A Billingsley-type theorem for the pressure of an action of an amenable group

. This paper extends the deﬁnition of Bowen topological entropy of subsets to Pesin-Pitskel topological pressure for the continuous action of amenable groups on a compact metric space. We introduce the local measure theoretic pressure of subsets and investigate the relation between local measure theoretic pressure of Borel probability measures and Pesin-Pitskel topological pressure on an arbitrary subset of a compact metric space.


1.
Introduction. Entropy is undoubtedly among the most essential characteristics of dynamical systems. The classical measure-theoretic entropy for an invariant measure and the topological entropy were introduced in [13] and [1] respectively. The basic relation between measure theoretic entropy and topological entropy is the variational principle [10,9]. Since then the subjects involving definition of new measure-theoretic and topological notions of entropy and studying the relation between them have gained a lot of attention in the study of dynamical system.
Topological pressure is a nontrivial and natural generalization of topological entropy. One of the most fundamental dynamical invariants that associate to a continuous map is the topological pressure with a potential function. It roughly measures the orbit complexity of the iterated map on the potential function. The notion of topological pressure was brought to the theory of dynamical systems by Ruelle [22] and Walters [24]. Ruelle [21] introduced topological pressure of a continuous function for actions of the groups Z n on compact spaces in this context when the action is expansive and satisfies the specification condition. Later, the variational principle was formulated by Walters in [24].
The theory related to the topological pressure and variational principle plays a fundamental role in statistical mechanics, ergodic theory and dynamical systems [4,22,24,8,12]. Since the works of Bowen [5] and Ruelle [22], the topological pressure turned into a basic tool in the dimension theory related to dynamical systems. From a viewpoint of dimension theory, Pesin and Pitskel [20] introduced another way to define topological pressure for continuous functions on noncompact sets in the case of Z-actions, which was based on the Carathéodory structure [7], which we call the Pesin-Pitskel topological pressure. In [20], Pesin and Pitskel proved the variational principle under some supplementary conditions..
Brin and Katok showed in [6] the interrelations between a measure-theoretic entropy and dimension-like characteristics of smooth dynamical systems. Ma and Wen [16] applied a dimensional type characteristic of the entropy to obtain the following relation between local measure entropy and the dimensional type entropy: Theorem A.(Theorem 1 in [16]) Let f be a continuous map on a compact metric space X. Let µ be a Borel probability measure on X, E be a Borel subset of X and 0 < s < ∞. 1.
If h µ (f, x) ≤ s for all x ∈ E, then h(f, E) ≤ s.

2.
If h µ (f, x) ≥ s for all x ∈ E and µ(E) > 0, then h(f, E) ≥ s. Ma and Wen's result was an analogue of the Billingsley's Theorem [2]. In 2012, B. Liang and K. Yan [15] introduced the topological pressure for any sub-additive potentials of a countable discrete amenable group action and any given open cover, and established a local variational principle for it. In 2013, A. Bis [3] generalized the notion of local measure entropy for the case of a group or a pseudogroup of homeomorphism of a metric space. He obtained an analogue of the variational principle for group and pseudogroup actions. Later, Tang, Cheng and Zhao [23] proved that Bowen topological pressure is bounded by measure theoretic pressure of Borel probability measures, which extended the result in [16] for Bowen topological pressure of integer group action.
In this paper, we generalize Ma-Wen's result [16] to dynamical systems acting by a countable discrete amenable group. We define the Pesin-Pitskel topological pressure and the local measure theoretic pressure for amenable group action and establish an analogue of the Billingsley's Theorem between local measure theoretic pressure and the topological pressure.
We organize the paper as follows: we begin in Section 3 by setting up our Pesin-Pitskel topological pressure definition and giving some important properties for amenable group actions. In Section 4 we give the local measure theoretic pressure of arbitrary subsets for amenable group action and we prove that the Pesin-Pitskel topological pressure can be bounded by the local measure theoretic pressure for actions of amenable groups. Finally, in Section 5 we calculate the Pesin-Pitskel topological pressure of some subsets of Bernoulli shifts for amenable groups.
