Entropy formulae of conditional entropy in mean metrics

. In this paper, we construct the Brin-Katok formula of conditional entropy for invariant measures of continuous maps on a compact metric space by replacing the Bowen metrics with the corresponding mean metrics. Additionally, this paper is also devoted to establishing the Katok’s entropy formula of conditional entropy for ergodic measures in the case of mean metrics.


1.
Introduction. Let a triple (X, d, T ) (or pair (X, T ) for short) be a topological dynamical system (TDS for short) in the sense that T : X → X is a continuous map on the compact metric space X with metric d. The terms M (X), M (X, T ) and E(X, T ) represent the sets of all Borel probability measures, T -invariant Borel probability measures, and T -invariant ergodic Borel probability measures, respectively. For x, y ∈ X and n ∈ N, the Bowen metric d n is given by d n (x, y) = max{d(T i (x), T i (y)) : i = 0, 1, · · · , n − 1}.
Given ε > 0, let B dn (x, ε) = {y ∈ X : d n (x, y) < ε} denote the d n -ball centered at x with radius ε. We also write B n (x, ε) for convenience, when there is no confusion.
In classical ergodic theory, measure-theoretic entropy and topological entropy are important determinants of complexity in dynamical system. The relationship between these two quantities is the well-known variational principle. Brin-Katok formula and Katok's entropy formula are two important formulas in the study of entropy theory.
x ∈ X, h µ (T, x) = h µ (T ). It was shown in [19,20] that the Brin-Katok formula above in the case of random dynamical systems. The Brin-Katok formula for the measure theoretic r-entropy was constructed in [17].
Given a TDS (X, d, T ). Let B X be the Borel σ-algebra of X. Then each µ ∈ M (X, T ) induces a measure preserving dynamical system (X, B X , µ, T ). Let B µ be the completion of B X , and I = {E ∈ B µ : T −1 E = E} is the σ-algebra of T -invariant sets of B µ . Then µ can be decomposed into a generalized combination of ergodic measures µ = µ I x dµ(x), where µ I x denotes the conditional measure of µ at x with respect to I. There is a well-known result that the ergodic decomposition of entropy [6], i.e.
where h µ (T |I) denotes the conditional entropy with respect of I (see section 2 for details). Let A be a T -invariant sub-σ-algebra of B µ , i.e.
and µ = µ A x dµ(x) is the disintegration of µ over A, where µ A x denotes the conditional measure of µ at x with respect to A. Nevertheless, µ A x is only Borel probability measure but maybe not T -invariant for µ-a.e. x ∈ X. Furthermore, for any f ∈ L 1 (X, B X , µ), the following equation holds [4] Particularly, for any B ∈ B X , we have In 2016, Zhou [18] established a conditional version of Brin-Katok formula and obtained a analogous conclusion with (1) for A.
In 1980, Katok [11] introduced the Katok's entropy formula: for any µ ∈ E(X, T ), where N µ (n, ε, δ) denotes the minimal number of d n -balls with radius ε which cover the set of µ-measure more than or equal to 1 − δ. Katok's entropy formula is an equivalent definition of the measure-theoretic entropy in a manner analogous to the definition of the topological entropy. In 2004, using spanning sets, He, Lv and Zhou [9] introduced a definition of measure-theoretic pressure of additive potentials for ergodic measures, and obtained a pressure version of Katok's entropy formula. In 2009, Zhao and Cao [15] gave a definition of measure-theoretic pressure of sub-additive potentials for ergodic measures, and generalized the above results in [11] and [9]. Moreover, we refer to [3,2] for more pressure versions of Katok's entropy formula. In 2009, Zhu [20] established Katok's entropy formula in the case of random dynamical systems. Very recently, in order to establish large deviations bounds for countable discrete amenable group actions, Zheng, Chen and Yang [16] introduced an amenable version of Katok's entropy formula.
All of the above studies were carried out in the case of Bowen metrics. However, in this paper, we consider the mean metrics. For any x, y ∈ X, n ∈ N, the mean metricd n [8] is given byd For x ∈ X, ε > 0, let Bd n (x, ε) := {y ∈ X :d n (x, y) < ε} denote thed n -ball centered at x with radius ε.
