Faces of simplices of invariant measures for actions of amenable groups

We extend the result of Downarowicz (Israel J Math 165:189–210, 2008) to the case of amenable group actions, by showing that every face in the simplex of invariant measures on a zero-dimensional dynamical system with free action of an amenable group G can be modeled as the entire simplex of invariant measures on some other zero-dimensional dynamical system with free action of G. This is a continuation of our investigations from Frej and Huczek (Groups Geom Dyn 11:567–583, 2017), inspired by an earlier paper (Downarowicz in Israel J Math 156:93–110, 2006).


Introduction
Let X be a Cantor space i.e. a compact, metrizable, zero-dimensional perfect space, and let G be a countable amenable group acting on X via homeomorphisms ϕ g , g ∈ G.
Amenability of G means that there exists a sequence of finite sets F n ⊂ G (called a Følner sequence, or the sequence of Følner sets), such that for any g ∈ G we have lim n→∞ |gF n F n | |F n | = 0, where gF = {g f : f ∈ F}, |·| denotes the cardinality of a set, and is the symmetric difference. The action of G is free if the equality gx = x for any g ∈ G and x ∈ X implies that g is the neutral element of G. It is well known that one can represent the system (X, G) as an inverse limit lim ←− X j ⊂ j∈N X j where each X j is a group subshift on finitely many symbols i.e. a closed G-invariant subset of some Λ j G , |Λ j | < ∞, with the action of G defined by gx(h) = x(hg). Indeed, if U = {U j : j ∈ N} is a base for topology in X consisting of clopen sets then we define Λ i to be the set of all elements of the cover of X by the sets of the form V 1 ∩ · · · ∩ V i , where either V j = U j or V j = U c j for every 1 ≤ j ≤ i. The space X i is an image of X by the map π i : X → Λ G i defined by the formula The inverse system whose inverse limit is conjugate to (X, G) is then given by the sequence of the spaces X i with bonding maps defined by coordinatewise inclusions. We will often refer to this inverse limit as a so called array system-an element of X in this interpretation is a map x(·, ·) on G × N, where x(·, j) ∈ X j . We will call such a map an array and from now on we will assume that our system is in array representation. By an (F, k)-block we mean a map B : F × [1, k] → j Λ j , where F is a finite subset of G (which will occasionally be called the shape of a block), k is a positive integer and [1, k] is an abbreviation for {1, . . . , k}. If E is a subset of the domain of a block B then by B[E] we will denote a restriction of B to E. By abuse of the notation, we will mean by |B| the cardinality of the shape of B. We will use the same letter to denote both a block and a cylinder set induced by this block-the exact meaning is always clear from the context. A block B occurs in X if B is a restriction of some x ∈ X . Let K be an abstract metrizable Choquet simplex, i.e. it is a compact convex set of a locally convex metric vector space, such that for each v ∈ K there is a unique Borel probability measure supported on the set of extreme points of K with barycenter in v (see [9] for an exhaustive course on the theory of Choquet simplices). Following [3] we define: Definition 1. 1 1. An assignment on K is a function Φ defined on K such that for each p ∈ K , the value of Φ( p) is a measure-preserving group action (X p , Σ p , μ p , G p ), where (X p , Σ p , μ p ) is a standard probability space.
2. Two assignments Φ on K and Φ on K are equivalent if there exists an affine homeomorphism π : K → K such that Φ( p) and Φ (π( p)) are isomorphic for every p ∈ K . 3. If (X, G) is a continuous group action on a compact metric space X then the set of all G-invariant measures supported by X , endowed with the weak* topology of measures, is a Choquet simplex, and the assignment by identity Φ(μ) = (X, Bor X , μ, G) (where Bor X is the Borel sigma-field) is the natural assignment of (X, G).
By a face of a simplex S we mean a compact convex subset of S which is a simplex itself and whose extreme points are also the extreme points of S. If K is a face of a simplex M G (X ) of all G-invariant probability measures on X then by the identity assignment on K we mean the restriction of the natural assignment on M G (X ) to K .
In the current article we aim to prove the following: Theorem 1.2 Let X be a Cantor system with free action of an amenable group G and let K be a face in the simplex M G (X ) of G-invariant measures of X . There exists a Cantor system Y with free action of G, such that the natural assignment on Y is equivalent to the identity assignment on K .
In case of actions of Z the theorem was proved in [3] (even with weaker assumptions; see Sect. 4) and the key tool used there was approximation of an arbitrary ergodic measure by a block (periodic) measure, i.e. a measure supported on a finite orbit. Density of periodic measures in the set of all invariant measures is usually a desired property and was proved to be true in various cases, e.g. for systems with specification property (see [1]). In case of a one-dimensional subshift one can construct a periodic measure by choosing a block B occurring in a system and uniformly distributing a probability mass on the orbit of a sequence obtained by periodic repetitions of B. Such a sequence need not be an element of a subshift (and the measure need not belong to its simplex of invariant measures), still it may give a useful approximation of a measure under consideration. For actions of groups other than Z (even Z d ) this procedure usually cannot be performed, roughly saying, because of irregular shapes of blocks, and the notion of a block measure seems to be obscure. We devote the next section to implementing it in our setup, but before we proceed, we recall a few facts about Følner sequences.
In any amenable group there exists a Følner sequence with the following additional properties (see [6]): 1. F n ⊂ F n+1 for all n, 2. e ∈ F n for all n (e denotes the neutral element of G), 3. n∈N F n = G, 4. F n = F −1 n for all n. Following [10] we say that a Følner sequence F n is tempered if for some C > 0 and all n,