2. Preliminaries. In this section, we recall some basic properties of amenable groups.
Let G be a discrete infinitely countable group. Denote by F (G) the set of all nonempty finite subsets of G. For K ∈ F (G) and δ > 0, denote by B(K, δ) the set of all F ∈ F (G) satisfying |KF \F | < δ|F |. The group G is called amenable if B(K, δ) is nonempty for every (K, δ). This is equivalent to the existence of a sequence of nonempty finite subsets {F n } of G which are asymptotically invariant, i.e., lim n→∞ |F n ∆gF n | |F n | = 0 for all g ∈ G.
Such sequences are called Følner sequences. For details on actions of amenable groups, one may refer to Ornstein and Weisss pioneering paper [18] or Kerr and Li's book [12]. The collection of pairs Λ = {(K, δ) : K ∈ F (G), δ > 0} forms a net where (K , δ ) (K, δ) means K ⊇ K and δ ≤ δ. For an R-valued function ϕ defined on Finally, we state the Følner property of amenable groups which is one of fundamental characterizations of amenability. The Følner property is useful for exhibiting amenable groups which are not locally finite.
Definition 2.2. Let ψ be a real-valued function on the set of all nonempty finite subsets of G. We say that ψ(F ) converges to a limit L as F becomes more and more invariant if for every > 0 there are a nonempty finite set K ⊂ G and a δ > 0 such that |ψ(F ) − L| < for every F ∈ B(K, δ).
At the end of this section, we give some notations that will appear in the later paper.
In this paper, we always assume that the group G is a discrete infinitely countable amenable group and X is a compact metrizable space. Denote by G X a Gaction topological dynamical system. Meanwhile, we denote by M(X) the space of all Borel probability measures on X and by C(X) the Banach space of all continuous functions from X into R equipped with the supremum norm · .
3. Topological pressures of the subsets for the amenable group action.
3.1. Pesin-Pitskel topological pressure for subsets. Let G act on a compact metrizable space X continuously. Consider a finite open cover U of X. For F ∈ F (G), we call a map U : F → U a string of length m(U) = |F |. We denote by dom(U) the domain of the string U : F → U , i.e., dom(U) = F.
For a given string U we associate the set = {x ∈ X : gx ∈ U(g) for all g ∈ dom(U)}.

XIAOJUN HUANG, YUAN LIAN AND CHANGRONG ZHU
For each f ∈ C(X) and each string U, we define Given a set F ∈ F (G), we denote by W F (U ) the set of all strings U : Let δ > 0 and F ∈ F (G). For s ∈ R, we define where the infimum is taken over all collections of strings This quantity was first introduced by Ruelle in [22]. It is well known (see, e.g. ( [19], is an outer measure on X. Note that the quantity M s (K,δ) (G, U , f, Z) does not decrease as (K, δ) increases. Thus the limit of the net {M s (3.1) Similar to Proposition 1.2 (p.13) in [19], we now describe a crucial property of the function M (·) (G, U , f, Z) for a fixed set Z. For the completeness, we give a proof here. Proof. Let > 0 and M = M s (G, U , f, Z) + 1. So there is a positive real number 0 < δ 0 < 1 and a nonempty finite set K 0 ∈ F (G) with 2M (t−s)|K0| < which satisfy that, for each (K, δ) ∈ Λ with (K, δ) (K 0 , δ 0 ), there is a collection which covers Z and satisfies Note that (K, δ) (K 0 , δ 0 ) implies δ < 1 and Briefly, for a string U ∈ Γ (K,δ) , set F = dom(U). Then one has Thus, it follows that |K|/2 ≤ |KF |/2 ≤ |F | = m(U). So m(U) ≥ |K|/2 for each U ∈ Γ (K,δ) .
Therefore, for s < t, we have The above inequality leads to So the statement of (1) is proved. This contradiction gives M t (G, U , f, Z) = ∞ whenever M s (G, U , f, Z) > 0 and t < s.
Given a subset Z of X, by Fact 3.1, there exists a unique s such that if t > s.

