Bowen Lemma in the Countable Symbolic Space

We consider the sets of quasi-regular points in the countable symbolic space. We measure the sizes of the sets by Billingsley-Hausdorff dimension defined by Gibbs measures. It is shown that the dimensions of those sets, always bounded from below by the convergence exponent of the Gibbs measure, are given by a variational principle, which generalizes Li and Ma’s result and Bowen’s result.


Introduction
Let (X, T) be a topological system, where T : X → X is a continuous transformation on a metric space X.Denote the set of Borel probability measures on X by M(X) and endow it with weak * -topology induced by C b (X), which stands for the set of all bounded continuous functions on X.For any x ∈ X, and n ≥ 1, define the orbit measure Write V(x) for the collection of all accumulation points of the sequence orbit measures {∆ x,n } n≥1 in the sense of weak * -topology of M(X).Denote by M T (X) the set of T-invariant Borel probability measures on X.For µ ∈ M T (X), the set G µ of µ-generic points is defined by When X is a compact metric space, Bowen [1] proved that for any µ ∈ M T (X), the topological entropy h top (G µ ) does not exceed the measure-theoretic entropy h µ .In his proof, Bowen consider the following quasi-regular point set: for t ≥ 0, define QR(t) := x ∈ X : ∃µ ∈ V(x) with h µ ≤ t .
Then he obtained h top (QR(t)) ≤ t which implied that h top (G µ ) ≤ h µ .As a basic dynamic model, the symbolic system always attract much attention for its intrinsic property.In the setting of finite symbolic space, it is well known that there is a close relationship between the indexes topological entropy and dimension, i.e., topological entropy is different from Hausdorff dimension by a multiplier under the usual metric.However, as a generalization of Hausdorff dimension, the index Billingsley dimension (see the definition in Section 2) provides more ways to measure the fractal sets.In this note, we only consider Billingsley dimensions defined by Gibbs measures.Denote by X N = {1, • • • , N} N (N ≥ 2) the finite symbolic space endowed with the product topology and define the shift map T : Let ϕ : X N → R be a potential with summable variations.It is well known that ϕ admits a unique Gibbs measure ν.Li and Ma in [2] consider the quasi-regular point set Q ν (t) with respect to a Gibbs measure ν : where P ϕ denotes the Gurevich pressure of ϕ and 0 ≤ t ≤ 1.

Theorem 1 ([2]
). Assume that ϕ has summable variations and denote the unique Gibbs measure by ν.Then for any 0 t 1 we have When ν is the uniform distribution on X N Billingsley dimension is just Hausdorff dimension under the usual metric on X N .Hence, the above dimension formula generalizes Bowen's result [3].
When X is the countable symbolic space, the situation changes.One key point is that X is non-compact space under the product topology, which leads to a phenomenon that the potential ϕ which admits a Gibbs measure has upper bound but no lower bound.Therefore, there exists T-invariant measure µ such that the integral ϕdµ is infinite.In the sequel, unless otherwise stated, we always denote by X the countable symbolic space.
We still denote by ϕ a potential on X and by ν the corresponding Gibbs measure.For any µ ∈ M T (X), define the entropy dimension of ν with respect to µ by , where Σ k N = {1, 2, . . ., N} k and [ω] denotes a cylinder (see the definition in Section 2).We define the convergence exponent of ν by For 0 ≤ t ≤ 1, define the quasi-regular point set with respect to ν by Our main result in this note is the following.Theorem 2. Let ϕ be a potential function of summable variations admitting a unique Gibbs measure ν with convergence exponent α ν .For 0 ≤ t ≤ 1, we have We give four remarks: (1) In the setting of finite symbolic space, according to Propositions 3 and 4 (see Section 2) we have h µ P ϕ − ϕdµ , thus the set QR ν (t) on the above is the same as the set Q ν (t) in Theorem 1.By a fact that the convergence exponent of Gibbs measure is zero in the finite symbolic space, we have generalized Theorem 1 and Bowen 's result in [1].
(2) The dimension formula (1) implies the dimension upper bound of generic point set.In fact, for any µ ∈ M T (X), we have G µ ⊂ QR ν (t) for t = β(ν|µ).Thus, we obtain dim ν G µ ≤ max{α ν , β(ν|µ)}, which was proved by Fan, Li and Ma in [3].More precisely, they showed (3) There are two main reasons to choose Gibbs measure to define Billingsley dimension.One reason is that Gibbs measure has quasi-Bernoulli property (see Lemma 1).On the other hand, note a fact that Lebsgue measure is boundedly equivalent with Gauss measure in the continued fraction system on [0, 1).Hence, by taking a suitable Gibbs measure on X, one can apply Theorem 2 to the Gauss map related to the continued fractions.Moreover, by transferring dimension result from the symbolic space to the interval [0, 1) [3], we can obtain the corresponding dimension formula for quasi-regular point set in expanding Markov Rényi dynamical system ([4], p. 148).( 4) When (X, T) is a topologically mixing countable Markov shift with BIP property ( [5], p. 97), the corresponding dimension formula can be proved by a parallel argument.
Let us present the idea of the proof.We first use set operation to describe the limit property possessed by the point in QR ν (t).Next, we adopt the finite-symbol-approximation technique developed in [3] to obtain the dimension upper-bound.There are two main problems to deal with.One problem is that the relative entropy h(ν|µ) may be infinite.The other one is that we need approximate a class of sets depending on infinite T-invariant measures.However, the authors in [3] approximated a class of sets depending on only one T-invariant measure.In fact, we prove Theorem 3 by combining the ideas in [2,3].For the lower bound, we draw support from the dimension formula (2).
The article is organized as follows.In Section 2, we give some preliminaries.Section 3 is devoted to the main result of this article.