Proposition 1.3 ([8]) Every Følner sequence F n has a tempered subsequence.
Standing assumption Throughout this paper, we will assume that the Følner sequence which we use is tempered and has all the above properties.
We recall the pointwise ergodic theorem for amenable groups.

Theorem 1.4 ([8])
Let G be an amenable group acting ergodically on a measure space (X, μ), and let F n be a tempered Følner sequence. Then for any f ∈ L 1 (μ), If F and A are finite subsets of G and 0 < δ < 1, we say that F is Observe that if A contains the neutral element of G, then (A, δ)-invariance is equivalent to the simpler condition and, equivalently, If (F n ) is a Følner sequence, then for every finite A ⊂ G and every δ > 0 there exists an N such that for n > N the sets F n are (A, δ)-invariant.

Lemma 1.7
Let (X, G) be a Cantor system in the array representation and let μ be an ergodic measure on X . Denote by e the neutral element of G. Let ϕ : X → X, ψ : X → X be continuous maps which commute with the action of G. Then for μ-almost every x then Note that (ϕ(x))(g) = gϕ(x) (e) = ϕ(gx)(e) (and similarly for ψ), so g ∈ S(x) is equivalent to gx ∈ B. Then, Taking the lower limits we obtain by Theorem 1.4 that D(S(x)) ≥ μ(B).

Block measures
We will explicitly define a metric consistent with the weak* topology on the set of probability measures on X , represented as an array system. First, let B k be the family of all blocks with domain F k × [1, k], occurring in X , and let Note that we may assume that B k consists only of blocks which yield cylinders of positive measure for some ergodic measure μ. Lemma 2.1 Let X be an array system and let (F n ) be a Følner sequence. For every t ∈ N there exists ε t > 0 such that if μ and ν are ergodic measures and x, y satisfy: Let n be large enough to ensure that Then, for B as above Thus, for k ≤ t + 1 we have For the sake of convenience, we introduce a notion of "distance" between a block and a measure. Let B be a block occurring in X , with domain F × [1, k] for some F ⊂ G and k ∈ N. For any block C with domain F j × [1, j], where j ≤ k, we can define the frequency of C in B in the following way: let By a standard argument we can draw from the pointwise ergodic Theorem 1.4 the following corollary.

Lemma 2.2 Let μ be an ergodic measure on X . For every ε and j we can find n and
Moreover, n and η can be chosen so that the union of all blocks C not satisfying the approximation rule has measure smaller than ε.
for almost every x. Since there are only finitely many blocks with such domain, we may assume that the above equality is satisfied simultaneously for all such blocks D on a subset of X having measure 1. Hence we can find n such that for every m ≥ n the inequality holds, for all such D, on a set of measure X ε at least 1 − ε. Pick x from this set.
Additionally, increasing n we may demand that each Fix m ≥ n and let C be a block which appears in x on the domain F m . By (1.2), Furthermore, using (1.1) Hence, Combining it with (2.1) we obtain |fr C (D) − μ(D)| < 3ε 4 for blocks C having domain exactly equal to F m × [1, m]. Now suppose that F is a (1 − η)-subset of F m and that C is a block which appears in x on the domain F, while C is x restricted to F m .
The left hand is greater than while the right hand is bounded from above by Hence which can be made smaller than ε 4 by choosing apprioprately small η, irrespective of m, C and D.
Finally, note that if |fr C (D) − μ(D)| < ε is not true for some D as above and as defined by 2.1, which means that the union of all C for which the approximation fails is less than ε.
We can now define the distance between a block and a measure: let Indeed, in this case we have d k (B, ν) < ε 2 j and

