On large deviations for amenable group actions

By proving an amenable version of Katok's entropy formula and handling the quasi tiling techniques, we establish large deviations bounds for countable discrete amenable group actions. This generalizes the classical results of Lai-Sang Young.


Introduction
The large deviation theory focuses on convergence properties of stochastic systems. The stochastic systems usually can occur in various stochastic processes, differential equations or dynamical systems. In the case of dynamical systems, the usual setting is a continuous transformation on a compact metric space with a reference measure. Then one of interesting questions is to obtain the exponential convergence rate for the reference measure of some subset related to Birkhoff average.
Many results on the large deviation theory for dynamical systems are related to entropy or pressure. Usually the desired exponential convergence rates are functions in terms of entropy or pressure. And these results can have applications to many aspects such as statistical mechanics, non-uniformly hyperbolic systems, shift spaces, etc.( [7,9,16,21]).
With the development of the theory of dynamical systems, Z d − actions are considered as the generalization for Z−actions. Some large deviation results are proved for these situations [6,10]. These results rely on the entropy and pressure theory for Z d −actions. But now the theory of entropy and pressure goes much further. It is established for actions of groups such as amenable groups and sofic groups beyond Z d [3,11,12,13,14,15,18,20]. In this paper, we will consider the large deviation results of Lai-Sang Young [21] for actions of countable discrete amenable groups.
Throughout this paper, we let (X, G) be a G−action topological dynamical system, where X is a compact metric space with metric d and G a countable discrete amenable group. A group G is said to be amenable if there exists a sequence of finite subsets {F n } of G which are asymptotically invariant, i.e., lim n→+∞ For a finite subset F in G, > 0 and a function f of X, we denote the Bowen ball associated to F with radius by B F (x, ) = {y ∈ X : d F (x, y) < } = {y ∈ X : d(gx, gy) < , for any g ∈ F } and denote the summation and average of f along F by and respectively. Let M (X), M (X, G) and E(X, G) denote the collection of Borel probability measures, the G−invariant and G−ergodic Borel probability measures of X respectively.
A Følner sequence {F n } in G is said to be tempered (see Shulman [17]) if there exists a constant C which is independent of n such that Let (X, G, µ) be a measure-theoretic G−action dynamical system where G is a countable discrete amenable group and µ is a G−ergodic Borel probability measure on X. The ergodic theorem( [11,20]) states that, if {F n } is a tempered Følner sequence in G and f ∈ L 1 (X, B, µ), then almost everywhere and in L 1 .
Let {F n } be a Følner sequence in G. For µ ∈ M (X), ϕ ∈ C(X, R) and E ⊂ R, we will consider the exponential growth rate of µ({x ∈ X : A Fn ϕ(x) ∈ E}). Then we have the following large deviations results. Theorem 1.1. Let (X, G) be a compact metric G−action topological dynamical system and G a discrete countable amenable group. Let {F n } be a tempered Følner sequence in G with lim n→+∞ |Fn| log n = ∞. Assume h top (X, G) < ∞. Then for any µ ∈ M (X), ϕ ∈ C(X) and c ∈ R, where the supremum is taken over all ν ∈ E(X, G) with ϕ dν > c.
We note here that h top (X, G) and h ν (X, G)are the topological and metric entropy of (X, G) respectively and they do not depend on the choice of the Følner sequences. h µ ({F n }; ν) is the ν−relative entropy with respect to µ for Følner sequence {F n }. We will give the detailed definitions of these entropies in Section 2.