Recently, the mean metrics attract a lot of attention. In 2016, using separated sets, Gröger and Jäger [7] gave a definition of topological entropy of the whole system in mean metrics, and proved that the topological entropy defined by mean metrics is equivalent to the topological entropy defined by Bowen metrics. Huang, Wang and Ye [10] introduced the notion of measure complexity for a TDS. They established Katok's entropy formula for ergodic measures in the case of mean metrics and this result is exactly the corollary of our main results (see Corollary 2). Meanwhile, this formula showed that one can replace the d n -balls withd n -balls when studying the measure complexity for a TDS. Since the advantage to use mean metrics is that it is an isomorphic invariant [10], they discussed the measure complexity for a TDS by mean metrics and showed that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity.
In this paper, inspired by the ideas of Brin and Katok [1], Zhou, Zhou and Chen [17] and Zhou [18], we construct the Brin-Katok formula of conditional entropy h µ (T |A) for invariant measures of continuous maps on a compact metric space by replacing the Bowen metrics with the corresponding mean metrics. Furthermore, following the idea of Katok [11], we establish the Katok's entropy formula of conditional entropy h µ (T |A) for ergodic measures in the case of mean metrics.
The following theorems present the main results of this paper. Let (X, d, T ) be a TDS, µ ∈ M (X, T ) and B µ be the completion of B X under µ. Suppose A is a T -invariant sub-σ-algebra of B µ , i.e. T −1 A = A (mod µ) and the measure disintegration of µ over A is Then, for µ-a.e. x ∈ X,h Then for any 0 < δ < 1, we have for µ-a.e. x ∈ X, whereÑ µ A x (n, ε, δ) denotes the minimal number ofd n -balls with radius ε whose union has µ A x -measure more than or equal to 1 − δ.
Let (X, d, T ) be a TDS, B X be the Borel σ-algebra of X, and µ ∈ M (X, T ). Suppose B µ is the completion of B X , and A = {∅, X}. Clearly A is a T -invariant sub-σ-algebra of B µ , and for each x ∈ X, µ A x = µ, where µ A x denotes the conditional measure of µ at x with respect to A. Also, we observe that h µ (T |A) = h µ (T ), where h µ (T ) is the measure theoretic entropy. Obviously, we have the following two corollaries of Theorem 1.1 and Theorem 1.2, respectively.
The remainder of this paper is organized as follows. Section 2 gives some preliminaries. Section 3 provides the proofs of the main results. Finally, in section 4, we consider the one-sided symbolic space as an example of Theorem 1.1.

2.
Preliminaries. Given a TDS (X, T ). A partition of X is a disjoint collection of elements of B X whose union is X. Let P X denote the collection of all finite Borel partitions of X. Suppose ξ, η ∈ P X . We write ξ ≤ η to mean that each element of ξ is a union of elements of η (i.e. η is a refinement of ξ). Let ξ = {A 1 , · · · , A k }, η = {B 1 , · · · , B m } be two finite partitions of (X, T ). Their join is the partition For a measurable partition ξ of X and x ∈ X, denote by ξ(x) the element of ξ containing x, and set Let µ ∈ M (X, T ). Given a T -invariant sub-σ-algebra A of B µ and ξ ∈ P X . Define (see [14], Definition 4.8) is a sub-additive sequence, the conditional entropy of ξ with respect to A is given by Furthermore, the conditional entropy of T with respect to A is defined by Then the conditional entropy of T with respect to A can be computed by The following theorem and lemma will be used in proving the main results.
for µ-a.e. x ∈ X and in L 1 (µ), where is the conditional informational function of ξ n−1 0 with respect to A. Moreover, if µ ∈ E(X, T ) then h µ (ξ|A, x) = h µ (T, ξ|A) for µ-a.e. x ∈ X. Lemma 2.2. [18] Let (X, T ) be a TDS, µ ∈ M (X, T ), ξ ∈ P X and A be a Tinvariant sub-σ-algebra of B µ . Then for µ-a.e. x ∈ X, there exists W is the measure disintegration of µ over A and h µ (ξ|A, x) is the function obtained in Theorem 2.1.
Following the ideas of Brin and Katok [1] as well as Feng and Huang [5], we replace the Bowen metric d n with the mean metricd n and give a definition of measure-theoretic lower and upper entropies. Then, the measure-theoretical lower and upper entropies of ν in mean metrics are defined respectively byh In order to prove the main theorems, we need the relationship between thed n -ball Bd n (x, ε) and the mistake dynamical ball B n (g; x, ε), where the mistake function g and mistake dynamical ball B n (g; x, ε) are defined in Definition 2.4 and Definition 2.5, respectively. The function g is called a mistake function. Note that g is a special mistake function and the definition of the function g is different from the definition of the mistake function in [12] or [13]. In fact, the mistake funcionĝ(n) in [12] should satisfy lim n→∞ĝ (n) n = 0, while the mistake function g * (n, ε) in [13] should satisfy lim n→∞ g * (n,ε) n = 0. However, since in this paper lim n→∞ g(n,ε) n = ε, we only have lim ε→0 lim n→∞ g(n,ε) n = 0 .
