\subseteq\bigcup_{k=1}^{\ell}\sigma_{n}((F'_{\ell'})^{-1}F_{k}){\mathcal{C}}_{i, k}\). Since the family \(\bigcup_{k=1}^{\ell}\{\sigma_{n}(F_{k})c: c\in{\mathcal{C}}_{i, k}\}\) is η-disjoint, one has and thus
$$|{\mathcal{V}}_i|= |{\mathcal{B}}|-|{\mathcal{B}}\setminus{\mathcal{V}}_i|\ge(1-\theta)d_n. $$
Take δ>0 with 2ηδ+2ηδℓ′+2δℓ′≤η. Taking \({\mathcal{V}}={\mathcal{V}}_{i}\) in Lemma 5.9, when nV is large enough, we find \({\mathcal{C}}_{i, 1}', \dots, {\mathcal{C}}_{i, \ell '}'\subseteq{\mathcal{V}}_{i}\) such that
  1. (1)

    for every k=1,…,′ and \(c\in{\mathcal{C}}_{i, k}'\), the map sσn,s(c) from \(F'_{k}\) to \(\sigma_{n}(F'_{k})c\) is bijective,

     
  2. (2)

    the sets \(\sigma_{n}(F'_{1}){\mathcal{C}}_{i, 1}', \dots, \sigma_{n}(F'_{\ell '}){\mathcal{C}}_{i, \ell'}'\) are pairwise disjoint and the family \(\bigcup_{k=1}^{\ell'}\{\sigma_{n}(F'_{k})c: c\in {\mathcal{C}}_{i, k}'\}\) is η-disjoint and (1−θη)-covers {1,…,d},

     
  3. (3)

    \(\sum_{k=1}^{\ell'}||\sigma_{n}(F'_{k}){\mathcal{C}}_{i, k}'|/d_{n}-\lambda_{k}|<\delta\).

     
Note that
$$\sum_{k=1}^{\ell'}\lambda_k\le \frac{1}{d_n} \bigg|\bigcup_{k=1}\sigma_n \bigl(F'_k\bigr){\mathcal{C}}_{1, k}' \bigg|+\delta\le1+\delta. $$
For each 1≤k′, take \({\mathcal{C}}''_{1, k}\subseteq {\mathcal{C}}'_{1, k}\) and \({\mathcal{C}}''_{2, k}\subseteq{\mathcal{C}}'_{2, k}\) with
$$\big|{\mathcal{C}}''_{1, k}\big|=\big|{\mathcal{C}}''_{2, k}\big|=\min\bigl(\big|{\mathcal{C}}'_{1, k}\big|, \big|{\mathcal{C}}'_{2, k}\big| \bigr). $$
Take a bijection \(\varphi'_{k}: {\mathcal{C}}''_{1, k}\rightarrow {\mathcal{C}}''_{2, k}\). We have and similarly \(|{\mathcal{C}}'_{2, k}|\cdot|F'_{k}|-|{\mathcal{C}}'_{1, k}|\cdot|F'_{k}|\le (2\eta\lambda_{k}+2\eta\delta+2\delta)d_{n}\). Thus for each i=1,2 one has and hence Therefore for i=1,2.

Since the families \(\bigcup_{k=1}^{\ell}\{\sigma_{n}(F_{k})c: c\in {\mathcal{C}}_{i, k}\}\) for i=1,2 are η-disjoint, we can find Fi,cFk with |Fi,c|≥(1−η)|Fk| for all i=1,2, k=1,…,, and \(c\in{\mathcal{C}}_{i, k}\) so that for each i=1,2, the sets σn(Fi,c)c for \(c\in \bigcup_{k=1}^{\ell}{\mathcal{C}}_{i, k}\) are pairwise disjoint. For every k=1,…, and \(c\in{\mathcal{C}}_{1, k}\), set \(\bar {F}_{c}=F_{1, c}\cap F_{2, \varphi_{k}(c)}\) and \(\hat{F}_{c}=\{s\in\bar{F}_{c}: Fs\subseteq\bar{F}_{c}\}\). Then \(|\bar{F}_{c}|\ge(1-2\eta)|F_{k}|\) and \(|\hat{F}_{c}|\ge|\tilde{F}_{k}|-2\eta|F|\cdot|F_{k}|\ge (1-(2|F|+1)\eta)|F_{k}|\).

Similarly, for every 1≤k′ and \(c\in{\mathcal{C}}''_{1, k}\), we find some \(\tilde{F}'_{c}\subseteq F'_{k}\) with \(|\tilde{F}'_{c}|\ge (1-2\eta)|F'_{k}|\) such that the sets \(\sigma_{n}(\tilde{F}'_{c})c\) for \(c\in\bigcup_{k=1}^{\ell}{\mathcal{C}}''_{1, k}\), as well as the sets \(\sigma_{n}(\tilde{F}'_{c})\varphi'_{k}(c)\) for \(c\in\bigcup_{k=1}^{\ell}{\mathcal{C}}''_{1, k}\), are pairwise disjoint. Setting \(\hat{F}'_{c}=\{s\in\bar{F}'_{c}: Fs\subseteq\bar{F}'_{c}\}\), we have \(|\hat{F}'_{c}|\ge(1-(2|F|+1)\eta)|F'_{k}|\).