Consequently, for any ε > 0 and sufficiently large j there exists δ such that if F is an (F j , δ)-invariant set and B is a block with domain F × [1, j], then there exists a probability measure μ B such that d(B, μ B ) < ε.
Proof Let us first observe that the second assertion follows from the first by Remark 2.3. Therefore, it suffices to prove the first statement.
Let Δ j = Λ 1 × · · · × Λ j . The full shift Δ G j is a Cantor set on which we have the uniform Bernoulli probability measure λ which assigns equal measures M j to all cylinders with domain F j × [1, j]. We shall define μ B by specifying its density f B with respect to λ. f B will be constant on cylinders with domain F j × [1, j]: on each such cylinder associated with a block C let f B (x) = 1 M j fr B (C). Obviously d j (μ B , B) = 0. We will now estimate d i (μ B , B) for i < j. Let D be any block from B i . Let C j be the family of (distinct) blocks from B j such that D = C∈C j C. We have: and we need to show that the latter quantity is close to Taking cardinalities and dividing by N F (F i ), we obtain If δ is small enough then the expression can be arbitrarily close to 0, while can be arbitrarily close to 1, so we can assume that Note that in the above lemma δ may be as small as we want.
Corollary 2.5 Let X be a zero-dimensional dynamical system with the action of an amenable group G and let μ be an ergodic measure on X . For any ε > 0 and any sufficiently large j there exists For actions of Z, it is a well-known fact that if a sufficiently long block C is a concatenation of shorter blocks B 1 , B 2 , . . . , B n of equal length, then the probability measure μ C (which for actions of Z can easily be assumed to be shift invariant) can be arbitrarily close to the arithmetic average of the measures μ B i . An analogous claim can be made for the action of any amenable group G; however the lack of a natural way to decompose a subset of G into smaller sets requires the use of quasitilings. Note that every quasitiling can be seen as a symbolic element T ∈ {0, 1, . . . , n} G , such that T (g) = i if g ∈ C i for some i, and T (g) = 0 otherwise.

Definition 2.7 A quasitiling T is:
1. disjoint, if the tiles are pairwise disjoint; 2. α-covering, if the union of all tiles has lower Banach density at least α. 3. congruent with a quasitiling T , if for any two tiles T ∈ T , T ∈ T we have either T ⊃ T or T ∩ T = ∅.
Any (T, k)-block whose shape T belongs to a quasitiling T will be called a (T , k)block.
Let (X, G) be a topological dynamical system. Suppose we assign to every x ∈ X a quasitiling T (x) of G, with the same set of shapes S 1 , . . . , S n for all x. This induces a map x → T (x) which can be seen as a map from (X, G) into {0, 1, . . . , n} G with the shift action. If such a map is a factor map (i.e. if it is continuous and commutes with the dynamics), we call it a dynamical quasitiling. A dynamical quastiling is said to be disjoint and/or α-covering, if T (x) has the respective property for every x. Note that though the set of shapes is common, the collection of centers depends on x so C i (x) become functions assigning to each x a subset of G. We introduce the following new definition.
Definition 2. 8 We will say that a dynamical quasitiling T consisting of shapes S(T ) = {S 1 , S 2 , . . . , S n } and centers C(T ) = {C 1 , C 2 , . . . , C n } has restricted block distribution if for every x ∈ X any (T, k)-block B which occurs in x on some domain S i c may occur in x only on domains of this form for c ∈ C i (x).

Remark 2.9
Clearly, if T has restricted block distribution then for any block D occuring in x on some domain S i c 0 and any B being a block occuring in x on a (disjoint) union F of tiles we have

Lemma 2.10 For any ε > 0 there exist j ∈ N and δ > 0 such that if T is a disjoint quasitiling by (F j , δ)-invariant sets, and C is a block with domain H × [1, j] such that some disjoint union of tiles T
Proof Applying Lemma 2.4, for any j there is δ j such that for any block B with domain F × [1, j], where F is a (F j , δ j )-invariant set, and for any block D with Let H be a subset of G and let T be a quasitiling of G by (F j , δ)-invariant sets for some δ > 0. Suppose that the union n i=1 T i is a (1 − δ)-subset of H for some pairwise disjoint tiles T 1 , T 2 , . . . , T n belonging to T . For every k ≤ j let us define the set ). Clearly, we can demand that δ < δ j (further restrictions will follow). Note that since each T ∈ T is (F j , δ)invariant, for any block B whose domain is a tile of T the measure μ B is well-defined.
Therefore, using the traingle inequality, We can estimate that 1−δ , so the whole expression can be made smaller than ε 8 j by appropriate choice of a small δ. Now, for every B i we have fr B i (D) is approximately equal to μ B i (D) with error ε 8 j , and this approximation is preserved by the weighted average we have obtained, therefore If j is sufficiently large, Remark 2.3 implies that and since d(C, μ C ) < ε 2 , we also have

