THE KATOK’S ENTROPY FORMULA FOR AMENABLE GROUP ACTIONS

. In this paper we generalize Katok’s entropy formula to a large class of inﬁnite countably amenable group actions.


Introduction. Topological entropy was introduced in [1] by Adler, Konheim and McAndrew who formulated it in terms of open covers, in analogy with the
Kolmogorov-Sinai measure entropy picture. Here too topological entropy turned out to be a very basic invariant with many applications. Topological entropy is related to measure entropy by the variational principle which asserts that for a continuous map on a compact metric space the topological entropy equals the supremum of the measure entropy taken over all the invariant probability measures. For other related work, see [2,4,5,7,8,9,12,15,16].
Definitions of topological entropy based on separated and spanning sets with respect to a metric were given independently by Bowen [3] and Dinaburg [6]. In [10], for Z-action dynamical system, Katok introduce an entropy with respect to Borel probability T -invariant ergodic measure µ by a similar manner. The metric entropy turns out to be the asymptotic value of the same kind with some subsets of positive measure instead of the whole space X. The Katok's entropy has the advantage to simplify some computations and has many applications in diffeomorphic dynamical systems.
In order to state Katok's result we proceed to define the relevant quantities. Let (X, d) be a compact metric space and T : X → X a homeomorphism of X, and d n an increasing system of metrics on X defined by: Denote by M T (X) the space of T -invariant probability measures on X and B(X) the set of all Borel subsets of X. Let > 0. A set Z ⊆ X is called (d n , )-spanning if for any x ∈ X there exists some z ∈ Z with d n (x, z) < . Denote by span(X, d n , ) the minimal cardinality of (d n , )-spanning subsets of X.
In [10] Katok obtained an entropy formula as follows: In this paper, we shall discuss extensions of Katok's entropy formula to the class of infinitely countable discrete amenable groups. The class of amenable groups includes all finite groups, solvable groups and compact groups, and actions of these groups on a compact metric space are a natural extension of the Z-actions considered in Katok's theorem: the foundations of the theory of amenable group actions were laid by Ornstein and Weiss in their pioneering paper [14].
Lindenstrauss [13] established the Shannon-McMillan-Breiman theorem for countably amenable group actions along tempered Følner sequences (with some mild growth conditions). Based on Lindenstrauss's result, we obtain the extension of Katok's entropy formula to a dynamic system of a countable discrete amenable group actions.
Main Theorem. Let G be a countably infinite amenable group. Let G X be a continuous action on the compact metric space (X, d) and µ be an ergodic and G-invariant Borel probability measure. For every 0 < δ < 1, one has For the definitions of imsup and liminf in Theorem 1.1, please refer to (1) and (2) in Section 2. For the definition of the number N µ (F, , δ), please refer to (6) in Section 3.
2. Amenable group. In this section, we give some basic properties of countably amenable group.
Let G be a countable discrete group. Denote by F (G) the set of all nonempty finite subsets of G. For K ∈ F (G) and δ > 0 write B(K, δ) for the collection of all The collection of pairs (K, δ) forms a net Λ where (K , δ ) (K, δ) means K ⊇ K and δ ≤ δ. For an R-valued function ϕ defined on F (G), we define and From the definition of the partial order ' ' it is clear that Furthermore, we state the Følner property of the infinitely countable amenable group which is one of fundamental characterizations of amenability. The Følner property is useful for exhibiting amenable groups which are not locally finite.
A key point in proving pointwise convergence results for amenable group action is working with appropriate Følner sequences. Now we introduce the following condition introduced by A. Shulman. Definition 2.2. A sequence {F n } of nonempty finite subsets of G is said to be tempered if there is a b > 0 such that | n−1 k=1 F −1 k F n | ≤ b|F n | for every n > 1. In this paper, we need the following proposition which is stated in [13]. Proposition 1. Every Følner sequence {F n } has a tempered subsequence. In particular, every amenable group has a tempered Følner sequence.