Preliminaries
In this section, we introduce notations and some useful known facts.We denote by X the countable symbolic space N N endowed with the product topology and define the shift map T : X → X by n=0 N n stand for the set of all finite words, where N 0 denotes the set of empty word.Given We will denote by C n the set of all n-cylinders for n 0. There is a one-to-one correspondence between N n and C n .Let C * = ∞ n=0 C n denote the set of all cylinders.For j, N 1 we will write

Billingsley Dimension
Let A ⊂ X and υ ∈ M(X) be a non-atomic Borel probability measure.Define Thus, there exists s 0 such that H s (A) = ∞ for s < s 0 and H s (A) = 0 for s > s 0 .This s 0 we define to be Billingsley dimension of A with respect to υ and denote it by dim υ A. In fact, Billingsley dimension dim υ is just Hausdorff dimension dim H under the metric ρ υ : for any x, y ∈ X, if x = y, we define ρ υ (x, y) = 0; otherwise where n = min{k 0 : x k+1 = y k+1 }.One can show that ρ υ is a ultrametric.Let n 1 be an integer.Define The following proposition means that the ρ υ -distance of two points uniformly tends to zero when they approach each other in the sense of Bowen.

Proposition 1 ([3]
).For {δ n } n 1 defined on the above, one has The reader may consult [6] for more information on Billingsley dimension.

Metrization of the w * -Topology
Recall that we endow M(X) with the w * -topology induced by C b (X).In this subsection, we introduce a metric to describe the w * -topology of M(X).
For every cylinder [ω] ∈ C * , we choose a positive number a [ω] so that where the sum is taken over all cylinders.For µ, υ ∈ M(X), define The following proposition shows that the metric d * is compatible with the w * -topology of M(X).
According to [7], we call a collection F of Borel subset of X a convergence-determining class if, for every µ ∈ M(X) and {µ n } ⊂ M(X), convergence µ n (A) → µ(A) for all µ−continuity set in F implies µ n w * −→ µ.As a corollary of Proposition 2, the cylinder set C * is a convergence-determining class.

Gibbs Measure
We use Gibbs measures to induce metrics on X.The following facts about Gibbs measures can be found in [5].
Recall that for a function ϕ : X → R, called potential function, the n-order variation of ϕ is defined by We say that a potential ϕ has summable variations if It is easy to see that a potential ϕ with summable variations is uniformly continuous on X.The Gurevich pressure of ϕ with summable variations is defined to be the limit where a ∈ N and it can be shown that the limit exists and is independent of a (see [8]).
An invariant probability measure ν is called a Gibbs measure associated to a potential function ϕ if it satisfies the Gibbsian property: there exist constants C > 1 and holds for any n 1 and any x ∈ X.It is known ( [9]) that a potential function ϕ with summable variations admits a unique Gibbs measure ν iff the Gurevich pressure P ϕ < +∞.Assume that ϕ admits a unique Gibbs measure ν ϕ .Then the constant P in (3) is equal to the Gurevich pressure P ϕ .Let ϕ * = ϕ − P ϕ , we have Hence, without loss of generality, we always suppose P ϕ = 0 in the rest of this paper.A trivial fact is that the Gibbsian property (3) implies that ϕ has no lower bound but upper bound, i.e., ∀x ∈ X, ϕ(x) ln C.
The Gibbsian property implies the quasi-Bernoulli property which is a key point in the proof of dimension upper bound in Section 3.

Lemma 1 ([3]
).Let ν be a Gibbs measure associated to potential ϕ.For any k words ω 1 , • • • , ω k , we have For any T-invariant Borel probability measure µ, define the relative entropy of ν with respect to µ by It is trivially true that h(µ|µ) = h µ .

Proposition 3 ([3]
). Assume that ϕ has summable variations and admits a unique Gibbs measure ν.Then for any invariant measure µ ∈ M T (X), we have By Proposition 3, we can rewrite the variational principle ([5], p. 86) in the following form Recall that we assume that P ϕ = 0.It is known that the supremum in the variational principle ( 5) is attained only by a Gibbs measure ν with h ν < ∞ if such Gibbs measure exits ( [5], p. 89).It follows that when ν = µ, we have h(ν|µ) > h µ , which implies h(ν|µ) > 0. Now we introduce two exponents.Define the convergence exponent of ν by For any µ ∈ M T (X), define the entropy dimension of ν with respect to µ by , where The convergence exponent α ν reflects the mass distribution of ν.Moreover, α ν has the following property.