Note that Similarly, Set \({\mathcal{W}}= (\bigcup_{k=1}^{\ell}\bigcup_{c\in{\mathcal{C}}_{1, k}}\sigma_{n}(\hat{F}_{c})c )\cup (\bigcup_{k=1}^{\ell'}\bigcup_{c\in {\mathcal{C}}''_{1, k}}\sigma_{n}(\hat{F}'_{c})c )\). Then
$$|{\mathcal{W}}|\ge \bigl(1-\bigl(2|F|+1\bigr)\eta \bigr) (1-\eta)\frac{1-\tau'}{24}d_n+ \bigl(1-\bigl(2|F|+1\bigr)\eta \bigr) (1-\theta-4\eta)d_n. $$
Take a φ∈Sym(dn) such that φ(σn,s(c))=σn,s(φk(c)) for all k=1,…,, \(c\in {\mathcal{C}}_{1, k}\), and \(s\in\bar{F}_{c}\), and \(\varphi(\sigma_{n, s}(c))=\sigma_{n, s}(\varphi'_{k}(c))\) for all k=1,…,′, \(c\in{\mathcal{C}}''_{1, k}\), and \(s\in\bar {F}'_{c}\). For every sF, note that σn,sφ=φσn,s on \({\mathcal{W}}\), and hence when η is small enough. We also have when η is small enough. □

Question 5.14

Does every countable sofic group G admit a sofic approximation sequence Σ such that the action of G′ on \((\prod_{\mathfrak{F}}\{1, \dots, d_{i}\}, {\mathfrak{B}}, \mu)\) is ergodic?

6 IE-tuples and Algebraic Actions

By an algebraic action we mean an action of a countable discrete group G on a compact metrizable Abelian group X by (continuous) automorphisms. The structure of such an action is captured by the Pontryagin dual \(\widehat{X}\) viewed as a module over the integral group ring ℤG. The ring ℤG consists of the finitely supported ℤ-valued functions on G, which we write in the form ∑sGfss, with addition (∑sGfss)+(∑sGgss)=∑sG(fs+gs)s and multiplication \((\sum_{s\in G} f_{s} s )(\sum_{s\in G} g_{s} s) = \sum_{s\in G} (\sum_{t\in G} f_{t} g_{t^{-1} s} )s\).

Given a matrix A in Mn(ℤG), the left action of G on (ℤG)n/(ℤG)nA gives rise via Pontryagin duality to the algebraic action \(G\curvearrowright X_{A} := \widehat{({\mathbb{Z}}G)^{n}/({\mathbb{Z}}G)^{n}A}\). Write A for the matrix in Mn(ℤG) whose (i,j) entry is the result of applying the involution (∑sGfss)=∑sGfss−1 to the (j,i) entry of A. Viewing \(\widehat{({\mathbb{Z}}G)^{n}}\) as ((ℝ/ℤ)G)n, we can then identify XA with the closed G-invariant subset
$$\bigl\{ x\in\bigl(({\mathbb{R}}/{\mathbb{Z}})^G \bigr)^n : xA^* = 0_{(({\mathbb{R}}/{\mathbb{Z}} )^G)^n} \bigr\} $$
of ((ℝ/ℤ)G)n equipped with the action of G by left translation. In the case that A is invertible in Mn(1(G)) the action GXA is expansive, and in fact such actions and their restrictions to closed G-invariant subgroups constitute precisely all of the expansive algebraic actions [7, Theorem 3.1]. When G is amenable, given an action of the form GXA with A invertible in Mn(1(G)), every tuple of points in X is an IE-tuple (see Lemma 5.4 and Theorems 7.3 and 7.8 in [7]). We will extend this result in two ways in Theorems 6.6 and 6.7, which demonstrate that in broader contexts independent behavior similarly saturates the structure of actions of the form GXA with A invertible in Mn(1(G)).

First however we examine orbit IE-tuples in the context of actions GX on a compact metrizable (not necessarily Abelian) group by automorphisms. It was shown in [7, Theorem 7.3] that, when G is amenable, the IE-tuples for such an action are determined by a closed G-invariant normal subgroup of X called the IE group. We now proceed to record some observations showing that the basic theory of the IE-group from [7] can be extended from amenable G to general G using orbit IE-tuples. Thus G will be an arbitrary countable discrete group until we turn to the sofic setting in Theorem 6.7.

The proof of Lemma 3.11 in [21] shows the following:

Lemma 6.1

LetGact continuously on a compact metrizable space X. LetAbe a Borel subset of X, andμaG-invariant Borel probability measure on X. ThenAhas independence density at leastμ(A) over G.

From Lemma 6.1 we immediately obtain:

Lemma 6.2

LetGact continuously on a compact metrizable spaceX. Letμbe aG-invariant Borel probability measure onX. Then every point in the support ofμis an orbit IE-1-tuple.

We now suppose that G acts continuously on a compact metrizable group X by automorphisms. From Lemma 6.2 we have:

Lemma 6.3

Every point ofXis an orbit IE-1-tuple.

Denote by IE(X) the set of all xX such that (x,eX) is an orbit IE-pair, where eX is the identity element of X. The proof of Theorem 7.3 in [7] shows the following.

Theorem 6.4

IE(X) is a closedG-invariant normal subgroup ofX. For everyk∈ℕ the set IEk(X,G) of all orbit IE-k-tuples is a closedG-invariant subgroup of the groupXkand

Now we suppose that X is Abelian. In this case a point xX is said to be 1-homoclinic if the function sφ(sx)−1 on G lies in 1(G) for every φ in the Pontryagin dual \(\widehat{X}\). The set of 1-homoclinic points is written Δ1(X). This set was studied in [29, 30, 37] in the case G=ℤd and in [7] for more general G. From the proof of Theorem 7.8 in [7] we obtain the following.

