c Journal of Dynamical and Control Systems, Vol. 13, No. 3, July 2007, 363–371 (2007) DOI: 10.1007/s10883-007-9019-3
A BREIMAN TYPE THEOREM FOR GIBBS MEASURES B. M. GUREVICH and A. A. TEMPELMAN Abstract. For a translation invariant Gibbs measure on a suitable translation invariant configuration set X ⊂ S G , where G is an amenable group and S is a finite set, we prove the convergence of the Shannon–McMillan–Breiman ratio on a specific subset of “generic” configurations. Provided that the above Gibbs measure exists, we also prove the convergence in the definition of the pressure and the fact that this Gibbs measure is an equilibrium measure.
1. Introduction. Let G be a countable amenable group with the unit e, and let S be a finite set. The following notation will be used throughout the paper: S G is the set of S-valued functions (configurations) on G endowed with the product topology; τ = {τg , g ∈ G} is the group of translations τg : S G → S G defined by (τg x)(g ) = x(g g), g, g ∈ G; X is a τ -invariant compact subset of S G ; I(X) is the set of all τ -invariant Borel probability measures on X; E(X) is the set of all ergodic measures in I(X). We denote by |A| the cardinality of A and set A−1 = {a−1 , a ∈ A}, Ac = T \ A, AB = {ab, a ∈ A, b ∈ B}, where A, B ⊂ G. Let {Tn } be a sequence of finite subsets of G. It is called a Følner sequence if, for each g ∈ G, |Tn |−1 |gTn Tn | → 0 as n → ∞, and it is n Tk called a regular sequence if sup |Tn |−1 |T˜−1 Tn | < ∞, where T˜n = n
n∈N
k=1
−1 (see [15]). A weaker condition, namely: sup |Tn |−1 |T˜n−1 Tn | < ∞, was n∈N
introduced in [12, 13]; following [7], we call {Tn } satisfying this condition a tempered sequence. Every countable amenable group contains tempered Følner sequences (see [7, 8]). For V ⊂ G, x ∈ X, we denote by xV the restriction of x to V , and let XV = {xV : x ∈ X},
CV (x) = {y ∈ X : yV = xV }.
2000 Mathematics Subject Classification. Primary 28D15, 28D20, 37A15, 37A30. Secondary 60G60, 82B20. Key words and phrases. Amenable group, Breiman theorem, Gibbs measure, Følner sequence, tempered sequence. The work of the first author is partially supported by the Russian Foundation for Basic Research (project No. 05-00-00449).
363 c 2007 Springer Science+Business Media, Inc. 1079-2724/07/0700-0363/0
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The classical Shannon–McMillan–Breiman theorem for G = Z says that if ν ∈ E(S G ) and Tn = [0, n − 1] ∩ Z, then lim −|Tn |−1 ln ν(CTn (x)) = h(ν), (1) n→∞
where h(ν) is the entropy of ν (the convergence is in L1 (ν) and a.e.). The process of generalizing the Breiman part of this theorem, the a.e. convergence (see [9, 10]), was culminated in [8], where (1) was proved for every amenable group G and every tempered Følner sequence {Tn } such that |Tn | ≥ n. The Breiman theorem has many applications, and for some of them one cannot be content with the fact that the convergence set has a full measure, but needs a specific information on this set. In the present paper we show, in particular, that such additional information can be obtained for a Gibbs measure (see statement 3 of Theorem 1 below). This refinement of the Breiman theorem even though is rather simple, is of great importance, for example, for evaluating the Hausdorff dimension of the set, where the Birkhoff average of a continuous function on X takes a given value (see [5] for the particular case G = Zd , X = S G ). Let us note that the first generalization of the Breiman theorem (for G = Z d ) was made in [1] precisely for Gibbs measures. According to [14] (see also [15, Theorem 8.7.3]), if ν is a τ -invariant Gibbs measure and μ ∈ E(S G ), then the limit in (1) exists μ-a.e. for every regular Følner sequence {Tn }; this limit is equal to h(ν) if μ = ν. 2. Preliminaries. Proceeding to precise statements, we start from some notions and facts from the Dobrushin–Lanford–Ruelle thermodynamic formalism (see, e.g., [2, 11]) that are used in the sequel. Let F be the family of all finite subsets of G and let F(V ) = {U : U ∈ F, U ∩ V = ∅} and F(g) = {U : U ∈ F, g ∈ U }. Fix a τ -invariant closed subset X ⊂ S G . A function Φ : F × X → R is called a potential if (a) Φ(A, x) = Φ(A, y) whenever xA = yA ; (b) Φ(Ag, x) = Φ(A, τg x), x ∈ X, g ∈ G, A ∈ F; (c) the norm Φ := sup |Φ(A, x)| A∈F (e)
(2)
x∈X
is finite. Let us fix a potential Φ. For a set A ∈ F and configurations x, y ∈ X, we denote by xA ∨ yAc the configuration z ∈ S G such that zA = xA , zAc = yAc . The energy of the configuration xA , given the configuration yAc , is defined
A BREIMAN TYPE THEOREM FOR GIBBS MEASURES
as follows: E(xA |yAc ) :=
⎧ ⎨ B∈F (A)
Φ(B, xA ∨ yAc ) if xA ∨ yAc ∈ X,
⎩ +∞
365
(3)
otherwise.