Now we denote by
Theorem 1.2. Let (X, G), {F n }, µ, ϕ and c be as in Theorem 1.1. Then for ψ ∈ V + , lim sup where the supremum is taken over all ν ∈ M (X, G) with ϕ dν ≥ c.
Let (X, G), {F n }, µ, ϕ and c be as in Theorem 1.1 and in addition, assume that (X, G) has weak specification property. Then for ψ ∈ V − , where the supremum is taken over all ν ∈ M (X, G) with ϕ dν > c.
We will give the definition of weak specification property in next section. We also remark that when G = Z, the weak specification property here coincides with the specification property in [21].
The above theorems are the amenable group action version of Lai-Sang Young's classical results in [11]. But for the proof, we need to handle the techniques for amenable group actions. For the proof of Theorem 1.1, we need to prove an amenable group version of Katok's entropy formula [9]. For the proof of Theorem 1.2, we need to employ the idea and technique of the proof of variational principle for amenable pressure (see for example, [13]). For the proof of Theorem 1.3, we need a lemma via the quasi tiling of amenable groups. We also should remark here that the referee mentioned to us the recent advances on the tilings of amenable groups by Downarowicz etc [5]. It says that each amenable group can be tiled by finite many shapes. Compare with the quasi tiling theory, this result can provide a better and simplified understanding on the structure of amenable groups.

Entropy.
Let (X, G, µ) be a measure-theoretic G−action dynamical system where G is a discrete countable amenable group and µ is a G−invariant Borel probability measure on X. Let P and Q be two finite measurable partitions of X and denote by P ∨ Q = {P ∩ Q : P ∈ Q and Q ∈ Q}. For a finite subset F in G, we denote by P F = g∈F g −1 P. Then the classical measure-theoretical entropy of P is defined by where {F n } is any Følner sequence in G and the definition is independent of the specific Følner sequence {F n }. The measure-theoretical entropy of the system (X, G, µ), h µ (X, G), is the supremum of h µ (G, P) over P.
Let's recall the classical Shannon-McMillan-Breiman theorem for amenable group actions. [11,20]). Let (X, G, µ) be an ergodic G−system. For any tempered Følner sequence {F n } in G with lim n→+∞ |Fn| log n = ∞ and any finite measurable partition P of X, for µ almost every x ∈ X, where P(x) denotes the atom in P that contains x.
Suggested by the SMB theorem, we may define the entropy at x ∈ X with respect to µ ∈ M (X) for {F n } by In [22], the authors showed that h µ ({F n }, x) = h µ (X, G) for µ-a.e. x ∈ X whence µ ∈ E(X, G) and the Følner sequence {F n } is tempered and satisfies lim n→+∞ |Fn| log n = ∞. For ν ∈ M (X), we define the ν−relative entropy with respect to µ for Følner sequence {F n } by Now we return to the topological case. Let (X, G) be a compact metric G−action topological dynamical system and G a discrete countable amenable group. Let {F n } be a Følner sequence. Then the topological entropy of (X, G) is defined in the following way.
Let U and V be two open covers of X and denote by U ∨V = {U ∩V : U ∈ U and V ∈ V}. For a finite subset F in G, we denote by U F = g∈F g −1 U. Let N (U) denote the number of sets in a finite subcover of U with smallest cardinality. Then the topological entropy of U is It is shown that h top (G, U) is not dependent on the choice of the Følner sequences {F n }.
And the topological entropy of (X, G) is where the supremum is taken over all the open covers of X.

Weak specification.
The following definition generalizes the traditional specification property to general group actions(see [4]).
Let (X, G) be a compact metric G−action topological dynamical system and G a discrete countable group (need not be amenable). (X, G) has weak specification if for any > 0 there exists a nonempty finite subset F of G with the following property: for any finite collection F 1 , · · · , F m of finite subsets G with and for any collection of points x 1 , · · · , x m ∈ X, there exists a point y ∈ X such that

Quasi tiling.
The quasi tiling theory for amenable groups, set up by Ornstein and Weiss(see [15]), is a useful tool and technique in the study of dynamical systems for amenable group actions. Let F (G) denote the collection of finite subsets of G. Let A, K ⊂ F (G) and We say that The following theorem states a fundamental quasi tiling property of amenable groups.
In our setting, we do not restrict the Følner sequence to be increasing and contain e. Let {F n } be a Følner sequence in G. For any integer N > 0, as in the proof of Theorem 2.4 of [19], choose k > 0 and δ such that (1 − 2 ) k < and 6 k δ < 2 . And choose N ≤ n 1 < n 2 < · · · < n k such that F n i+1 is (F n i F −1 n i , δ)-invariant and Then due to the proof of Theorem 2.4 of [19], any (F n kF −1 can also be -quasi-tiled by F n 1 , F n 2 , · · · , F n k . Hence we get the following proposition. Proposition 2.3. Let G be an amenable group and {F n } be a Følner sequence in G. Then for any 0 < < 1 4 and any integer N > 0, there exist integers N ≤ n 1 < n 2 < · · · < n k such that any F M (M sufficiently large) can be -quasi-tiled by F n 1 , F n 2 , · · · , F n k .