Lindenstrauss [13] established the Shannon-McMillan-Breiman theorem for countably amenable group actions along tempered Følner sequences {F n } with |F n |/ log n → ∞. Such tempered Følner sequences with that mild growth condition always exist, as one can start with any Følner sequence and recursively construct a subsequence with these properties. For completeness, we will illustrate the existence of the tempered Følner sequences {F n } with |F n |/ log n → ∞ for an infinitely countable amenable group G in the next few paragraphs. Definition 2.3. Let K, F be nonempty finite subsets of G and δ > 0. We say that Remark 1. The above definition is a less intuitive formulation of approximate invariance than the one implicitly expressed by Definition 2.1. More precisely, the sequence {F n } is Følner if and only if for every finite set K ⊆ G and δ > 0 there is an N ∈ N such that F n is (K, δ)-invariant for all n ≥ N (see [11] p.92-93).
With above definitions and arguments, we have the following results. Proof. Otherwise, there is a constant M > 0 such that |F n | ≤ M for all n ∈ N. Since G is infinite, we can choose a nonempty finite subset K of G with |K| > M +1. Thus, for any n ∈ N, one has {s ∈ F n : Ks ⊆ F n } = ∅.
Therefore, for every n ∈ N, F n is NOT (K, 1/2)-invariant which contradicts the conclusion in Remark 1.
Hence Fact 2.4 is obtained.
Proposition 2. Let G be an infinitely countable amenable group G. Then every Følner sequence of G has a tempered Følner subsequence {F n } with |F n |/ log n → ∞.
Proof. Let {F n } be a Følner sequence of G. From Proposition 1, we may assume that the Følner sequence {F n } is tempered. By Fact 2.4, we know that lim n→∞ |F n | = +∞. Thus, for each n ∈ N, there exist positive integer m n with m n > m n−1 (m 0 = 0) such that |F mn |/ log n > n. Set F n = F mn . Then the sequence {F n } is our desired.
Let G be a countably infinite amenable group. Thus we can find a family of finite subsets {G n } ∞ n=1 of G which satisfies that e ∈ G 1 ⊆ G 2 ⊆ · · · and G = ∞ n=1 G n . For the sake of our following proofs, we present some simple facts and give the proofs for completeness here.
Proof. We only to prove the first equation. The proof of the second equation is similar.
From (3) in this section, it follows that Thus, one has lim sup From the arbitrariness of (K, δ) we get Hence the first equation is proved.
Fact 2.6. Let G be a countably infinite amenable group and ϕ an R-valued function defined on F (G). Then there exist two Følner sequences {F m } and {F m } such that Proof. We only to prove the first equation. By Fact 2.5, we have We may assume that lim sup F is finite. For any m ∈ N, there exists n m ∈ N such that Hence the fact is proved.
3. The shannon entropy for the amenable group action. Let (X, d) be a compact metric space and G be a countably amenable group with the identity element e. Throughout G X is a continuous action on a compact metric space (X, d).
We write M (X) for the set of Borel probability measures on X, which is a weak * compact subset of the dual space C(X) * . We write M G (X) for the weak * closed set of G-invariant measures in M (X).
Let µ ∈ M (X) be a probability measure and P = {A 1 , · · · , A m } a finite partition of X. The information function of P is defined as The (Shannon) entropy of P is then defined as We also give a condition version of Shannon entropy with respect to a second finite partition Q = {B 1 , · · · , B l }. The conditional information function is defined as The Conditional (Shannon) entropy of P given Q is then defined by

XIAOJUN HUANG, JINSONG LIU AND CHANGRONG ZHU
Proposition 3. Let P, Q be finite partitions of X, and let µ ∈ M (X) be a probability measure. Then Notation. For a finite partition P of X and a empty finite set F ⊆ G, we write P F for the join s∈F s −1 P, unless F is empty in which case we interpret this as the trivial partition {X}.
Let µ ∈ M G (X). For each finite partition P of X the function F → H P F on the collection of finite subsets of G satisfies two conditions: G-invariant and Shearer's inequality. Thus where F ranges over nonempty finite subsets of G and {F n } is a Følner sequence of G.
Definition 3.1. Let µ ∈ M G (X). For a finite partition P of X we define h µ (P) to be the above limit. The entropy of the action G (X, µ) is then defined as Let P = {A 1 , · · · , A m } a finite partition of X. We denote by where diam(A i ) denotes the diameter of the set A i and ∂(A i ) denote the boundary of the set A i . In order to prove our result, we need the following lemmas. For the detailed proofs of these lemmas, please refer to [11].