So we define
2) Let d be a compatible metric on the compact metrizable space X and U be a finite open cover of X. Thus we can consider the diameter of the subset of X. Let diam(U ) = max{diam(U i ) : U i ∈ U } be the diameter of the cover U . So we define lim sup Proof. We only need to prove the first equality. By the symmetry of the compatible metrics and the equality (3.3), it is sufficient to prove that Let > 0. By Remark 3.2, we know that there is δ 0 > 0 such that, for any subset The arbitrariness of implies the inequality (3.5). Hence the fact is obtained.
According to the above definitions and arguments, we have the following limit existence result for dynamical systems acting by a countable discrete amenable group. Our result is a generalization of Theorem 11.1 (p.69) in [19] or Proposition 1 (p.309) in [20]. Proposition 3.4. Let X be a compact metrizable space and f ∈ C(X). Then the limit lim Proof. Let d be a compatible metric on X. It is clear that we only need to show lim sup Let U be a finite open cover of X and V be a finite open cover of X with diameter smaller than the Lebesgue number L(U ) of U . Thus V is finer than U . In what follows, we will prove that Let K ∈ F (G) and δ > 0.
Let Ω (K,δ) ⊆ F ∈B(K,δ) W F (V ) be any collection of strings which covers Z. Now, we construct a collection of strings Γ (K,δ) ⊆ F ∈B(K,δ) W F (U ) which covers Z.
For each string V ∈ Ω (K,δ) , we know that V is a mapping from F into V , where F = dom(V) ∈ B(K, δ). For every g ∈ F , V(g) is an open subset of X. Owing to V is finer than U , we can find an open subset U V (g) ∈ U which satisfies that V(g) ⊆ U V (g). Therefore, we get a string U V from F into U . Furthermore, it is clear that m(U V ) = |F | = m(V) and Consequently, we define a collection of strings as follows Then (3.9) implies that the collection Γ (K,δ) covers Z. We define Ω (K,δ) to be the set of V in Ω (K,δ) satisfying X(V) = ∅. Then we define Γ (K,δ) to be U V : V ∈ Ω (K,δ) . It is clear that the collection Γ (K,δ) also covers Z.
That is

XIAOJUN HUANG, YUAN LIAN AND CHANGRONG ZHU
Using the fact that m(U V ) = m(V), we have The arbitrariness of the collection Ω (K,δ) which covers Z implies that . Taking the limit for the net (K, δ) ∈ Λ, we get Combining with (3.8), it follows that Note that V is any finite open cover of X with diameter smaller than the Lebesgue number L(U ) of U . Therefore, one has Taking the limsup as the diameter of the open cover U tends to zero, we get that Since X is compact and f is uniformly continuous on X, it is easy to see that Due to Proposition 3.4, we can define the Pesin-Pitskel topological pressure of the action G X as follows: For f ∈ C(X) and each string U, we write For each string U, we choose any real number where the infimum is taken over all collections Similar to the definitions of (3.1) and (3.2), we can define For the relation between the numbers h P top G, U , f * , Z and h P top G, U , f, Z , we have the following theorem.
Proof. Let d be a compatible metric on X and U be a finite open cover of X.
According to the definitions, it is clear that In what follows, we will show that where We may assume that h P top G, U , f * , Z < ∞. Let s be any positive real number is any collection of the strings which covers Z. For each string U ∈ Ω (K,δ) , by the definition of string, it is clear that So, we deduce that Since X is compact and f is uniformly continuous on X, we have Taking the liminf as the diameter of the open cover U tends to zero, one has Therefore, combining the inequalities (3.13) and (3.16), we get that Hence the theorem is proved.
By the above theorem, we denote In order to get the following results, we need to define the infimum function f * as follows: For each string U, we define Corollary 3.6. For the above function f * , one has where V runs over all finite open covers of X.
Proof. Let d be a compatible metric on X. By Theorem 3.5, we know that the limit Furthermore, according to the definitions it is clear that Thus it suffices to prove that In what follows, we show that So we get that Consequently, we define a collection of strings as follows Hence we get that Since Hence the proof is completed.
where V runs over all finite open covers of X.