Katok's entropy formula and proof of Theorem 1.1
In this section, we will prove Theorem 1.1. For this aim, we need the Katok's entropy formula [9] for amenable group action dynamical systems.
Theorem 3.1. Let (X, G) be a compact metric G−action topological dynamical system and G a discrete countable amenable group. Let µ ∈ E(X, G) and {F n } a tempered Følner sequence in G with lim n→+∞ |Fn| log n = ∞, then where N (F, , δ) is the minimal number of B F (x, ) balls that cover a set of measure no less than 1 − δ.
In the following we will show that Let ξ be a finite measurable partition of X such that µ(∂ξ) = 0 and denote bỹ h = h µ (G, ξ). Then for sufficiently small γ > 0, the γ−neiborghhood of ∂ξ (denoted by U γ ) has measure less than . By the ergodic theorem, 1 |Fn| g∈Fn χ Uγ (gx) converges to µ(U γ ) a.e.. For any > 0, notice that and E N increases. For sufficiently large N , whence n > N , we have By SMB Theorem, − 1 |Fn| log µ(ξ Fn (x)) converges toh a.e.. Hence by the same argument as above, for sufficiently large N , whence n > N , Then for any n > N , µ(E) > 1 − 2 .
Denote the total number of such (ξ, F n )−names by L n , then when n is large enough, For an upper bound of L n , one may refer to [9] or [2]. Here we will give a brief estimate. Applying the Stirling formula n! = √ 2πn( n e ) n e αn , 1 12n + 1 < α n < 1 12n , L n can be estimated by: We note that η > 0 is a constant only dependent on #ξ and but independent of x and n and moreover, η tends to 0 as tends to 0. Let D be a subset of X with measure no less than 1 − δ. If we let < 1−δ 4 , then µ(D ∩ E) > 1−δ 2 . We will estimate a lower bound for the number of B Fn (x, γ) balls needed to cover the set D ∩ E. Suppose D ∩ E is covered by the family of balls {B Fn (y, γ) : y ∈ S}. From the discussion above, the intersection of D ∩ E with each ball in this family can be covered by no more than L n many atoms of ξ Fn and any atom of ξ Fn that intersects D ∩ E has measure at most exp(−|F n |(h − )). Hence which allows us to deduce that lim inf Letting goes to 0 and then taking a sequence of ξ's whose diameters tend to 0, we obtain (3.2).

