Periodic measures are dense in invariant measures for residually finite amenable group actions with specification

We prove that for certain actions of a discrete countable residually finite amenable group acting on a compact metric space with specification property, periodic measures are dense in the set of invariant measures.


1.
Introduction. Let G be a discrete countable residually finite amenable group acting on a compact metric space X. Denote by M(X, G) the set of G-invariant measures and M e (X, G) the set of ergodic G-invariant measures. For a point x ∈ X, we call x a periodic point if |orb(x )| < ∞. Define the periodic measure µ x as a probability measure with mass |orb(x)| −1 at each point of orb(x) and we denote by M P (X) the set of all such periodic measures.
Specification property, introduced by Bowen [3] in 70's for Z−actions, is a basic property used in smooth and topological dynamical systems to obtain maximal entropy measure, exponential growth of periodic orbits, density of periodic or ergodic measures, multifractal analysis etc. Specification property seems a very strong property, but there are many examples of dynamical systems satisfying this property, including subshifts of finite type, sofic shifts, the restriction of an Axiom A diffeomorphism to its non-wondering set, expanding differential maps and geodesic flows on manifold with negative curvature. Readers may refer [9,Chapter 21] for more details of specification. For non-uniformly hyperbolic dynamical systems, several versions of specification-like were introduced, including [2,11,13,16]. Pfister and Sullivan [18] also introduced a weak specification property called g-almost product property, which was renamed as the almost specification by Tompson [22]. In [19], Ruelle introduced the notion of weak specification for Z d actions and called the definition in [3] as strong specification. Recently, Chung and Li [6] generalized specification to general countable group actions. We will give the details in next section.
For smooth dynamical systems, the problem of density of periodic measures is well studied. For instance, Sigmund [21] proved that for uniformly hyperbolic diffeomorphisms with specification property, each invariant measure can be approximated by periodic measures. Hirayama [11] proved that each invariant measure supported by the closure of a Pesin set of a topological mixing measure is approximated by periodic measures. Liang, Liu and Sun [13] improved Hirayama's result by weakening the assumption of mixing measure to that of hyperbolic ergodic measure.
Our main results are as follows.
Theorem 1.1. Let G be a discrete countable residually finite amenable group acting on a compact metric space X with specification property. Then M P (X, G) is dense in M(X, G) in the weak * topology. Moreover M e (X, G) is residual in M(X, G).
Theorem 1.2. Let Γ be a countable discrete group and f an element of ZΓ invertible in l 1 (Γ, R). Then the action of Γ on X f which is the Pontryagin dual of ZΓ/ZΓf has specification property.

2.
Preliminary. In this section, we will recall some notions and basic facts about amenable groups and residually finite groups. Also we will give the definition of specification.
where F(G) is the collection of all finite subsets of G. Such sequences are called Følner sequences. The quasi-tiling-theory is a useful tool for actions of amenable groups which is set up by Ornstein and Weiss in [17].
We say that The subsets C 1 , C 2 , · · · , C k are called the tiling centres. The next proposition is [7, Lemma 9.4.14].
For our proof, we also need the Mean Ergodic Theorem for amenable group actions.
Lemma 2.1 (Mean Ergodic Theorem). Let G be an amenable group acting on a probability measure space (X, B, µ) by measure preserving transformation, and let {F n } n∈N be a Følner sequence. For any f ∈ L 2 (µ), set A n (f )(x) = 1 |Fn| g∈Fn f (gx).

Residually finite group.
A group is residually finite if the intersection of all its normal subgroups of finite index is trivial. Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups, polycyclic-by-finite groups, finitely generated linear groups and fundamental groups of 3-manifolds. For more information about residually finite groups, readers can refer [5,Chapter 2].
Let (G n , n ≥ 1) be a sequence of finite index normal subgroups in G. We say Clearly such sequence exists if G is countable and residually finite.
If G ⊂ G is a subgroup with finite index, we say that Q ⊂ G is a fundamental domain of the right coset space G \G, i.e. a finite subset such that {G s | s ∈ Q} is a partition of G.
The following proposition is [8, Corollary 5.6] and we will use it to control the periodic orbits we get. The original proof is a version of the Ornstein-Weiss quasitiling lemma. Also there is an algebraic proof from [1, Theorem 6].
Proposition 2. Let G be a countable discrete residually finite amenable group and let (G n , n ≥ 1) be a sequence of finite index normal subgroup with lim n→∞ G n = {e G }. Then there exists a Følner sequence (Q n , n ≥ 1) such that Q n is a fundamental domain of G/G n for every n ≥ 1.

