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Katok's Entorpy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

Katok's entropy formula is an important formula in entropy theory. This paper is devoted to establishing Katok's entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.


Introduction
Let a triple (X, d, f ) be a topological dynamical system in the sense that f : X → X is a continuous map on the compact metric space X with metric d. For x, y ∈ X and n ∈ N, the Bowen metric d n is given by d n (x, y) = max{d(f i (x), f i (y)) : i = 0, 1, · · · , n − 1}.
Given ε > 0, x ∈ X and n ∈ N, let B n (x, ε) = {y ∈ X : d n (x, y) < ε} denote the d n -ball centered at x with radius ε.
In classical ergodic theory, measure-theoretic entropy and topological entropy are very important determinants of complexity in dynamical systems. And the variational principle reveals the relationship between the measure-theoretic entropy and the topological entropy. Katok's entropy formula which plays an important role in the study of entropy theory is an equivalent definition of the measure-theoretic entropy in a manner analogous to the definition of the topological entropy.
In 1980, Katok [13] introduced the Katok's entropy formula: for any f -invariant ergodic Borel probability measure µ, and 0 < δ < 1, where N µ (n, ε, δ) denotes the minimal number of d n -balls with radius ε whose union is a set of µ-measure more than or equal to 1 − δ.
In 2004, using spanning sets, He, Lv and Zhou [6] introduced a definition of measuretheoretic pressure of additive potentials for ergodic measures, and obtained a pressure version of Katok's entropy formula. In 2009, Zhao and Cao [19] gave a definition of measure-theoretic pressure of sub-additive potentials for ergodic measures, and generalized the above results in [13] and [6]. Moreover, we refer to [3,2] for more pressure versions of Katok's entropy formula. In 2009, Zhu [21] established a random version of Katok's entropy formula. In 2017, Zheng, Chen and Yang [20] introduced an amenable version of Katok's entropy formula for countable discrete amenable group actions. Huang, Wang and Ye [12] established Katok's entropy formula for ergodic measures in the case of mean metrics. In 2018, Huang, Chen and Wang [10] constructed Katok's entropy formula of conditional entropy in mean metrics, where the conditional entropy is with respect to a T -invariant sub-σ-algebra. In 2019, Huang and Wang [11] established a pressure version of Katok's entropy formula in the case of mean metrics.
Let M be an n-dimensional smooth, connected and compact Riemannian manifold without boundary and f : M → M be a C 1 -diffeomorphism. f is said to be partially hyperbolic if there exists a nontrivial Df -invariant splitting T M = E s ⊕ E c ⊕ E u of the tangent bundle into stable, center, and unstable distributions, such that all unit vectors and for some suitable Riemannian metric on M. The stable distribution E s and unstable distribution E u are integrable to the stable and unstable foliations W s and W u respectively such that DW s = E s and DW u = E u . In 1985, from the measure theoretic point of view, Ledrappier and Young [14] gave a definition of unstable metric entropy for partially hyperbolic diffeomorphisms. And the entopy defined in [14] can be regarded as that given by H µ (α|f α), where α is an increasing partition (i.e. α ≥ f α) subordinate to the unstable leaves. In 2008, from the topological point of view, Hua, Saghin and Xia [9] provided the unstable volume growth. In 2017, Hu, Hua and Wu [7] introduced the definition of unstable metric entropy h u µ (f ) for any invariant measure µ, and gave the definition of the unstable topological entropy h u top (f ). Similar to that in the classical entropy theory, the corresponding versions of Shannon-McMillan-Breiman theorem, and the variational principle relating h u µ (f ) and h u top (f ) are given. The unstable metric entropy h u µ (f ) for an invariant measure µ is defined by using where ξ is a finite measurable partition of the underlying manifold M, and η is a measurable partition consisting of local unstable leaves that can be obtained by refining a finite partition into pieces of unstable leaves. In [7], they showed that the unstable metric entropy h u µ (f ) is identical to the metric entropy defined in [14]. Hu, Wu, Zhu [8] introduced the notion of unstable topological pressure for a C 1 -partially hyperbolic diffeomorphism f : M → M with respect to a continuous function on M, obtained a variational principle for this pressure. And they investigatedthe corresponding so-called u-equilibriums. Motivated by the work of Bowen [1] and Pesin [15], Tian and Wu [16] established a concept called Bowen unstable topological entropy which is the unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory dimension characteristic. In particular, they proved the Bowen unstable topological entropy of the whole space coincides with the unstable topological entropy of the system in [7]. And they also constructed some basic results in dimension theory for Bowen unstable topological entropy including a variational principle for any compact (not necessarily invariant) subset between its Bowen unstable topological entropy and unstable metric entropy of probability measures supported on this set. Wang, Wu and Zhu [17] introduced the unstable entropy and unstable pressure for a random dynamical system with u-domination. For random diffeomorphisms with domination, they also gave a version of Shannon-McMillan-Brieman theorem for unstable metric entropy, and obtained a variational principle for unstable pressure. In 2019, Wu [18] introduced two notions of local unstable metric entropies and the notion of local unstable topological entropy relative to a Borel cover U of M, and showed that when U is an open cover with small diameter, the entropies coincide with the unstable metric entropy and unstable topological entropy, respectively. And the unstable tail entropy and the unstable topological conditional entropy were also defined in [18].
In this paper, inspired by the ideas of Katok [13], for ergodic measures, we establish the Katok's entropy formula of unstable metric entropy h u µ (f ) for partially hyperbolic diffomorphisms.
The following theorems present the main results of this paper.
Theorem 1.1. (Katok's entropy formula of unstable metric entropy) Let M be an n-dimensional smooth, connected and compact Riemannian manifold without boundary and f : M → M be a C 1 -partially hyperbolic diffeomorphism. Suppose µ is an ergodic measure of f . Let η ∈ P u ε 0 , and the measure disintegration of µ over η is Then for any 0 < δ < 1, we have x (n, ε, δ) denotes the minimal number of d u n -balls with radius ε whose union has µ η x -measure more than or equal to 1 − δ.
See Section 2 for the definitions of d u n -ball, P u ε 0 and h u µ (f ). Similarly, we have the following result: Let M be an n-dimensional smooth, connected and compact Riemannian manifold without boundary and f : M → M be a C 1 -partially hyperbolic diffeomorphism. Suppose µ is an ergodic measure of f . Let η ∈ P u ε 0 , and the measure disintegration of µ over η is Let ξ ∈ P ε 0 satisfying H µ (ξ|η) < ∞. Then for any 0 < δ < 1, we have x (n, ξ, δ) denotes the minimal number of elements of the partition ξ n−1 0 whose union has µ η x -measure more than or equal to 1 − δ. See Section 2 for the definitions of ξ n−1 0 and P ε 0 . The remainder of this paper is organized as follows. Section 2 gives some preliminaries. Section 3 provides the proof of Theorem 1.1. Finally, we prove Theorem 1.2 in Section 4.