Lemma 3.2. Let (X, d) be a compact metric space, µ ∈ M (X) and κ > 0. Then there is a finite partition of X whose member all have diameter less than κ and boundary of µ-measure zero. Lemma 3.3. Let P be a finite partition of X and let > 0. Then there exists a η > 0 such that, for every finite partition Q of X with the property that for all A ∈ P there is a set B in the σ-algebra generated by Q satisfying µ(A B) < η, one has H(P|Q) < . where The conclusion of the following lemma is well known, we give a proof here for completeness.
Proof. Let > 0. By the definition of h µ (X, G), there is a finite partition Q = {A 1 , · · · , A r } of X such that µ(A i ) > 0(i = 1, · · · , r) and Let 0 < η = η( /2) < be as in Lemma 3.3. Since every probability measure on the compact metric space is regular, there exist the compact sets Denote by By Lemma 3.2, there is a finite partition P of X such that diam(P) < κ/2 and µ ∂(P) = 0. For 1 ≤ i < r, we define Thus, we get a finite partition P * = {B 1 , · · · , B r }. It is clear that P * ≤ P.
Suppose that 1 ≤ i < r. Due to K i ⊆ A i , we have x ∈ K i . Assume that x ∈ p for some p ∈ P with p ∩ K i = ∅. If x ∈ K j for some j = i, then p ∩ K j = ∅. Thus one has dist(K i , K j ) ≤diam(p) < κ/2 which contradicts that dist(K i , K j ) ≥ κ. Hence x ∈ X\ r t=1 K t . Suppose that x ∈ B r \A r . Since K r ⊆ A r , it follows that x ∈ K r . Combing with the assumption x ∈ B r and B r = X\ Hence the claim is obtained. For 1 ≤ i ≤ r, noting K i ⊆ B i and Claim, it is not hard to see that
Definition 3.6. Let P be a finite partition of X and F be a nonempty finite subset of G. We define the Hamming pseudometric H P,F (·, ·) on X as follows: For any x, y ∈ X, we denote by F x,y = {s ∈ F : the points of sx and sy are NOT in the same member of P}.
The Hamming pseudometric H P,F (x, y) is then defined as Notation. Let P be a finite partition of X. For x ∈ X we write P(x) for the member of P which contains x.
Actually, we have an alternative view on the quality H P,F . Let x, y ∈ X. where ϕ, ψ : F → P are two maps from the finite set F into the finite set P. Thus it follows that F x,y = {s ∈ F : ϕ(s) = ψ(s)}.
Denote by B H P,F (x, ) the ball of radius in the pseudometric H P,F around x. It is not hard to see that B H P,F (x, ) is the union of some members of P F . Denote by P F x, = c ∈ P F : c ⊆ B H P,F (x, ) . Lemma 3.7. For x ∈ X and 0 < < 1/2, one has The assumption c ∈ P F x, implies that H P,F (x, y) < for any y ∈ c, i.e.
We define a set Map P,F which consists of some maps from F into P as follows: .
It is clear that Let L be the collection of subsets of F with the cardinality at most |F |, that is, It is easy to see that For each D ∈ L , we write Map D for the collection of some maps from F into P as follows: Thus, the inequality (4) implies that Hence, by a simple computation, we have Hence, the lemma is proved.
The following theorem is the classical Shannon-McMillan-Breiman theorem for amenable groups. For the details, please refer to [13].  X, B, µ). Let P be a finite partition of X, and {F n } be a tempered Følner sequence for G with |F n |/ log n → ∞. Then pointwise a.e. and in L 1 . For every > 0 we denote by B d F x, the open Bowen ball of radius in the metric d F around x, i.e.,

XIAOJUN HUANG, JINSONG LIU AND CHANGRONG ZHU
Let 0 < δ < 1, > 0 and µ ∈ M (X). We define the quantity N µ (F, , δ) as follows: Therefore, according to the notations of Section 2, we have the following two quantities: Now we prove the following theorem which implies our main result.