Theorem 6.5

Suppose that\(\widehat{X}\)is a finitely generated leftG-module. Then Δ1(X)⊆IE(X).

From Theorem 6.5 and [7, Lemma 5.4] we obtain:

Theorem 6.6

Letn∈ℕ, and letAbe an element ofMn(ℤG) which is invertible inMn(1(G)). Then for the actionGXAone has IE(XA)=XA.

Now we let G be a countable sofic group and \(\varSigma=\{\sigma_{i}: G\rightarrow\mathrm{Sym}(d_{i})\}_{i=1}^{\infty}\) a sofic approximation sequence for G.

Theorem 6.7

Letn∈ℕ, and letAbe an element ofMn(ℤG) which is invertible inMn(1(G)). Consider the actionGXA. Then, for eachk∈ℕ, everyk-tuple of points inXAis aΣ-IE-tuple.

Before proceeding to the proof of Theorem 6.7, we give an application to a problem of Deninger. For an invertible element f in the group von Neumann algebra \({\mathcal{L}}G\) of a countable discrete group G the Fuglede-Kadison determinant is defined by \(\det_{{\mathcal{L}}G} f = \exp\mathrm{tr}(\log|f|)\) where tr is the canonical trace on \({\mathcal{L}}G\). In [9, Question 26] Deninger asked whether \(\det_{{\mathcal{L}}G} f>1\) whenever f∈ℤG is invertible in 1(G) and has no left inverse in ℤG. An affirmative answer was given by Deninger and Schmidt in the case that G is residually finite and amenable [10, Corollary 6.7] and more generally by Chung and the second author in the case G is amenable [7, Corollary 7.9]. Using Theorem 6.7, Proposition 4.16(3), Theorem 7.1 in [23], and the argument in the proof of Corollary 7.9 in [7], we obtain an answer to Deninger’s question for all countable residually finite groups:

Corollary 6.8

Suppose thatGis residually finite and thatfis an element ofGwhich is invertible in1(G) and has no left inverse inG. Then\(\det_{{\mathcal{L}}G} f>1\).

Let n∈ℕ. For A=(Aij)1≤i,jnMn(1(G)), we set
$$\|A\|_1=\sum_{1\le i, j\le n}\|A_{ij} \|_1. $$
For (a1,…,an)∈ℝd, we set ∥(a1,…,an)∥=max1≤jn|aj|. For ξ:{1,…,d}→ℤn, we set
$$\|\xi\|_\infty=\max_{1\le j\le d}\big\|\xi(j)\big\|_\infty. $$
Denote by P the natural quotient map (ℝn)G→((ℝ/ℤ)n)G. Denote by ρ the canonical metric on ℝ/ℤ defined by
$$\rho(t_1+{\mathbb{Z}}, t_2+{\mathbb{Z}}):= \min_{m\in{\mathbb{Z}}}|t_1-t_2-m|. $$
By abuse of notation, we also use ρ to denote the metric on (ℝ/ℤ)n defined by
$$\rho\bigl((a_1, \dots, a_n), (b_1, \dots, b_n)\bigr):=\max_{1\le j\le n}\rho (a_j, b_j). $$
Via the coordinate map at the identity element of G, we will think of ρ as a continuous pseudometric on ((ℝ/ℤ)n)G.

Lemma 6.9

Letn∈ℕ, and letAbe an element ofMn(ℤG) which is invertible inMn(1(G)). Consider the actionGXA. LetFbe a nonempty finite subset ofGand letM,δ>0. For everyd∈ℕ, good enough sofic approximationσ:G→Sym(d), andξ:{1,…,d}→ℤnwithξM, if we defineh:{1,…,d}→(ℤn)Gandφ:{1,…,d}→XAby
$$\bigl(h(a)\bigr)_{t^{-1}}=\xi(ta) \quad \text{\textit{for all}}\ t\in G $$
and
$$\varphi(a)=P\bigl(\bigl(h(a)\bigr) \bigl(A^*\bigr)^{-1}\bigr) $$
thenφ∈Map(ρ,F,δ,σ).

Proof

Since (A)−1Md(1(G)), there exists a nonempty finite subset K of G such that for all z1,z2∈(ℤn)G such that ∥z1,∥z2M and z1,z2 coincide on K, one has ∥(z1(A)−1)e−(z2(A)−1)e<δ/2, which implies that ρ(P(z1(A)−1),P(z2(A)−1))<δ/2.