We also consider the energy E(xA ) :=
Φ(B, x)
B⊂A
of xA , x ∈ X, and the local energy ϕ(x) := |V |−1 Φ(V, x),
x ∈ X.
(4)
V ∈F (e)
The partition function Z(A|yAc ), A ∈ F, with the boundary condition yAc and the partition function with the empty boundary condition are defined, respectively, as follows: Z(A|yAc ) := exp[−E(z|yAc )], (5) z∈XA
Z 0 (A) :=
exp[−E(z)],
(6)
z∈XA
A generalized version of the van Hove theorem (see [14] and [15, Theorem 8.7.2]) claims that for X = S G and every Følner sequence {Tn }, the limit P (Φ) := lim |Tn |−1 log Z 0 (Tn ) (7) n→∞
exists and does not depend on {Tn }; it is called the pressure corresponding to Φ. For a more general case X ⊂ S G , see Theorem 1 below. A measure ν ∈ I(X) is said to be a Gibbs measure with the potential Φ if, for every V ∈ F and x ∈ X, the function y → γV,x , where γV,x (y) =
exp[−E(xV |yV c )] , Z(V |yV c )
y ∈ X,
is a version of the conditional probability ν(CV (x)|BV c ) of the cylinder CV (x) with respect to the σ-algebra BV c generated by the cylinders CW , where W ⊂ V c . We denote by G(X, Φ) the set of all such measures. Let us note that an arbitrary τ -invariant compact subset X ⊂ S G by no means supports a Gibbs measure. A sufficient condition for such a measure to exist is that X is a Markov set (see [11, Chap. 1]). We say that a τ -invariant set X is a Markov set if there are M ∈ F and Y ∈ S M such that X = {x ∈ S T : (τg x)M ∈ Y for all g ∈ G}. It is clear that every Markov set is compact and that the whole space S G is Markov.
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3. Statement of main results. In the sequel, we have to impose the following condition (see [11]) on X and {Tn }. (D) For each n ∈ N, there exists a set Tˆn such that Tn ⊂ Tˆn , lim |Tn |/|Tˆn | = 1, and for every pair x, y ∈ X, there exists z ∈ X n→∞
such that zTn = xTn , zTˆc = yTˆc . n
n
Remark 1. Obviously, condition (D) is satisfied if X = S G and {Tn } is an arbitrary Følner sequence. It is also satisfied if G = Z, X is Markov, τ is topologically mixing on X (i.e., there exists n ∈ N such that C[k,l] (x) ∩ C[k+r,l+r](y)=∅ for all x, y ∈ X, k ≤ l, r ≥ n), and if each Tn is an interval in Z. Another simple example is the Markov set consisting of all {0, 1}configurations on G = Zd , d > 1, that contain no pair of neighboring ones. Now it is possible to state our main results. For a Følner sequence T = {Tn } and a function f on X, we denote by Xf,T the set of points x ∈ X such that there exists limit lim |Tn |−1 f (τg x) =: f˜T (x). (8) n→∞
g∈Tn
Theorem 1. Let X be a compact τ -invariant subset of S G , T = {Tn } be a Følner sequence such that condition (D) is satisfied, and let Φ be a potential (defined on F × X) such that G(X, Φ) = ∅. Then 1. there exists a limit in (7) that does not depend on the choice of T , and, therefore, the pressure P (Φ) is well defined by (7); 2. for each ν ∈ G(X, Φ), we have