Similar to the Z-action case (see, e.g. Theorem 11.2, p.70 in Pesin's book [19]), we now give some properties of the Pesin-Pitskel topological pressure of G on the set Z ⊆ X.
Proposition 3.8. Let G X be a continuous action. Then (1) follows directly from the definition of the Pesin-Pitskel topological pressure.
For (2), by Theorem 3.5 we only need to show that By Corollary 3.6, we know that, Now we define a collection of strings as follows It is clear that the collection Γ (K,δ) covers Z. Furthermore, one has < .
It follows that Thus we have that Since s is any real number with sup i≥1 h P top G, f * , Z i < s, we get that For (3), it suffices to prove h P top G, f, gZ ≤ h P top G, f, Z by the symmetries of two sets Z and gZ. We may assume that h P top G, f, Z < +∞. Let s = h P top G, f, Z and κ > 0. Let U be a finite open cover of X, K ∈ F (G) and δ > 0. Since M s+κ (G, U , f, Z) = 0, there exist a set K 1 ∈ F (G) with K 1 ⊇ K and a real number δ 1 with 0 < δ 1 < δ such that From the definition of M s+κ (K1,δ1) (G, U , f, Z), there exists a collection Ω = {U i : i ∈ I} of strings which satisfies that The collection Ω covering Z means that Z ⊆ i∈I t∈dom(Ui) t −1 U i (t). Since g : X → X is a homeomorphism, it follows that Consequently, we define the new string U i : dom(U i )g −1 → U by means of U i as follows: For the nonempty finite set F i g −1 , we have Meanwhile, it is easy to see that Thus, we deduce that The arbitrariness of κ implies the desired conclusion.

3.2.
Bowen pseudometric pressure of subsets. In mathematics, a pseudometric is a generalized metric space in which the distance between two distinct points can be zero.
A pseudometric space (X, ρ) is a set X together with a non-negative real-valued function ρ : X × X → R ≥0 (called a pseudometric) such that, for every x, y, z ∈ X, Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have ρ(x, y) = 0 for distinct points x = y.
Let (X, T ) be a topological space and ρ be a pseudometric on X. Then ρ is said to be continuous if the map ρ : X × X → R ≥0 is continuous. Here the topology of the space X × X is the product topology.
Throughout G X is a continuous action on a compact metrizable space and ρ is a continuous pseudometric on X. Definition 3.9. Let F be a nonempty finite subset of G. Define on X the pseudometric ρ F (x, y) = max s∈F ρ(sx, sy).
For every > 0 we denote by B ρ F x, the Bowen ball of radius in the pseudometric ρ F around x, i.e., is an open subset of X. Definition 3.10. The pseudometric ρ is said to be dynamically generating if for all distinct x 1 , x 2 ∈ X there is an s ∈ G for which ρ(sx, sy) > 0.
It is well known that one can obtain a compatible metric from any dynamically generating continuous pseudometric ρ, see for example [14,12]. So we have the following result and omit the details here.
Lemma 3.11. Let ρ be a dynamical generating continuous pseudometric on X.
Enumerate the elements of G as s 1 = e, s 2 , · · · . Define a new continuous pseudometric ρ on X by ρ(x, y) = ∞ j=1 2 −j+1 ρ(s j x, s j y) for all x, y ∈ X. Then ρ is a compatible metric on X.
For the dynamically generating pseudometric, from a simple compactness argument one has the following result (see p.250 in [12]). Lemma 3.12. Let ρ be a dynamical generating continuous pseudometric on X and ρ be the compatible metric determined by ρ as in Lemma 3.11. Let > 0. Then there exist a finite set K ⊆ G and a δ > 0 such that ρ(x, y) < if ρ(sx, sy) < δ for all s ∈ K.
The Pesin-Pitskel topological pressure can be defined in the following alternative way.
Let f ∈ C(X), x ∈ X and F ∈ F (G). Denote Let s ∈ R and (K, δ) ∈ Λ. For > 0, put where the infimum is taken over all collections Note that the quantity h s  With a similar argument as in Fact 3.1, we have the following propositions.