Proof of Theorem 1.2
Proof of Theorem 1.2.
Let ψ ∈ V + and denote V n = {x ∈ X : A Fn ϕ(x) ≥ c}. We will show that there exists ν ∈ M (X, G) with ϕ dν ≥ c such that lim sup By the definition of V + , we may let C and be such that for any x ∈ X and n ∈ N, µ(B Fn (x, )) ≤ C exp(−S Fn ψ(x)). For each n, let E n be a maximal (F n , )-separated set contained in V n . Let σ n ∈ M (X) be the atomic measure concentrated on E n by the formula σ n = x∈En e −S Fn ψ(x) δ x z∈En e −S Fn ψ(z) . Define ν n ∈ M (X) by We note here that S Fn ψ dσ n = ψ dν n .
For convenience we denote Z n = z∈En e −S Fn ψ(z) . Assume that the upper limit of 1 |Fn| log Z n is attained along a sequence {n j }, i.e. lim sup Let ν be a weak limit of ν n j . We may assume ν n j → ν by taking a subsequence and still denote this subsequence by {n j }. Then ν ∈ M (X, G). Also notice that we have that ψ dν ≥ c. Let β be a finite Borel partition of X each element of which has diameter less than . Moreover, we require that ν(∂β) = 0. Recall that for any finite subset F of G, β F = g∈F g −1 β. Then each element of β Fn contains at most one point in E n . Hence = log Z n .
By Lemma 3.1 (3) of [8], the multi-subadditivity of H σn (β • ), we have: Because ν(∂β F ) = 0, we have that lim sup which allows us to deduce that lim sup by taking F over all the finite subsets of G.
Since V n ⊆ x∈En B Fn (x, ), we have lim sup
Let L be the maximal cardinality of an -separated set in X. And then choose N ∈ N sufficiently large such that the following three conditions hold.
(C2) For all n ≥ N , |Fn gFn| |Fn| < γ kM L|F | , where F is the finite subset of G associated with γ in the definition of weak specification of (X, G).
We note that (C2) can be satisfied since {F n } is a Følner sequence, (C3) holds by the ergodic theorem and (C4) holds by Theorem 3.1.
By Proposition 2.3, there exists N 1 > N large enough, such that for any n ≥ N 1 , F n can be γ kM L|F | -quasi tiled by {F n 1 , F n 2 , · · · , F n l } with tiling centers {C 1 , C 2 , · · · , C l } and all n j 's are no less than N . We may require N 1 large enough to ensure that the family of translations F = {F n j c j : 1 ≤ j ≤ l, c j ∈ C j } can be partitioned into k Hence whence γ is sufficiently small.
Claim. For each F n j c j there exists a subset T c j of F n j such that Proof of the Claim. To obtain T c j 's, we first notice that since the translations in F are γ |F | -disjoint, there must exists for each F n j c j a subsetT c j ⊆ F n j with such thatT c j c j 's are pairwise disjoint. We then let T c j = {t ∈ F n j : F t ⊆T c j } ∩T c j . Obviously T c j ⊆ F n j and T c j c j 's are pairwise disjoint. Moreover, F T c j ⊆T c j implies that F T c j c j 's are also pairwise disjoint. From the construction of T c j , Thus which implies that This finishes the proof of the claim.
We should mention here that C j 's may not be pairwise disjoint, hence T c j may not be associated with a unique index c j . But if we replace the index c j by a pair (c j , j), this situation can be avoided.
Denote byF = {T c j c j : F n j c j ∈ F} andF i = {T c j c j : F n j c j ∈ F i }. Then Since λ i (E D ) > 1 2 , by (C4), there exist at least exp(|F n j |(h λ i − γ)) -many (F n j , 4 )separated points in E D . And hence at least exp(|Fn j |(h λ i −γ)) L |Fn j \Tn j | -many (T n j , 4 )-separated points in E D . Denote the collection of these separated points byẼ D . Note that for any two different elements inF, say D 1 and D 2 , we have that F D 1 ∩ D 2 = ∅. We then can apply the weak specification property. To each tuple (x D ) D∈F ∈ XF with x D ∈ c −1 j E i,j for D = T c j c j ⊂ F n j c j , there is a shadowing point y ∈ X such that d(gy, gx D ) < , for any g ∈ D. Then B ∪F (y, )'s are disjoint for different (x D )'s. This can be seen by the following. Let (x 1 D ) and (x 2 D ) be any two different such tuples with x 1 D = x 2 D for some D ∈F and let y 1 and y 2 be the corresponding shadowing points. Since x 1 D ∈ B D (y 1 , ) and This contradicts with the fact that c j x 1 D = c j x 2 D ∈Ẽ D . Note that ∪F ⊂ F n , hence B Fn (y, )'s are also disjoint for different (x D )'s. Now let us give a lower bound for the total number Q of different (x D )'s. Since for each D = T c j c j ∈F i , the total number of choices of x D , denoted by Q D , is at least exp(|Fn j |(h λ i −γ)) L |Fn j \Tn j | , we have (5.1)).
For the first term of the righthand side of the above inequation, we have ≥ |F n | φ dλ − |F n |5γ (by (5.2)).
Thus A Fn φ(z) > c, which can deduce that B Fn (y, ) ⊆ V Fn .