P (Φ) = sup h(μ) − ϕdμ = h(ν) − ϕdν, μ∈I(X)
i.e., ν is an equilibrium measure with respect to the potential Φ; 3. for each x ∈ Xϕ,T and each ν ∈ G(X, Φ), lim −|Tn |−1 ln ν(Cn (x)) = ϕ˜T (x) + P (Φ). n→∞
(9)
Remark 2. If X is an arbitrary compact τ -invariant subset of S G , it is unknown, to our knowledge, whether there exists a Følner sequence for which the limit in (7) exists. Analogously, in the general case it is unknown whether there exists a Φ-equilibrium measure. Some sufficient conditions for both facts to take place are given in [3, 14, 15]. In particular, these facts hold for X = S G . Items 1 and 2 of Theorem 1 can be considered as a generalization of this result. Remark 3. It is not difficult to show that, under the assumptions of Theorem 1, uniformly in y ∈ X, lim |Tˆn |−1 ln Z(Tˆn |y ˆc ) = P (Φ), n→∞
Tn
A BREIMAN TYPE THEOREM FOR GIBBS MEASURES
367
where Tˆn is the set mentioned in condition (D). 4. The proof. We begin with two auxiliary statements. For a sequence of sets Tn ∈ F and for a potential Φ, we set bn (x) = |Φ(V, x)|, x ∈ X, n = 1, 2, . . . .
(10)
g∈Tn V ∈F (g)∩F (Tnc )
Lemma 1. If {Tn } is a Følner sequence, then lim |Tn |−1 sup bn (x) = 0.
n→∞
Proof. Let
a(W ) = sup x∈X
(11)
x∈X
|Φ(U, x)|,
W ∈ F.
U ∈F (e)∩F (W c )
Since the series in (2) converges, we have sup a(W ) < ∞. Moreover, given W ∈F
We denote ε > 0, there Wε ∈ F such that a(W
exists ) < ε for−1all W ⊃ Wε . −1 w−1 Tn . Then Tn \ Dn = (Tn \ w Tn ) and Tn g ⊃ Wε if Dn = w∈Wε
w∈Wε
g ∈ Dn . Using these facts and the invariance of the potential Φ, we obtain, for every x ∈ X, |Φ(V, x)| bn (x) = =
g∈Tn V :V g −1 ∈F (e)∩F (Tnc g −1 )
|Φ(U g, x)| =
g∈Tn U ∈F (e)∩F (Tnc g −1 )
≤
g∈Tn
a(Tn g −1 ) =
|Φ(U, τg x)|
g∈Tn U ∈F (e)∩F ((Tn g −1 )c )
a(Tn g −1 ) +
g∈Tn ∩Dn
a(Tn g −1 )
g∈Tn \Dn
≤ ε|Tn | + sup a(W ) W ∈F
|Tn (w−1 Tn )|.
w∈Wε
Taking into account the Følner property of {Tn }, we make sure that the right-hand side of the last inequality is less than 2ε|Tn | for sufficiently large n. The following lemma is, in fact, not new (see, e.g., [14, 15]); we present a simplified proof. Lemma 2. If {Tn } is a Følner sequence, then, uniformly in x ∈ X, 1 E(xTn |xTnc ) − E(xTn ) = 0, lim n→∞ |Tn | 1 lim ϕ(τt x) = 0. E(xTn |xTnc ) − n→∞ |Tn | t∈Tn
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Proof. From (3) and (4) we obtain, respectively, E(xTn |xTnc ) = E(xTn ) + V
and
ϕ(τg x) =
g∈Tn
=
|V |−1 Φ(V, τg x) =
g∈Tn V ∈F (e)
−1
|V |
g∈Tn V :g∈V ⊂Tn
=
|V |−1
V ⊂Tn
where Sn (x) =
Φ(V, x) +
Φ(V, x),
∈F (Tn )∩F (Tnc )
|V |−1 Φ(V t, x)
g∈Tn V ∈F (e)
|V |−1 Φ(V, x)
g∈Tn V ∈F (g)∩F (Tnc )
Φ(V, x) + Sn (x) = E(xTn ) + Sn (x),
g∈V
|V |−1 Φ(V, x).