Proposition 3.13. For any s ∈ R, the function h s (ρ, G, f, ·) satisfies the following properties: We define where the infimum is taken over all collections If a family of Bowen balls with radius 1 covers Z, then the family of Bowen balls with the same centers which have radius 2 also covers Z. At the same time, it is clear that Thus one has h s (K,δ), 1 (ρ, G, f * , Z) ≥ h s (K,δ), 2 (ρ, G, f * , Z). Similar to topological pressure, we also define (3.28) The proof idea of the following theorem comes from oral communication with Professor Hanfeng Li. Proof. It is clear that h P (ρ, G, f * , Z) ≤ h P (ρ, G, f, Z). Thus it suffices to prove that h P (ρ, G, f, Z) ≤ h P (ρ, G, f * , Z). We may assume that h P (ρ, G, f * , Z) < ∞. Let κ > h P (ρ, G, f * , Z) be any real number. Due to the equation (3.28), one has h (ρ, G, f * , Z) < κ for all 0 < . Let γ > 0. Then there is θ > 0 such that By Lemma 3.12, for above positive number θ and the dynamically generating pseudometric ρ, there exist a nonempty finite subset K 0 ⊂ G and η > 0 such that ρ(x, y) < θ if ρ(sx, sy) < η for all s ∈ K 0 . (3.32) Choose a nonempty finite subset K ⊂ G with Let 0 < < η/2 and 0 < δ < γ/(4|K|M f ).

Claim. We have
Thus, since K −1 = K, one has

By (3.31) it follows that
Combining with (3.35) one has, for any Therefore, we get It follows that The arbitrariness of the collection Γ (K,δ) which covers Z implies that So the claim is obtained. From (3.25) we know that Let (K , δ ) ∈ Λ. We choose the (K, δ) satisfying By Claim we have (3.37) The arbitrariness (K , δ ) ∈ Λ implies that Combining with (3.30), it follows that Taking the upper limit as → 0 we get h P (ρ, G, f, Z) ≤ κ + γ. The arbitrariness of γ and κ > h P (ρ, G, f * , Z) imply that which is our desired.
Theorem 9.38 in [12] said not only that we may compute topological entropy using separated sets, but that we can do it using dynamically generating continuous pseudometrics, and not merely compatible metrics. Kerr and Li studied deeply the properties of dynamically generating continuous pseudometric and obtained many interesting results in [12]. Similar to Theorem 9.38 in [12], we have the following result.
Theorem 3.17. Let ρ be a dynamically generating continuous pseudometric on X.
To establish the proof of Theorem 3.17, it suffices to carry out the following two steps: Step 1. We will show h P (ρ, G, f, Z) ≤ h P top G, f, Z . Due to Theorem 3.16 we know that h P (ρ, G, f, Z) = h P (ρ, G, f * , Z). So we only need to prove h P (ρ, G, f * , Z) ≤ h P top G, f, Z . Let ε > 0, U be a finite open cover of X such that diam (U ) < ε. In what follows, we will show that Claim. h s (K,δ),ε ρ, G, f * , Z ≤ M s (K,δ) G, U , f, Z . Let Γ (K,δ) ⊆ F ∈B(K,δ) W F (U ) be a collection of strings which covers Z. We may assume that X(U) = ∅ for each string U ∈ Γ (K,δ) . Fix a point x U ∈ X(U). Note that Thus for any point y ∈ X(U), one has gx U , gy ∈ U(g) for all g ∈ dom(U). Therefore, we have ρ(gx U , gy) ≤ diam(U ) < ε (for any g ∈ dom(U)), that is, So we get a collection of Bowen balls .
Clearly, the collection Ω (K,δ) of Bowen balls covers Z. Note that |dom(U)| = m(U) and (3.39), we have From the arbitrariness of the collection Γ (K,δ) which covers Z, it follows that h s (K,δ),ε ρ, G, f * , Z ≤ M s (K,δ) G, U , f, Z . Taking the limit for the net (K, δ) ∈ Λ, we get Since s is any real number with s > h P top G, U , f, Z , we have that Letting ε → 0 one has h P ρ, G, f * , Z ≤ h P top G, f, Z . Hence the conclusion of Step 1 is obtained.