Remark 2.11
In the above lemma, we can increase j and decrease δ without spoiling the approximation error ε, because if j ≥ j and δ ≤ δ then (F j , δ )-invariant set is also (F j , δ)-invariant and a (1 − δ )-subset of any H is a (1 − δ)-subset of H .
The next two lemmas concerning the existence of quasitilings were proved in [4]. The following lemma is analogous to the case of classical one-dimensional subshifts.

Lemma 2.14 For every ε > 0 there exist J and δ such that if for some j > J the set F is a (F j , δ)-invariant and B is a (F, j)-block, then d(B, M G (X )) < ε and d(μ B , M G (X )) < ε.
Proof We will prove the assertion in the language of blocks, i.e. we will show that d(B, M G (X )) < ε. The assertion for measures will follow from Corollary 2.5.
Suppose that there is ε > 0 such that for every J and δ there exists an integer j > J and a (F, j)-block B J,δ on domain D J,δ × [1, j], which is (F j , δ)-invariant and d(B J,δ , M G (X )) ≥ ε. For ε 2 and every J we choose δ = δ J via Lemma 2.4 with the additional requirement that δ J |F j | < ε J . We denote B J = B J,δ J and D J = D J,δ J . Let μ be a limit point of the sequence μ B J . Clearly, d(μ, M G (X )) ≥ ε/2.
We will obtain a contradiction by showing that μ is G-invariant. It suffices to prove that μ(C) = μ(g(C)) for every (F i , i)-block C and every g ∈ G, where Fix γ > 0, g ∈ G and a cylinder set C on F i × [1, i]. Let J be large enough to ensure that Then, The first condition guarantees that the sum of the first and the last terms are less than γ 3 . The choice of B J was made with use of Lemma 2.4, so both the second and the fourth summands are smaller than ε 4 j . By the second requirement above, their sum is again less than γ 3 . We only need to show that the middle term is bounded by γ 3 .
In particular, Thus, an occurrence of C in B J 'at position h' yields the occurrence of g(C) in B J 'at gh', if only gh ∈ D J . By a similar argument, occurrences of g(C) force occurrences of C. The number of pairs (h, gh) such that only one of these elements belongs to D J is smaller than δ J |F j ||D J |, so On the other hand, which ends the proof.
Finally, we prove our last tool.

Claim 1 V α is an open set.
We will prove that its complement V c α is closed. Let μ be the weak* limit of a sequence μ k of elements of V c α . Then μ k = M(X) νdξ k for some ξ k supported by the closed αneighborhood of M G (X ) with ξ k (F) ≥ ε. The sequence ξ k has a subsequence which converges in the weak* topology to some measure ξ -let us assume that ξ k itself is already convergent. By the portmanteau lemma, ξ(F) ≥ lim k→∞ ξ k (F) ≥ ε. By the same lemma, ξ assigns to the closed α-neighborhood of M G (X ) the value 1, so it is supported by this neighborhood.
The only thing left to show is the equality μ = M(X) νdξ . For any function f ∈ C(X ) the map ν → ν( f ) = f dν is a real continuous map of M(X ). Therefore, by the definition of weak* convergence (used both in spaces M(X ) and M(M(X ))), which is the desired equality.