g∈Tn V ∈F (g)∩F (Tnc )
Obviously (see (10)),
≤ bn (x). Φ(V, x) V ∈F (Tn )∩F (Tnc )
|Sn (x)| ≤ bn (x), Hence
sup E(xTn |xTnc ) − E(xTn ) ≤ sup bn (x)
x∈X
and
x∈X
sup E(xTn |xTnc ) − ϕ(τt x) ≤ 2 sup bn (x).
x∈X
t∈Tn
x∈X
It remains to apply Lemma 1. Let X and {Tn } satisfy condition (D). Then for every Gibbs measure ν ∈ G(X, Φ), there exists a sequence of positive numbers rn such that rn /|Tn | → 0 as n → ∞ and e−rn ≤ ν(CTn (x))ZT0n exp[E(xTn )] ≤ ern ,
x ∈ X.
(12)
This fact is established in the proof of [3, Theorem 2.1]. One can observe that there are conditions (A)–(D) in the statement of this theorem. But the inspection of the proof immediately shows that conditions (A) and (C) are not used for proving (12), while condition (B) is automatically fulfilled in our case (see [3, Lemma 4.1]). By (12), for all x ∈ X, |Tn |−1 [−E(xTn ) − ln ν(CTn (x)) − rn ] ≤ |Tn |−1 ln Z 0 (Tn ) ≤ |Tn |−1 [−E(xTn ) − ln ν(CTn (x)) + rn ].
(13)
A BREIMAN TYPE THEOREM FOR GIBBS MEASURES
369
The integration with respect to ν yields
−1 |Tn | − E(xTn )ν(dx) + HTn (ν) + rn ≤ |Tn |−1 ln Z 0 (Tn )
−1 ≤ |Tn | − E(xTn )ν(dx) + HTn (ν) − rn , (14) where HTn (ν) is the entropy of the partition constituted by the cylinders of the form CTn (x), x ∈ X. Taking into account Lemma 2 and the converwe see that both the first and third gence of HTn (ν)/|Tn | to h(ν) (see [6]), expressions in (14) tend to h(ν) − ϕdν. Hence (15) lim |Tn−1 | ln Z 0 (Tn ) = h(ν) − ϕdν n→∞
and, therefore, the limit does not depend on T . Hence P (Φ) is well defined by (7), and we have P (Φ) = h(ν) − ϕdν. Let us now transform (12) as follows: |Tn |−1 E(xTn ) + ln Z 0 (Tn ) − rn ≤ −|Tn |−1 ln ν(CTn (x)) ≤ |Tn |−1 E(xTn ) + ln Z 0 (Tn ) + rn . For x ∈ Xϕ,T , we immediately obtain (9) from (15) and Lemma 2. The theorem is proved. 5. Corollaries. For a measure μ ∈ I(X), we set ⎧ ⎫ ⎨ ⎬ Xϕ,T ,μ = x ∈ X : lim |Tn |−1 ϕ(τg x) = ϕdμ . n→∞ ⎩ ⎭ g∈Tn
Corollary 1. Under the conditions of the proved theorem, for every μ ∈ I(X), ν ∈ G(X, Φ), and x ∈ Xϕ,T ,μ ,
1 log ν(Cn (x)) = ϕdμ + P (Φ) =: h(ν, μ). lim − (16) n→∞ |Tn | Remark 4. If X = S T , then by the variational principle (see [14, 15]), h(ν, μ) ≥ h(μ) and h(ν, μ) = h(μ) if and only if μ ∈ G(X, Φ). If {Tn } is a tempered Følner sequence and μ ∈ E(X), then by the pointwise ergodic theorem ( [12, 13]; see also [7, 8, 15]) we have μ(Xϕ,T ,μ ) = 1, and, therefore, Corollary 1 implies the following statement. Corollary 2. If {Tn } is a tempered Følner sequence and if the conditions of Theorem 1 are satisfied, then for every μ ∈ E(X) and every ν ∈ G(X, Φ), (16) holds μ-a.e. Acknowledgment. The authors are grateful to S. A. Pirogov for a number of useful remarks.
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(Received June 15 2006)
A BREIMAN TYPE THEOREM FOR GIBBS MEASURES
Authors’ addresses: B. M. Gurevich Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia E-mail:
[email protected] A. A. Tempelman Departments of Mathematics and Statistics, The Pennsylvania State University, University Park, PA 16802, USA E-mail:
[email protected] 371