Step 2. We will show h P top G, f, Z ≤ h P (ρ, G, f, Z). By Theorem 3.5 we only need to prove Let U be a finite open cover of X and κ > 0. Clearly it suffices to prove that Let ρ be the compatible metric on X which is determined by the dynamically generating continuous pseudometric ρ as in Lemma 3.11. Denote L = L ρ (U ) by the Lebesgue number of U under the metric ρ.
Let 0 < 0 < L/6. Since ρ is continuous and X is compact, there is a constant M with ρ(x, y) ≤ M for all x, y ∈ X. Consequently, there exists N ∈ N such that where |U | denotes the number of the elements of U . Take s = h 0 (ρ, G, f, Z). Thus So there exist K 1 ∈ F (G) and δ 1 > 0 with (K 1 , δ 1 ) (K 0 , δ 0 ) such that, for any (K, δ) (K 1 , δ 1 ), there is a countable collection For each i ∈ I, we denote A i = N j=1 g −1 j F i . Due to (K, δ) (K 0 , δ 0 ) we know that g j ∈ K for j = 1, · · · , N . Since F i ∈ B(K, δ) and g 1 = e, one has A i ⊆ F i and (3.46) Note that, for every t ∈ A i and g j (1 ≤ j ≤ N ), one has g j t ∈ F i . For any pair of points y 1 , y 2 ∈ B ρ Fi (x i , 0 ) (i ∈ I) and any t ∈ A i , we have ρ(ty 1 , ty 2 ) = ∞ j=1 ρ(g j ty 1 , g j ty 2 ) 2 j−1
For any i ∈ I and each ϕ ∈ Map i , we define the string U i,ϕ ∈ W Fi (U ) as follows: For fixed i ∈ I, from (3.47) it is easy to see that For i ∈ I, we denote Note that F i ∈ B(K, δ) for each i ∈ I. Now we define a set of strings Ω (K,δ) as follows: It follows that Note that m(U i,ϕ ) = |F i | for U i,ϕ ∈ Ω i , A BILLINGSLEY-TYPE THEOREM FOR THE PRESSURE   983 Thus, combining the inequality (3.46) one has Taking the upper limit as 0 → 0, the inequality (3.41) is obtained.

4.
Local measure pressures of subsets for actions of amenable groups. Let X be a compact metrizable space and M(X) denote the set of all Borel probability measures on X which is equipped with weak * topology.
Recall that G is a discrete countable infinite amenable group and e ∈ G is its unit. Fix 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 facts.
Let ρ be a dynamically generating continuous pseudometric on X.
Let F ∈ F (G), f ∈ C(X), δ > 0 and µ ∈ M(X). For x ∈ X, we define The local lower measure pressure is defined by Similarly, we define The local upper measure pressure is defined by So there exists a positive real number r 1 with 0 < r 1 < r such that µ (B ρ F (x 0 , r 1 )) > a. Set δ = r − r 1 . For any point y ∈ B ρ F (x 0 , δ), it is easily checked that B ρ F (y, r) ⊇ B ρ F (x 0 , r 1 ). Thus, we get µ (B ρ F (y, r)) > a which implies that y ∈ E, i.e., is Borel measurable.
Proof. It is immediately obtained from Fact 4.4.
Thus it suffices to prove that the function is Borel measurable. By Fact 4.1, one has is a family of finite subsets of G which satisfies that e ∈ G 1 ⊆ G 2 ⊆ · · · and G = ∞ n=1 G n . Hence we only need to prove that the function h n (x) = inf F ∈B(Gn,1/n) h µ (G, ρ, f, x, r, F ) is Borel measurable for given r > 0.
Since G is countable group, the set B(G n , 1/n) is countable. Thus we only need to show that h µ (G, ρ, f, x, r, F ) is Borel measurable. Furthermore, It is easy to see that the function x → 1 |F | f F (x) is Borel measurable as it is a continuous function. So it is enough to show that the function is Borel measurable. Fact 4.5 shows that the above function is Borel measurable. Hence the fact is obtained.