Denote by Λ the set of all a∈{1,…,d} satisfying t(sa)=(ts)a for all tK−1 and sF. When σ is a good enough sofic approximation for G, one has |Λ|≥(1−(δ/2)2)d. Let aΛ and sF. Then
$$s\bigl(\varphi(a)\bigr)=P\bigl(\bigl(s\bigl(h(a)\bigr)\bigr) \bigl(A^* \bigr)^{-1}\bigr) $$
and
$$\varphi(sa)=P\bigl(\bigl(h(sa)\bigr) \bigl(A^*\bigr)^{-1}\bigr). $$
For every tK−1 one has
$$\bigl(s\bigl(h(a)\bigr)\bigr)_{t^{-1}}=\bigl(h(a)\bigr)_{s^{-1}t^{-1}}= \xi\bigl((ts)a\bigr)=\xi \bigl(t(sa)\bigr)=\bigl(h(sa)\bigr)_{t^{-1}}. $$
Thus, by the choice of K, we have ρ(s(φ(a)),φ(sa))<δ/2. Note that ((ℝ/ℤ)n)G has diameter 1 under ρ. It follows that
$$\rho_2\bigl(s\varphi(\cdot), \varphi(s\cdot)\bigr)\le\bigl((\delta /2)^2+1-|\varLambda|/d\bigr)^{1/2}<\delta. $$
Therefore φ∈Map(ρ,F,δ,σ). □

We are ready to prove Theorem 6.7.

Proof of Theorem 6.7

Let x=(x1,…,xk) be a k-tuple of points in XA. Then for each 1≤jk there is a zj∈(ℤn)G such that ∥zj≤∥A1 and xj=P(zj(A)−1).

Let U1×⋯×Uk be a product neighborhood of x in Xk. Since the map from bounded subsets of (ℤn)G equipped with the pointwise convergence topology to XA sending z to P(z(A)−1) is continuous [10, Proposition 4.2], there is a nonempty finite subset K of G such that for every 1≤jk and z∈(ℤn)G with ∥z≤∥A1 and z|K=zj|K, one has P(z(A)−1)∈Uj.

Let F be a nonempty finite subset of G and δ>0. Let d∈ℕ and let σ be a map from G to Sym(d). Denote by Λ the set of all a∈{1,…,d} such that sata for all distinct s,tK−1. When σ is a good enough sofic approximation for G, we have |Λ|≥d/2. Let \({\mathcal{J}}\) be a maximal subset of Λ subject to the condition that the sets K−1a for \(a\in{\mathcal{J}}\) are pairwise disjoint. Then \(\varLambda\subseteq(\sigma(K^{-1}))^{-1}\sigma (K^{-1}){\mathcal{J}}\), and hence \(|\varLambda|\le|K|^{2}|{\mathcal{J}}|\). Therefore \(|{\mathcal{J}}|\ge d/(2|K|^{2})\).

We claim that \({\mathcal{J}}\) is a (ρ,F,δ,σ)-independence set for U=(U1,…,Uk) when σ is a good enough sofic approximation for G. Let ω be a map from {1,…,d} to {1,…,k}. Define ξ:{1,…,d}→ℤn by \(\xi(ta)=(z_{\omega(a)})_{t^{-1}}\) for all \(a\in{\mathcal{J}}\) and tK−1, and ξ(b)=0 for all b not in \(K^{-1}{\mathcal{J}}\). Then we have h:{1,…,d}→(ℤn)G and φ∈Map(ρ,F,δ,σ) defined in Lemma 6.9 for M=∥A1 when σ is a good enough sofic approximation. Let \(a\in {\mathcal{J}}\). For any tK−1, one has
$$\bigl(h(a)\bigr)_{t^{-1}}=\xi(ta)=(z_{\omega(a)})_{t^{-1}}. $$
By the choice of K we have φ(a)=P(h(a)(A)−1)∈Uω(a). This proves our claim, and finishes the proof of the theorem. □

7 Orbit IE-tuples and Untameness

Let G be a countably infinite group acting continuously on a compact Hausdorff space X.

Theorem 7.1

Letk∈ℕ and letAbe ak-tuple of subsets ofX. Suppose thatAhas positive independence density overG. ThenAhas an infinite independence set inG.

Proof

Denote by q the density of A over G.

Let F1 be a nonempty finite subset of G. Take s1,s2,… in G such that setting Fn+1=FnFnsn for all n∈ℕ one has FnFnsn=∅ for all n∈ℕ.

Let n∈ℕ. Take an independence set En of A contained in Fn with |En|≥q|Fn|. We will construct, inductively on m, nonempty finite subsets \(F^{(n)}_{m, k}\) and \(E^{(n)}_{m}\) of G for all 1≤mkn and \(t^{(n)}_{m}\in G\) for all 1≤m<n such that
  1. (1)

    \(F^{(n)}_{n, n}=F_{n}\) and \(E^{(n)}_{n}=E_{n}\);

     
  2. (2)

    \(t^{(n)}_{m}\) is equal to either e or \(s_{m}^{-1}\) for each 1≤m<n;

     
  3. (3)

    \(F^{(n)}_{m, k}=F^{(n)}_{m+1, k}t^{(n)}_{m}\) and \(F^{(n)}_{m, m}=F_{m}\) for all 1≤m<kn;

     
  4. (4)

    \(E^{(n)}_{m}=E^{(n)}_{m+1}t^{(n)}_{m}\) for each 1≤m<n;

     
  5. (5)

    \(|E^{(n)}_{m}\cap F^{(n)}_{m, k}|\ge q|F^{(n)}_{m, k}|\) for all 1≤mkn.