Claim 2
If α is small enough then V α contains K . If not then letting α tend to 0, we could find a measure in K that is a barycenter of a distribution ξ on M G (X ) with ξ(F) ≥ ε. This is not possible.
Returning to the main proof, let γ be small enough that the open γ -neighborhood of K is contained in V α . Using Lemma 2.10 choose η and j to obtain the the error of approximation equal γ /2 for any (F j , η)-quasitiling (i.e. in the lemma γ /2 and η play the role of ε and δ, respectively). Let T be such a quasitiling. By Lemma 2.14 and Remark 2.11, making j large enough, we can also assume that every block with domain S ×[1, j] (where S is a shape of T ) that occurs in X lies in the α-neighborhood of the set of invariant measures on X . Note that the union B of the collection of all elements of B (as defined in the statement of the lemma) is clopen, and thus the function μ → μ( B) is continuous on the set M(X ). Suppose that μ is an ergodic measure in K such that B∈B μ(B) |B| > ε. The function ν → B∈B ν(B) |B| is continuous, therefore if ν is close enough to μ, then B∈B ν(B) |B| > ε. In particular, by Corollary 2.5 we can find a block C occurring in X , such that d(μ C , μ) < γ 2 , and B∈B μ C (B) |B| > ε. By Lemma 2.4 we can demand that fr C B approximates each B ∈ B so well that also B∈B fr C (B) |B| > ε. Note that by the restricted block distribution, for elements B of the tiling fr C (B) is derived by calculating only the appropriate elements of the tiling of C. We can also assume that the union of tiles of T contained in the domain of C is a (1−η)-subset of C. By Lemma 2.10, μ C is closer than and B is a (F, k)-block occurring in X t−1 , then the distance between μ B and M G (X t−1 ) is less than ε t . For sufficiently large n t we can pick k > J t such that: We will show that the maps Φ t converge uniformly on K * . To this end, it suffices to uniformly estimate the distance between Φ t (μ) and Φ t−1 (μ) for ergodic μ ∈ K * by a summable sequence. By Lemma 2.15, for any μ ∈ K * we have the estimate where B denotes the family of all T * n t -blocks B such that d(B, K * ) > δ t . As we have already said, this implies that if x ∈ X * , then the set of coordinates in φ t−1 (x) belonging to tiles of T * n t that are domains of blocks from B has upper Banach density less than 2ε t . Sinceφ t only makes any changes on these coordinates, φ t (x) differs from φ t−1 (x) on a set of density less than 2ε t . If x is in the support of some invariant measure μ, then φ t−1 (x) and φ t (x) are in the support of Φ t−1 (μ) and Φ t (μ), respectively, and since the two points agree on a set of large upper Banach density, the measures are within distance less than 1 2 t (according to the choice of ε t with use of Lemma 2.1). This uniform convergence, together with the fact that Φ t (M G (X * )) is within the 2ε t -neighborhood of Φ t−1 (K * ), implies that Φ(M G (X * )) ⊂ Φ(K * ), and since the other inclusion is obvious, the two sets are equal. Now, define the set Y (which will support the desired assignment) as follows: Observe that Y is a closed, shift-invariant set, and that for any Følner set F and any k ∈ N every block with domain F × [1, k] in Y occurs in infinitely many of the sets X t . It follows that every invariant measure on Y can be approximated by invariant measures on the X t 's, and thus the set of invariant measures on Y is contained in Φ(M G (X * )) = Φ(K * ). The other inclusion is generally true: for any weakly* convergent sequence of measures μ t supported by X t , the limit measure μ is always supported by ∞ s=1 ∞ t=s X t . Therefore M G (Y ) = Φ(K * ). By Lemma 1.7 for every ergodic μ ∈ K * the set of points x ∈ X t−1 such that the column x(e) is modified byφ t has measure μ less than 2ε t , becauseφ t commutes with the shift map and for any x in the support of μ the set of modified coordinates has upper Banach density less than 2ε t . If this bound works for all ergodic measures it works for all measures in K * . Since the sequence ε t is summable, the Borel-Cantelli lemma implies that for almost every x ∈ X * the columns φ t (x)(e) are all equal from some point onwards. By shift-invariance, the same is true for φ t (x)(g) for any g, so ultimately we conclude that if μ ∈ K * , then for μ-almost every x ∈ X * every coordinate of x is only changed finitely many times. This means that a limit point φ(x) is then well-defined, and this map φ is invertible (since every φ t (x) retains the original contents of x in the bottom row). In other words φ is an isomorphism between the measure-theoretic dynamical systems (X * , μ) and (Y, Φ(μ)).

Concluding remarks
Firstly, we note that we can strengthen Theorem 1.2 by combining it with theorem 1.2 of [7], obtaining the following version: Theorem 4.1 Let X be a Cantor system with free action of an amenable group G and let K be a face in the simplex M G (X ) of G-invariant measures of X . There exists a Cantor system Y with minimal free action of G, such that the natural assignment on Y is equivalent to the identity assignment on K .
Secondly, note that the result of this paper is not strictly a strengthening of the main theorem 4.1 in [3], since while we gain the result for actions of amenable groups, we add the requirement that the action be free, whereas the original result merely requires that the face in question contain no periodic measures. Unfortunately, it is very much unclear how the machinery used to deal with periodic points would transfer to the group case, which is why the matter of directly extending the result of [3] remains open.