Specification.
In this subsection, we will recall specification property of general group actions, which is from [6, Section 6].
Let α be a continuous G-action on a compact metric space X with metric ρ. The action has specification property if there exist, for every ε > 0, a nonempty finite subset F = F (ε) of G with the following property : for any finite collection of finite subsets F 1 , F 2 , · · · , F m of G satisfying and for any subgroup G of G with and for any collection of points x 1 , x 2 , . . . , x m ∈ X, there is a point y ∈ X satisfying and sy = y for all s ∈ G .
In [6], this property is called strong specification. We will call it specification since there is no misunderstanding.
3. Proof of Theorem 1.1. Let (G n , n ≥ 1) be a sequence of finite index normal subgroups with lim n→∞ G n = {e G } and (Q n , n ≥ 1) be a Følner sequence such that Q n is a fundamental domain of G/G n as described in Proposition 2.
Let ν ∈ M(X, G), ε > 0 and W a finite subset of C(X), where C(X) is the set of all the continuous real valued functions on X. Uniformly continuity of the elements of W implies that there is δ ∈ (0, ε) such that |ξ(x) − ξ(y)| < ε 8 for all x, y ∈ X with d(x, y) < δ and all ξ ∈ W. By Mean Ergodic theorem, A n (ξ) converges to ξ * in L 2 for all ξ ∈ W, so we can choose a subsequence {A n k } k∈N such that A n k (ξ) converges to ξ * ν-a.e. for all ξ ∈ W. For convenience, we will write the subsequence (Q n k , k ≥ 1)as (Q n , n ≥ 1). Set We know ν(Q(G)) = 1. Denote by ξ * (x) the limit for each x ∈ Q(G). Next we will construct a finite partition of X as following: Since W is finite, Next we will construct F 1 , F 2 , · · · , F t and G m satisfying (1) and (2) in specification property. The idea is from [25, Theorem 1.3] but with very minor changes. Suppose η = {A 1 , A 2 , · · · , A l }. Let a i = ν(A i ) for i = 1, 2, · · · , l and a = min {a i : i = 1, 2, · · · , l}. By Egorov's Theorem, there exist a Borel subset 4|F | 2 l for all g ∈ F, n ≥ N 2 . By Proposition 1, there exist n k > n k−1 > · · · > n 1 ≥ N 2 and N 3 ∈ N s.t. Q m can be γ 4|F | 2 l -quasi-tiled by Q n1 , Q n2 , · · · , Q n k when m > N 3 . Also N 3 will be large enough such that the family of all the translations By the definition, we know Claim. {S nj (c j )c j |S nj (c j )c j ∈ F} and G m satisfy the conditions in specification property.
Using specification property, there is some y ∈ X such that ρ(gx j (c j ), gy) < γ < δ, for all g ∈ S nj (c j )c j . Denote by µ y the periodic measure supported on orb(y).
Claim. (K, ρ) is a convex compact metric set and denote by ext(K) the set of extreme points of K. Then ext(K) is a G δ subset of K. Let K n = {x ∈ K : there exist y, z ∈ K such that x = 1 2 (y + z) and d(y, z) ≥ 1 n }.
Obviously, K n is closed.
As a result, ext(K) = K \ K 0 is a G δ subset of K. We know that M e (X, G) is the set of extreme points of M(X, G). So by the claim, M e (X, G) is a G δ set. Thus we finish the proof.

4.
Proof of Theorem 1.2. For a countable group Γ and an element f = f s s in the integral group ring ZΓ, where ZΓ is the set of finitely supported Z-valued functions on Γ, consider the quotient ZΓ/ZΓf of ZΓ by the left ideal ZΓf generated by f. It is a discrete abelian group with a left Γ-action by multiplication. The Pontryagin dual of ZΓ/ZΓf which is denoted by X f is a compact metrizable abelian group with a left action of Γ by continuous group antomorphisms and denote by ρ some compliable metric on X f . Denote X = (R/Z) Γ . We denote by ρ 1 the canonical metric on R/Z defined by The left and right actions l and r on X are defined by (l s x) t = x s −1 t and (r s x) t = x ts for every s, t ∈ Γ and x ∈ X. We can extend those actions of Γ to commuting actions l and r of ZΓ on X by setting

It is easy to check
Denote by the restriction to X f of the Γ-action l on X.
Then v = r f (w) is what we need. Let e Γ and e X f be the unit elements of Γ and X f respectively. Set W = {e Γ } ∪ support(f * ) = ({e Γ }∪support(f )) −1 . Let ε > 0. Then we can find a nonempty finite subset W 1 of Γ and ε 1 ∈ (0, f −1 1 ) such that if x, y ∈ X f satisfy max s∈W1 |x s − y s | ≤ 2ε 1 , then ρ(x, y) ≤ ε. Take a finite subset W 2 of Γ containing e Γ satisfying By Lemma 4.1, for any finite collection of finite subsets F 1 , F 2 , · · · , F m of Γ satisfyingF F i ∩ F j = ∅, 1 ≤ i = j ≤ m and any collection of points x 1 , x 2 , . . . , x m , otherwise.
The following lemma is a version of [4, Lemma 1]. The same argument also appeared in the proof of [6, Lemma 6.2].
A point x ∈ X f is said to be homoclinic if sx → e X f as Γ s → ∞. The set of all homoclinic points, denoted by ∆(X f ), is a Γ-invariant normal subgroup of X f . Claim 2. Let d be the expansive constant of (X f , α f ) i.e. if x, y ∈ X f with ρ(sx, sy) ≤ d for all s ∈ Γ then x = y. For any ε ∈ (0, d),F = W 1 W 2 (W 1 W 2 ) −1 , any finite subset F 1 of Γ and x ∈ X f , there exists y ∈ ∆(X f ) s.t. max s∈F1 ρ(sx, sy) ≤ ε and sup s∈Γ\F F1 ρ(se X f , sy) ≤ ε.
To prove the above claim, we may assumeF =F −1 otherwise we can replacẽ F byF ∪F −1 . Let F 1 be a finite subset of Γ and x ∈ X f . For each finite set F 2 ⊂ Γ \F F 1 , from Claim 1, we can find y F2 ∈ X f such that ρ(sx, sy F2 ) ≤ ε for all s ∈ F 1 and ρ(se X f , sy F2 ) ≤ ε for all s ∈ F 2 . Note that the collection of the finite subsets of Γ \F F 1 has a partial order. Take a limit point y ∈ X f of {y F2 } F2 . Then ρ(sx, sy) ≤ ε for all s ∈ F 1 and ρ(se X f , sy) ≤ ε for all Γ \ F F 1 . By Lemma 4.2, we know y ∈ ∆(X f ).