In the following, we prove a lemma which is much like the classical covering lemma. For the Z-action case of this lemma, please refer to [16] (see Lemma 1, p.506). For each ω ∈ C , it is clear that ω ⊆ ω * . Now we show that ω * ∈ Ω. It is not hard to see that all Bowen balls of ω * are pairwise disjoint as C is a totally ordered sub-collection of Ω. Meanwhile, if a Bowen ball B ρ Fp (z, r) ∈ F meets some Bowen ball from ω * , then B ρ Fp (z, r) meets a Bowen ball from ω for some ω ∈ C . By the definition of ω, there is a Bowen ball B ρ Fm (y, r) ∈ ω ⊆ ω * such that F m ⊆ F p and B ρ Fm (y, r) ∩ B ρ Fp (z, r) = ∅. Hence ω * ∈ Ω. By Zorn's lemma, there exists a maximal element G in Ω. We claim that First, we prove that the intersection of each Bowen ball of F and the union of all Bowen balls from G is nonempty. Otherwise, there exists a Bowen ball B ρ Fm (x, r) ∈ F such that So we can define m 0 = min{m : B ρ Fm (x, r) ∈ F does NOT meet any Bowen ball from G}. We choose a Bowen ball B ρ Fm 0 (x 0 , r) ∈ F such that it does not meet any Bowen ball in G. We write G * = G ∪ B ρ Fm 0 (x 0 , r) . Now, we show that G * ∈ Ω. It is clear that all Bowen balls of G * ⊆ F are pairwise disjoint. Suppose that B ρ Fp (z, r) ∈ F meets some Bowen ball from G * . Then we divide into two cases into consideration.
• If B ρ Fp (z, r) meets some Bowen ball from G, owing to G ∈ Ω, then there is Fp (z, r) does NOT meet any Bowen ball from G, then B ρ Fp (z, r)∩B ρ Fm 0 (x 0 , r) = ∅. By the definition of m 0 , it follows that m 0 ≤ p, i.e., F m0 ⊆ F p .
According to above arguments we deduce that G ∪ B ρ Fm 0 (x 0 , r) ∈ Ω which contradicts that G is a maximal element in Ω.

Hence, each Bowen ball B ρ
Fn (x, r) ∈ F must meet some element from G. Since G ∈ Ω, there is a Bowen ball B ρ Fp (y, r) ∈ G such that F p ⊆ F n and B ρ Fn (x, r) ∩ B ρ Fp (y, r) = ∅. Since the metric ρ Fp ≤ ρ Fn , one has Since X is a compact metrizable space, it satisfies the second axiom of countability. Furthermore, owing to the fact that all elements of G are open and pairwise disjoint, the set G is countable.Hence the lemma is proved.
The following theorem establishes the relation between Pesin-Pitskel topological pressure and the local pressure of a Borel probability measure. It is a generalization of Ma-Wen's result [16] to dynamical systems acting by a countable discrete amenable group.
Theorem 4.8. Let G X be a continuous action, f ∈ C(X) and ρ a dynamically generating continuous pseudometric on X. Let µ be a Borel probability measure on X and Z ⊆ X a Borel subset. For s ∈ R, the following properties hold: 1. If h P µ (G, ρ, f, x) ≥ s for all x ∈ Z and µ(Z) > 0, then h P (ρ, G, f, Z) ≥ s; 2. If h P µ (G, ρ, f, x) ≤ s for all x ∈ Z then h P (ρ, G, f, Z) ≤ s. Proof. (1) Let ν > 0. We want to show that Let k ∈ N, {G k } ∞ k=1 be a family of finite subsets of G which satisfies that e ∈ G 1 ⊆ G 2 ⊆ · · · and G = ∞ k=1 G k . Set For each F ∈ B(G k , 1 k ), we denote To make the proof precise we proceed as follows.