     
To start with, we define \(F^{(n)}_{n, n}\) and \(E^{(n)}_{n}\) according to (1). If \(|E^{(n)}_{n}\cap F_{n-1}|\ge q|F_{n-1}|\), we set \(t^{(n)}_{n-1}=e\). Otherwise, since \(|E^{(n)}_{n}\cap F^{(n)}_{n, n}|\ge q|F^{(n)}_{n, n}|\) and \(F^{(n)}_{n, n}=F_{n}\) is the disjoint union of Fn−1 and Fn−1sn−1, we must have \(|E^{(n)}_{n}\cap F_{n-1}s_{n-1}|\ge q|F_{n-1}s_{n-1}|\), and we set \(t^{(n)}_{n-1}=s^{-1}_{n-1}\). Defining \(F^{(n)}_{n-1, k}\) for n−1≤kn and \(E^{(n)}_{n-1}\) according to (3) and (4) respectively, we have that (5) holds for m=n−1. Next, if \(|E^{(n)}_{n-1}\cap F_{n-2}|\ge q|F_{n-2}|\), we set \(t^{(n)}_{n-2}=e\). Otherwise, since \(|E^{(n)}_{n-1}\cap F^{(n)}_{n-1, n-1}|\ge q|F^{(n)}_{n-1, n-1}|\) and \(F^{(n)}_{n-1, n-1}=F_{n-1}\) is the disjoint union of Fn−2 and Fn−2sn−2, we must have \(|E^{(n)}_{n-1}\cap F_{n-2}s_{n-2}|\ge q|F_{n-2}s_{n-2}|\), and we set \(t^{(n)}_{n-2}=s^{-1}_{n-2}\). Defining \(F^{(n)}_{n-2, k}\) for n−2≤kn and \(E^{(n)}_{n-2}\) according to (3) and (4) respectively, we have that (5) holds for m=n−2. Continuing in this way, we define \(F^{(n)}_{m, k}, E^{(n)}_{m}\), and \(t^{(n)}_{m}\) satisfying the above conditions.
Note that if E′ is an independence set for A in G, then Es is an independence set for A in G for all sG. By induction on m, we find easily that \(E^{(n)}_{m}\) is an independence set for A in G for all n∈ℕ and 1≤mn. Also note that for any 1≤mkn, \(F^{(n)}_{m, k}\) depends only on \(F_{k}\cap E^{(n)}_{k}\). In particular, for any fixed k∈ℕ the number of sets appearing in \(F^{(n)}_{1, k}\) for all nk is finite. Thus we can find a strictly increasing sequence n1<n2<⋯ in ℕ such that for any fixed k∈ℕ the sets \(F^{(n_{l})}_{1, k}\) and \(E^{(n_{l})}_{1}\cap F^{(n_{l})}_{1, k}\) do not depend on lk. Set \(E=\bigcup_{k\in{\mathbb{N}}}(E^{(n_{k})}_{1}\cap F^{(n_{k})}_{1, k})\). Since \(|E^{(n_{k})}_{1}\cap F^{(n_{k})}_{1, k}|\ge q|F^{(n_{k})}_{1, k}|=q|F_{k}|=q|F_{1}|2^{k-1}\) for every k∈ℕ, the set E is infinite. For every k∈ℕ one has Thus the sequence \(\{E^{(n_{k})}_{1}\cap F^{(n_{k})}_{1, k}\}_{k\in{\mathbb{N}}}\) is increasing. Since the family of independence sets for A in G is closed under taking increasing unions, we conclude that E is an independence set for A in G. □

Recall that a tuple (x1,…,xk)∈Xk is an IT-tuple if for every product neighborhood U1×⋯×Uk of (x1,…,xk) the tuple (U1,…,Uk) has an infinite independence set [21].

Corollary 7.2

Every orbit IE-tuple of the actionGXis an IT-tuple.

Write C(X) for the Banach space of continuous complex-valued functions on X with the supremum norm. The action GX is said to be tame if no element fC(X) admits an infinite subset J of G such that, for s ranging in J, the family of functions xf(s−1x) in C(X) is equivalent to the standard basis of 1, meaning that there is a bijection between the two which extends to an isomorphism (i.e., a bounded linear map with bounded inverse) between the closures of their linear spans [14, 25]. The action is tame if and only if there is no nondiagonal IT-pair in X×X [21, Proposition 6.4]. Thus from the above corollary we see that a tame action has no nondiagonal orbit IE-tuples.

8 Σ-IE-tuples and Li-Yorke Chaos

Let G be a countably infinite sofic group and \(\varSigma=\{\sigma_{i}: G\rightarrow\mathrm{Sym}(d_{i})\}_{i=1}^{\infty}\) a sofic approximation sequence for G. We fix a free ultrafilter \({\mathfrak{F}}\) on ℕ and use it in the definitions of sofic topological entropy and Σ-IE-tuples, as in Sect. 5.

Let GX be a continuous action on a compact metrizable space. Let ρ be a compatible metric on X. We say that (x,y)∈X×X is a Li-Yorke pair if
$$\limsup_{G \ni s\to\infty}\rho(sx, sy)>0 \quad \text{and} \quad \liminf_{G \ni s\to\infty}\rho(sx, sy)=0, $$
where the limit supremum and limit infimum mean the limits of supsGFρ(sx,sy) and infsGFρ(sx,sy), respectively, over the net of finite subsets F of G. Note that the definition of Li-Yorke pair does not depend on the choice of the metric ρ. We say that the action GX is Li-Yorke chaotic if there is an uncountable subset Z of X such that every nondiagonal pair (x,y) in Z×Z is a Li-Yorke pair. These definitions adapt those for continuous ℕ-actions, which have their origins in [28]. In that setting Blanchard, Glasner, Kolyada, and Maass showed that positive entropy implies Li-Yorke chaos [3]. The following theorem demonstrates that, in our sofic context, positive topological entropy with respect to some sofic approximation sequence implies Li-Yorke chaos (cf. [21, Theorem 3.18]).