Theorem 3.12. Let G be a countably infinite amenable group and {F n } be a tempered Følner sequence for G with |F n |/ log n → ∞. Let G X be a continuous action on the compact metric space (X, d) and µ be an ergodic and G-invariant Borel probability measure. For every 0 < δ < 1, one has Proof. We divide our proof into two steps.
Step 1. We show that the quantity in the right part of the formula does not exceed h µ (X, G).
To prove this it is enough to show that: for every > 0 and 1 > δ > 0. Let 0 < δ < 1 and > 0. Let P be a finite partition of X with diam(P) < /2. By SMB Theorem and Egorov Theorem, we can deduce that there is a Borel set S 0 ⊆ X such that µ(S 0 ) > 1 − δ and So there is N 0 ∈ N such that, for all n ≥ N 0 and all x ∈ S 0 , one has µ P Fn (x) > exp(−|F n |(h µ (P) + )).
For each n ≥ N 0 , we denote that Thus the set A n is the union of some members of P Fn and µ(c) > exp(−|F n |(h µ (P) + )) for each c ∈ L n .
For each member p ∈ P Fn , we choose a point x p ∈ p. Since diam(P) < /2, by a simple computation, it is not hard to see that p is contained in a d Fn -ball center at x p and radius , i.e.
p ⊆ B d Fn (x p , ). Thus the set A n (or the set S 0 ) can be covered by at most |L n | Bowen balls with radius in the metric d Fn . Hence, Since can be taken arbitrarily small and h µ (P) ≤ h µ (X, G), we obtain the conclusion of Step 1.
Let us denote for brevity the characteristic function of the set U γ (P) by χ γ and let Claim 2. Suppose that x, y ∈ X and d Fn (x, y) < γ < . Then, for any s ∈ F n , either sx and sy belong to the same member of P or both of them belong to the set U γ (P).
If the pair points s x, s y ∈ X are in the deferent members of P for some s ∈ F n , then we claim that the two point are both in U γ (P).
Suppose that the pair points s x and s y are NOT in the same member of P, that is, s x ∈ p and s y ∈ X\p for some p ∈ P.
Note that the assumption d Fn (x, y) < γ implies that d(s x, s y) < γ . If s x ∈ U γ (P), then s x ∈ U γ (p) which implies that dist(s x, X\p) ≥ γ . Noting s y ∈ X\p, it follows that d(s x, s y) ≥ γ which contradicts that d(s x, s y) < γ . So s x ∈ U γ (P). With the same argument we can get s y ∈ U γ (P). So both of the points s x and s y belong to the set U γ (P). Hence Claim 2 is obtained.
Since µ ∈ M G (X) is a G-invariant measure, we have X χ(x)dµ = X χ(gx)dµ for any g ∈ G.
It is clear that F n = W n 1 W n 2 . Since x ∈ D Fn, , it follows that which implies that Let y ∈ B d Fn (x, γ ) and g ∈ F n . By Claim 2, we know that gx and gy belong to the same member of P or both of them belong to the set U γ (P). If g ∈ W n 2 (i.e. gx is not in U γ (P)), the above conclusion implies that both of the points gx, gy are in the same member of P. Thus we get |{s ∈ F n : sx and sy are NOT in the same member of P}| ≤ |W n 1 |. It follows that H P,Fn (x, y) = 1 |F n | |{s ∈ F n : sx and sy are NOT in the same member of P}| Hence Claim 3 is proved.
From the definition of the spanning set we know that the balls B d Fn (y, γ ) y∈En can cover the set D Fn,γ ∩ S n . By Claim 3, we know that the Hamming metric balls B H P,Fn (y, ) y∈En cover the set D Fn,γ ∩ S n . Denote by U n = c ∈ P Fn : c ⊆ B H P,Fn (y, ) for some y ∈ E n .
Thus one has D Fn,γ ∩ S n ⊆ c∈Un c.
By SMB Theorem and Egorov Theorem, we know that there is a Borel set T ⊆ X such that µ(T ) > 1+3δ 4 and − 1 |F n | log µ P Fn (x) ⇒ h µ (P) uniformly on T.
So there is N 0 ∈ N such that, for all n ≥ N 0 and all x ∈ T , one has µ P Fn (x) < exp(−|F n |(h µ (P) − )).
Combing ( Hence Main Theorem is proved.