Step 1. We will show that Z k is a Borel subset of X. Note that By Fact 4.5, we know that the function x → − 1 |F | log µ B ρ F x, 1 k is Borel. At the same time, the function x → 1 |F | f F (x) is continuous. Thus we get that the set Z F is a Borel subset of X. Meanwhile, it is easy to see that

XIAOJUN HUANG, YUAN LIAN AND CHANGRONG ZHU
Furthermore, we know that the set B(G k , 1 k ) is countable. Hence we deduce that Z k is a Borel subset of X since Z is a Borel set.
Step 2. {Z k } is an increasing sequence of Borel sets and ∞ k=1 Z k = Z. The monotonicity of G n implies that So there exists δ 0 > 0 such that Combining with Fact 4.1 we have Thus, we can find an N ∈ N such that We choose a positive integer m ∈ N with m > N and 1 m < δ 0 . Therefore, for every F ∈ B (G m , 1/m), we know that which implies x 0 ∈ Z m as desired. Hence Step 2 is proved. From Steps 1 and 2, we know that lim n→∞ µ(Z n ) = µ(Z). It follows that there is m ∈ N satisfying Step 3. We will show that Let K be a nonempty finite subset of G and δ > 0 with (K, δ) (G m , 1 m ). Let 0 < r < 1 2m . Suppose that Γ = {B ρ Fi (x i , r)} i∈I is a collection of Bowen balls of X with x i ∈ X and satisfies F i ∈ B(K, δ) and To finish the proof, we now define the index set I ⊆ I by It is clear that We now choose a point y i ∈ B ρ Fi (x i , r) ∩ Z m for each i ∈ I . Therefore, one has From (4.5) and (4.6), we then have So we get Consequently, we deduce that The arbitrariness of the collection Γ leads to We deduce by the definition of h r (ρ, G, f, Z) that Hence The arbitrariness of ν implies that h P (ρ, G, f, Z) ≥ s.
Hence we complete the proof.
So the fact is proved.
Using the fact that G is a countably infinite amenable group and Theorem 2.5 holds, there exists a nested Følner sequence {F n } of G. Let K ∈ F (G) and δ > 0. For each z ∈ Z with h P µ (G, ρ, f, z) ≤ s, by Fact 4.9, we have lim sup F 1 |F | log e f F (z) µ(B ρ F (z, )) −1 < s + ν.
5. An example of Bowen topological pressure. Let G be a countably infinite amenable group and A = {1, · · · , k} be a finite set. We equip A with the discrete topology and A G with the associated product topology. Let G A G be the left action of G on A G . This left action of G on A G is called the G-shift on A G . Define on A G the continuous pseudometric ρ(x, y) = 0, if x(e) = y(e) 1, if x(e) = y(e).
Then it is easy to see ρ is a dynamically generating pseudometric. Let µ be the uniform distribution on A, i.e., µ(i) = 1 k for i ∈ A. By Daniell-Kolmogorov extension theorem, there is a unique Borel probability measure µ G on A G which behaves as an ordinary product measure on Borel cylinder, i.e., if F is a nonempty finite subset of G, {A s } s∈F is a collection of Borel subsets of A, and π F : A G → A F is the coordinate restriction map then

XIAOJUN HUANG, YUAN LIAN AND CHANGRONG ZHU
Theorem 5.1. Let G be a countably infinite amenable group, k a positive integer and A = {1, · · · , k}. Let ρ be the dynamically generating continuous pseudometric on A G defined as above. Let µ be the uniform distribution on A and µ G be the Borel probability measure on A G which is determined by µ. Let f ≡ C be a constant function on A G . For any Borel set Z ⊆ A G , if µ G (Z) > 0, then h P top (G, C, Z) = log k + C. Proof. Let x ∈ A G and F be a nonempty finite subset of G and 0 < < 1. It can easily be verified that Since the Borel set Z satisfies µ G (Z) > 0 and Theorem 4.8 holds, we deduce that h P (ρ, G, C, Z) = log k + C. Z 0 := x ∈ A G : x(e) = x(g) = 1 .
It is clear that Z 0 is not G-invariant and µ G (Z 0 ) = (1/k) 2 > 0. From Theorem 5.1 we have h P top (G, C, Z 0 ) = log k + C.