Theorem 8.1

Suppose thatk≥2 andx=(x1,…,xk) is aΣ-IE-tuple inXkwithx1,…,xkpairwise distinct. For each 1≤jk, letAjbe a neighborhood ofxj. Then there exist Cantor setsZjAjforj=1,…,ksuch that the following hold:
  1. (1)

    every nonempty finite tuple of points inZ:=⋃jZjis aΣ-IE-tuple;

     
  2. (2)
    for allm∈ℕ, distincty1,…,ymZ, and\(y'_{1}, \dots, y'_{m} \in Z\)one has
    $$\liminf_{G\ni s\to\infty}\max_{1\le i\le m} \rho\bigl(sy_i, y'_i\bigr)=0. $$
     

We now set out to prove Theorem 8.1. We begin with the following lemmas.

Lemma 8.2

Letk≥2 andA=(A1,…,Ak) be a tuple of closed subsets ofXwith positive upper independence density overΣ. For eachj=1,…,kletUjbe an open set containingAj. LetEbe a finite subset ofG. Then there exists ansGEsuch that the tupleAconsisting ofAis−1Ujfor alli,j=1,…,khas positive upper independence density overΣ.

Proof

Take \(1<\lambda<\frac{k}{k-1}\). Then we have the constant c>0 in Lemma 4.5. Take a q>0 such that for every nonempty finite subset F of G and δ>0 the set VF,δ of all i∈ℕ for which A has a (ρ,F,δ,σi)-independence set of cardinality at least qdi is in \({\mathfrak{F}}\). Take a finite subset W of G such that cq|W|>8 and for any distinct s,tW one has \(s^{-1}t\not\in E\). When 0<|W|2κ<1/2, the number of subsets of {1,…,d} of cardinality no greater than |W|2κd is equal to \(\sum^{\lfloor|W|^{2}\kappa d\rfloor}_{j=0}\binom{d}{j}\), which is at most \(|W|^{2}\kappa d \binom{d}{|W|^{2}\kappa d}\), which by Stirling’s approximation is less than exp(βd) for some β>0 depending on κ but not on d when d is sufficiently large with β→0 as κ→0. Take cq/(2|W|2)>κ>0 such that for any 1≤jk and xXUj one has \(\rho(x, A_{j})>\sqrt{\kappa}\) and for all sufficiently large d∈ℕ the number of subsets of {1,…,d} of cardinality no greater than |W|2κd is at most \((\frac {k}{(k-1)\lambda} )^{q d}\).

Let F be a nonempty finite subset of G and δ>0. Set F′=FW and δ′=min(δ,κ). Let i∈ℕ be such that A has a (ρ,F′,δ′,σi)-independence set \({\mathcal{J}}_{i}\) of cardinality at least qdi. For each \(\omega\in\{1, \dots, k\}^{{\mathcal{J}}_{i}}\) take a φω∈Map(ρ,F′,δ′,σi) such that φω(a)∈Aω(a) for every \(a\in{\mathcal{J}}_{i}\). For each \(\omega\in\{1, \dots, k\}^{{\mathcal{J}}_{i}}\), there is some Λω⊆{1,…,di} with |Λω|≥(1−|W|2δ′)di such that \(\rho(\varphi_{\omega}(\sigma_{i}(s)a),s\varphi_{\omega}(a))<\sqrt {\delta'}\) for all sW−1W and aΛω. By the choice of κ, when i is large enough there is a subset Ωi of \(\{1, \dots, k\}^{{\mathcal{J}}_{i}}\) with \((\frac{k}{(k-1)\lambda} )^{q d_{i}}|\varOmega_{i}|\ge k^{|{\mathcal{J}}_{i}|}\) such that the set Λω is the same, say Θi, for every ωΩi, and |Θi|/di≥1−|W|2δ′>1−cq/2. Then
$$|\varOmega_i|\ge k^{|{\mathcal{J}}_i|} \biggl(\frac{(k-1)\lambda}{k} \biggr)^{q d_i} \ge k^{|{\mathcal{J}}_i|} \biggl(\frac{(k-1)\lambda}{k} \biggr)^{|{\mathcal{J}} _i|}=\bigl((k-1)\lambda\bigr)^{|{\mathcal{J}}_i|}. $$
By our choice of c, we can find a subset \({\mathcal{J}}'_{i}\) of \({\mathcal{J}}_{i}\) with \(|{\mathcal{J}}'_{i}|\ge c|{\mathcal{J}}_{i}|\ge cq d_{i}\) such that every map \({\mathcal{J}}'_{i}\rightarrow\{1, \dots, k\}\) extends to some ωΩi. When i is large enough, one also has \(|{\mathcal{W}}_{i}|\ge(1-cq/4)d_{i}\) for the set Note that \(|{\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i}|\ge cqd_{i}/4\) and every map \({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i}\rightarrow\{1, \dots , k\}\) extends to some ωΩi.
Denote by η the maximum of \(|\sigma_{i}(s)({\mathcal{W}}_{i}\cap \varTheta_{i}\cap{\mathcal{J}}'_{i})\cap\sigma_{i}(t)({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i})|/d_{i}\) for s,t ranging over distinct elements of W. Then for each sW there is a subset ϒi,s of \(\sigma_{i}(s)({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i})\) with cardinality at most η|W|di such that the sets \((\sigma_{i}(s)({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i}))\setminus\varUpsilon_{i, s}\) for sW are pairwise disjoint. It follows that On the other hand, we have
$$\sum_{s\in W}\big|\sigma_i(s) \bigl({\mathcal{W}}_i\cap\varTheta_i\cap{\mathcal{J}}'_i \bigr)\big|=|W|\cdot \big|{\mathcal{W}}_i\cap\varTheta_i\cap{ \mathcal{J}}'_i\big|\ge|W|cq d_i/4 \ge2d_i. $$
Thus η≥1/|W|2. Then we can find some distinct \(t_{i}, t'_{i}\in W\) with \(|\sigma_{i}(t_{i})({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i})\cap \sigma_{i}(t'_{i})({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i})|\ge d_{i}/|W|^{2}\). Set \(s_{i}=t_{i}^{-1}t'_{i}\). Then siW−1W∖{e}, and Take a maximal subset Ξi of \(({\mathcal{W}}_{i}\cap\varTheta_{i}\cap {\mathcal{J}}'_{i})\cap(\sigma_{i}(s_{i}))^{-1}({\mathcal{W}}_{i}\cap\varTheta_{i}\cap {\mathcal{J}}'_{i})\) subject to the condition that for any aΞi, neither σi(si)(a) nor (σi(si))−1(a) is in Ξi. Then \(\varXi_{i}\cup\sigma_{i}(s_{i})\varXi_{i} \cup(\sigma_{i}(s_{i}))^{-1}\varXi_{i}\supseteq({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i})\cap (\sigma_{i}(s_{i}))^{-1}({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i})\). It follows that \(|\varXi_{i}|\ge|({\mathcal{W}}_{i}\cap\varTheta_{i}\cap {\mathcal{J}}'_{i})\cap (\sigma_{i}(s_{i}))^{-1}({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i})|/3\ge d_{i}/(3|W|^{2})\). Note that Ξi and σi(si)Ξi are disjoint subsets of \({\mathcal{W}}_{i}\cap\varTheta_{i}\cap{\mathcal{J}}'_{i}\).