Then, we can have the following relationship which plays a crucial role in our proofs. Lemma 2.6. For any x ∈ X, n ∈ N + and 0 < ε < 1, we have Proof. Given y ∈ X. If d n (x, y) < ε, thend n (x, y) < ε. So B n (x, ε) ⊂ Bd n (x, ε). Set Noting that Thus, we obtain #{i ∈ Λ n : d( then y ∈ B n (g; x, √ ε). Therefore, we have Bd n (x, ε) ⊂ B n (g; x, √ ε). Proof. Let {ξ i } ∞ i=0 be a family of finite Borel partitions of X satisfying where ∂ξ i denotes the union of the boundaries ∂B of all elements B ∈ ξ i . In fact, using the Monotone Convergence Theorem it is sufficient to show that the following equation holdsh for µ-a.e. x ∈ X, where h µ (ξ i |A, x) is the function obtained in Theorem 2.1.
(1) Firstly, we want to show that for µ-a.e. x ∈ X,
For l ∈ N + , we define where χ U δ (ξ) is the characteristic function of the set U δ (ξ). Therefore, by the same method of the proof of Theorem 1.2 in [18], we can choose N 1 ∈ N such that for any l ≥ N 1 , we have Then also by the same method of the proof of Theorem 1.2 in [18], we have for any l ≥ N 1 , µ(Q l ) > 1 − ε 1 4 . Clearly, the sets A l are nested, i.e. A 1 ⊂ A 2 ⊂ · · · . Then fix some l 1 > N 1 , for any x ∈ Q l1 , l ≥ l 1 we have . According to Lemma 2.2, we can find a subset X 1 ⊂ X with µ(X 1 ) = 1 such that for any x ∈ X 1 , there exists W x ∈ B X with µ A x (W x ) = 1 such that for any y ∈ W x , lim n→∞ − log µ A x (ξ n−1 0 (y)) n = h µ (ξ|A, x).
Let I = X 1 ∩ Q l1 . Clearly, µ(I) > 1 − ε 1 4 . Fixx ∈ I. By the Egorov Theorem, we can find a number l 2 large enough such that µ Â For n ∈ N and given a point y ∈ X, we call the collection C(n, y) := (ξ(y), ξ(T (y)), · · · , ξ(T n−1 (y)) the (ξ, n)−name of y. Since each point in one element V of ξ n−1 0 has the same (ξ, n)-name, we can define C(n, V ) := C(n, y) for any y ∈ V , which is called the (ξ, n)-name of V . For n ∈ N and ξ, we give a metric d ξ n between (ξ, n)−names of y and z as follows: It can also be viewed as a semi-metric on X. If z ∈ B(y, δ), then either y and z belong to the same element of ξ or y ∈ U δ (ξ), z / ∈ ξ(y). Noting that Bd n (y, δ 2 ) ⊂ B n (g; y, δ), hence if y ∈ E, n > l and z ∈ Bd n (y, δ 2 ), the distance d ξ n between (ξ, n)-names of y and z does not exceed δ + 2 √ ε, i.e. d ξ n (C(n, y), C(n, z)) ≤ δ + 2 √ ε.
(2) Secondly, we will turn to prove the second part of the theorem: for every 0 < δ < 1, we have for µ-a.e. x ∈ X.
Clearly, the sets D l are nested and exhaust X up to a set of µ-measure zero. Therefore, there exists l 0 > 1 such that µ(D l ) ≥ 1 − 2 √ ε for any l ≥ l 0 .
}. Using Chebyshev's Inequality, we obtain for any l ≥ l 0 . Thus for any l ≥ l 0 , µ(M l ) > 1 − ε 1 4 . The sets D l are nested, i.e. D 1 ⊂ D 2 ⊂ · · · . Then fix some l 1 > l 0 , for any x ∈ M l1 , l ≥ l 1 we have According to Lemma 2.2, we can find a subset X 1 ⊂ X with µ(X 1 ) = 1 such that for any x ∈ X 1 , there exists W x ∈ B X with µ A x (W x ) = 1 such that for any y ∈ W x , lim n→∞ − log µ A x (ξ n−1 0 (y)) n = h µ (T, ξ|A).