Lemma 2 ([3]
).Let α ν be the convergence exponent of Gibbs measure ν associated to a potential function ϕ.Then for any > 0 there exist constants C 0 and M such that If µ = ν, it is clear that β(ν|µ) = 1.However, we have the following claim.

Proof of the Main Result
The proof is divided into two parts.First, we shall prove We introduce a technical lemma.For every word ω ∈ Σ n N of length n and every word u ∈ Σ k N of length k with k n, denote by p(u|ω) the frequency of appearances of u in ω, i.e., where τ u (ω) denotes the number of j with 1

Lemma 3 ([10]).
For any h > 0, δ > 0, k ∈ N and n ∈ N large enough, we have Theorem 3. Let ϕ be a potential function of summable variations admitting a unique Gibbs measure ν with convergence exponent α ν .For 0 ≤ t ≤ 1, we have Proof.Fix 1 > 0, we define, for It is clear that {M 1 (N 1 )} N 1 ≥1 monotonically converges to M T (X).Hence we have where Fix 2 > 0. According to the definition of β(ν|µ), we have where By the σ-stability and monotonicity of Hausdorff dimension, we have According to the definition of V(x) and Proposition 2, we have where For m large enough, by the uniformness of H k,N (•, •), we have where Clearly, the parameters n, m, N, k and N 1 successively go to the infinity.In the sequel, we will write C(k, N, n) for C 1 , 2 (N 1 , k, N, m, n) for simplicity.Now we shall give a suitable covering of A 1 , 2 (N 1 , k, N).Consider the (n + k − 1)-prefixes of the points in C(k, N, n): Assume that γ > max{α ν , t} and γ < 3/2 without loss of generality.By the definition of γ-Hausdorff measure, we have It suffices to dominate the outer sum in the above inequality by a constant.Recall that τ u (ω) denotes the number of times that the subword u appears in ω.Given a word ω ∈ Λ n (k, N), we say that (τ u (ω)) u∈Σ k N is the appearance distribution with respect to Σ k N of ω.Write D n (k, N) for the set of such appearance distributions of all elements of Λ n (k, N).Clearly we have D n (k, N) n N k since there are at most Thus, we have In order to dominate the sum on the right-hand side in the above inequality, we shall decompose D((τ u )) into disjoint union of some sets.Given On the other hand, we say that a subword between maximal (N, k)-run subwords is a "bad subword".Thus, the element of D((τ u )) is just like where B r i stands for "bad subword" with length r i and W n i stands for maximal (N, k)-run subword with length n i .Write It is clear that every element in D((τ u )) has at most s maximal (N, k)-run subwords, which implies that t s.Moreover, by setting K t := ∑ t i=1 n i , we have and For 1 t s, let D t be the set of words in D((τ u )) with t maximal (N, k)-run subwords.Clearly, D((τ u )) is a disjoint union of D t 's, i.e., Next, we continue to partition D t by the length pattern of "bad subword" and maximal (N, k)-run subword.Recall that every word ω ∈ D t has the form (10).We say that (r, n) := (r 1 , n 1 , r 2 , • • • , n t , r t+1 ) is the length pattern of "bad subword" and maximal (N, k)-run subword.Let L t stand for the set of all such length pattern of ω in D t .Given a length pattern (r, n) ∈ L t , we denote by B(r, n) the set of elements of D t with the length pattern (r, n).Thus, D t is partitioned into B(r, n)'s.It follows that We shall estimate the cardinal of L t and the sum on the right-hand side in the above inequality.First, note that every length pattern (r, n) ∈ L t is just corresponding to the integer solution of the following equation set By the elementary combinatorial theory, we have According to the definition of M 1 (N 1 ), for any µ ∈ M 1 (N 1 ) and k, N ≥ N 1 , we have m .Thus, when m is taken large enough, we have In other word, we have Note that the parameter n goes to the infinity before k.For any δ > 0, when n is taken large enough, by the Stirling formula we have On the other hand, denote by D t the set of finite words by deleting all "bad subwords" of ω in D t , i.e., By the quasi-Bernoulli property of Gibbs measure ν, we have Recall that γ > t.We can take 2 > 0 small enough such that γ > t + 2 2 .Then, we can take δ > 0 small enough and k, N large enough such that Observe that the function x ln x is bounded in (0, 1].Then we can take k large enough such that Thus, we have L(γ, n, k, (τ u )) ≤ −3δ, and∑ ω∈D t ν γ ([ω]) ≤ exp(−3nδ).In combination with (9), (12) and (17), this yields∑ ω∈Λ n (k,N) ν γ ([ω]) sn N k e −2nδ ≤ e −nδ .