Let ξ=(ξ1,ξ2):Ξi→{1,…,k}2. Define a map ξ′:Ξiσi(si)Ξi→{1,…,k} by ξ′(a)=ξ1(a) and ξ′(σi(si)(a))=ξ2(a) for all aΞi. Extend ξ′ to some ωΩi. Then φω∈Map(ρ,F,δ,σi), \(\varphi_{\omega}(a)\in A_{\omega(a)}=A_{\xi_{1}(a)}\) and \(\varphi_{\omega}(\sigma_{i}(s_{i})(a))\in A_{\omega(\sigma _{i}(s_{i})(a))}=A_{\xi_{2}(a)}\) for all aΞi. For any aΞi, since \(\rho(\varphi_{\omega}(\sigma_{i}(s_{i})a),s_{i}\varphi_{\omega}(a))<\sqrt{\delta'}\le\sqrt{\kappa}\), by the choice of κ we have \(s_{i}\varphi_{\omega}(a)\in U_{\xi _{2}(a)}\), and hence \(\varphi_{\omega}(a)\in A_{\xi_{1}(a)}\cap s_{i}^{-1}U_{\xi_{2}(a)}\). Therefore Ξi is a (ρ,F,δ,σi)-independence set of cardinality at least di/(3|W|2) for the tuple consisting of \(A_{l}\cap s_{i}^{-1}U_{j}\) for all l,j=1,…,k.

There is some sF,δW−1W∖{e} such that the set of iVF′,δ for which si is defined and si=sF,δ lies in \({\mathfrak{F}}\). It follows that we can find an sW−1W∖{e} such that for any nonempty finite subset F of G and δ>0 there are some nonempty finite subset \(\tilde{F}\) of G and \(\tilde{\delta}>0\) with \(F\subseteq\tilde{F}\) and \(\delta >\tilde{\delta}\) such that \(s_{\tilde{F}, \tilde{\delta}}=s\). Then the tuple A′ consisting of Als−1Uj for all l,j=1,…,k has upper independence density at least 1/(3|W|2) over Σ. From the choice of W we have \(s\not\in E\). □

From Lemma 8.2 by induction on m we have:

Lemma 8.3

Letk≥2 andA=(A1,…,Ak) be a tuple of closed subsets ofXwith positive upper independence density overΣ. For eachj=1,…,kletUjbe an open set containingAj. LetEbe a finite subset ofGandm∈ℕ. Then there exists1,…,smGEsuch that\(s_{i}^{-1}s_{j}\not\in E\)for all distinct 1≤i,jmand the tupleAconsisting of\(A_{i}\cap s^{-1}_{1}U_{\omega(1)}\cap\dots\cap s^{-1}_{m}U_{\omega(m)}\)for all 1≤ikandω∈{1,…,k}mhas positive upper independence density overΣ.

We are ready to prove Theorem 8.1.

Proof of Theorem 8.1

We may assume that the Aj are closed and pairwise disjoint. Take an increasing sequence E1E2⊆⋯ of finite subsets of G with union G. We shall construct, via induction on m, closed nonempty subsets Am,j of X for \(1\le j\le k^{2^{m-1}}\) with the following properties:
  1. (a)

    A1,j=Aj for all 1≤jk,

     
  2. (b)

    for every m≥2 and \(1\le i\le k^{2^{m-2}}\), Am−1,i contains exactly \(k^{2^{m-2}}\) of the Am,j for \(1\le j\le k^{2^{m-1}}\),

     
  3. (c)

    for every m≥2 and map \(\gamma:\{1, \dots, k^{2^{m-1}}\}\rightarrow\{1, \dots, k^{2^{m-2}}\}\) there exists a tγGEm−1 such that \(t_{\gamma}A_{m, j}\subseteq\overline{U_{m-1, \gamma(j)}}\) for all \(1\le j\le k^{2^{m-1}}\), where Um−1,i={xX:ρ(x,Am−1,i)<2mδm−1} for all \(1\le i\le k^{2^{m-2}}\) and δm−1=minρ(x,y) for x,y ranging over points in distinct Am−1,j,

     
  4. (d)

    when m≥2, diam(Am,j)≤2m for all \(1\le j\le k^{2^{m-1}}\),

     
  5. (e)

    for every m, the sets Am,j for \(1\le j\le k^{2^{m-1}}\) are pairwise disjoint,

     
  6. (f)

    for every m, the collection \(\{A_{m, j} : 1\le j\le k^{2^{m-1}}\}\), ordered into a tuple, has positive upper independence density over Σ.

     
Suppose that we have constructed such Am,j over all m. Properties (b), (d) and (e) imply that \(Z=\bigcap_{m\in{\mathbb{N}}}\bigcup_{j=1}^{k^{2^{m-1}}}A_{m, j}\) is a Cantor set. Property (a) implies that Zj:=ZAj is also a Cantor set for each 1≤jk. Condition (1) follows from properties (d) and (f). Condition (2) follows from properties (c) and (d).

We now construct the Am,j. Define A1,j for 1≤jk according to property (a). By assumption properties (e) and (f) are satisfied for m=1. Assume that we have constructed Am,j for all \(j=1,\dots, k^{2^{m-1}}\) with the above properties. Set \(n=1+(k^{2^{m-1}})^{k^{2^{m}}}\). By Lemma 8.3 we can find s1,…,snGEm such that the tuple consisting of \(A_{m,i}\cap s^{-1}_{1}U_{m, \omega(1)}\cap\dots\cap s^{-1}_{n}U_{m, \omega(n)}\) for all \(1\le i\le k^{2^{m-1}}\) and \(\omega\in\{1, \dots, k^{2^{m-1}}\}^{n}\) has positive upper independence density over Σ. Take a bijection \(\varphi: \{1, \dots, k^{2^{m-1}}\}^{\{1, \dots, k^{2^{m}}\}}\rightarrow\{2, \dots, n\}\). For each \(\gamma:\{1, \dots, k^{2^{m}}\}\rightarrow\{1, \dots, k^{2^{m-1}}\}\), set tγ=sφ(γ). For all \(1\le i, j\le k^{2^{m-1}}\), define \(\omega_{i, j}\in\{1, \dots, k^{2^{m-1}}\}^{n}\) by ωi,j(1)=j and \(\omega_{i, j}(\varphi(\gamma))=\gamma((i-1)k^{2^{m-1}}+j)\) for all \(\gamma\in\{1, \dots, k^{2^{m-1}}\}^{\{1, \dots, k^{2^{m}}\}}\), and set \(A_{m+1, (i-1)k^{2^{m-1}}+j}=A_{m,i}\cap s^{-1}_{1}\overline{U_{m, \omega_{i, j}(1)}}\cap\dots\cap s^{-1}_{n}\overline{U_{m, \omega_{i, j}(n)}}\). Then properties (b), (c), (e) and (f) hold for m+1. For each \(1\le j\le k^{2^{m}}\) write Am+1,j as the union of finitely many closed subsets each with diameter no bigger than 2−(m+1). Using Lemma 4.13 we may replace Am+1,j by one of these subsets. Consequently, property (d) is also satisfied for m+1. This completes the induction procedure and hence the proof of the theorem. □

Corollary 8.4

IfhΣ(X,G)>0 for some sofic approximation sequenceΣthen the action is Li-Yorke chaotic.

An action GX is said to be distal if infsGρ(sx,sy)>0 for all distinct x,yX. We refer the reader to [2] for the basics of distal actions. Since distal actions have no Li-Yorke pairs, from Corollary 8.4 we obtain the following consequence, which extends the result of Parry that distal integer actions on compact metrizable spaces have zero entropy [34]. For amenable G we write htop(X,G) for the classical topological entropy, which is equal to the sofic entropy hΣ(X,G) for every Σ [24].

Corollary 8.5

If the actionGXis distal, thenhΣ(X,G)=0 or −∞. In particular, ifGis amenable andGXis distal, thenhtop(X,G)=0.

We remark that every distal action has an invariant Borel probability measure [2, p. 125], [39, p. 496]. But we do not know whether one can conclude that hΣ(X,G)=0 in Corollary 8.5.

Notes

Acknowledgements

The first author was partially supported by NSF grants DMS-0900938 and DMS-1162309. The second author was partially supported by NSF grant DMS-1001625. We are grateful to Wen Huang, Liviu Paunescu and Xiangdong Ye for helpful comments.

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Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of MathematicsChongqing UniversityChongqingChina
  3. 3.Department of MathematicsSUNY at BuffaloBuffaloUSA

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