Preliminaries
Let M be an n-dimensional smooth, connected and compact Riemannian manifold without boundary and f : M → M be a C 1 -diffeomorphism. From now on we always assume that f is a C 1 -partially hyperbolic diffeomorphism of M, and µ is an f -invariant probability measure.
We say that α is a measurable partition of M if there exists some measurable set M 0 ⊂ M with full measure such that, restricted to M 0 , α = ∞ n=1 α n for some increasing sequence α 1 ≤ α 2 ≤ · · · ≤ α n ≤ · · · of countable partitions. By α i ≤ α i+1 we mean that every element of α i+1 is contained in some element of α i or, equivalently, every element of α i is a union of elements of α i+1 . Then we say that α i is coarser than α i+1 or, equivalently, α i+1 is finer than α i . Represent by ∞ n=1 α n the partition whose elements are the non-empty intersections of the form ∩ ∞ n=1 A n with A n ∈ α n for each n. Equivalently, this is the coarser partition such that It is easy to see that every countable partition is measurable. For a measurable partition ξ of M and x ∈ M, denote by ξ(x) the element of ξ containing x, and set Let ε 0 be small enough and P ε 0 denote the set of finite measurable partitions of M whose elements have diameters smaller than or equal to ε 0 , that is, diamξ := sup{diamA : denotes the local unstable manifold at x whose size is greater than the diameter ε 0 of β. Note that for each where d is the distance on M generated by the Riemannian metric on M. Note we have lim n→∞ d(f −n t, f −n y) = 0, so t ∈ W u (y). Noting that t ∈ β(y) and the size of W u loc (y) is greater than diam(β(y)), we obtain t ∈ W u loc (y). Therefore, t ∈ η(y) = β(y) ∩ W u loc (y). Then, we have η(x) ⊂ η(y). Similarly, η(y) ⊂ η(x). Hence, η(x) = η(y).
Clearly, η is a measurable partition satisfying β ≤ η. Let P u ε 0 denote the set of partitions η obtained by this way.
Denote by d u the metric induced by the Riemannian structure on the unstable manifold. Given ε > 0 and x ∈ M, let B u (x, ε) denote the open ball centered at x with radius ε in the unstable manifold W u (x) with respect to d u . For x ∈ M, y ∈ W u (x) and n ∈ N, let d u n (x, y) = max{d u (f i x, f i y) : i = 0, 1, · · · , n − 1}. Given ε > 0, x ∈ M and n ∈ N, let B u n (x, ε) = {y ∈ W u (x) : d u n (x, y) < ε} denote the d u n -ball centered at x with radius ε, i.e. an (n, ε) Bowen ball in W u (x) centered at x.
The following conclusion is used in the proof of the Lemma 4.4 in [7]. Let γ > 0 be small enough. There exists C > 1 such that for any x ∈ M, γ) is the open ball inside W u (x) centered at x with radius γ with respect to the metric d u and W u (x, γ) is the closure of W u (x, γ).
Then, we can obtain the following two lemmas.
for any y, z ∈ W u (x). Note that for γ > 0 small enough, there exists C > 1 such that for any x ∈ M, d(y, z) ≤ d u (y, z) ≤ Cd(y, z) for any y, z ∈ W u (x, γ). Therefore, for any z ∈ B n (y, ε C ) ∩ η(x), we have Since y, z ∈ W u (x), d u (y, z) ≤ ε 0 and ε 0 is small enough, Since ε 0 is small enough, λε 0 can be smaller than γ. And note that f (y), By induction, we obtain for i = 0, 1, · · · , n − 1. Therefore z ∈ B u n (y, ε) ∩ η(x). Thus Recall that for a measurable partition η of M and a probability measure ν on M, the canonical system of conditional measures for ν and η is a family of probability measures{ν η x : x ∈ M} with ν η x (η(x)) = 1, such that for every measurable set B ⊂ M, x → ν η x (B) is measurable and The following notions are standard. The information function of ξ ∈ P ε 0 with respect to f -invariant probability measure µ is defined as and the entropy of partition ξ as The conditional information function of ξ ∈ P ε 0 given η ∈ P u ε 0 with respect to µ is defined as I µ (ξ|η)(x) = − log µ η x (ξ(x)). Then the conditional entropy of ξ ∈ P ε 0 given η ∈ P u ε 0 with respect to µ is defined as Definition 2.1. [7] For an f -invariant probability measure µ, the conditional entropy of f with respect to ξ ∈ P ε 0 given η ∈ P u ε 0 is defined as The conditional entropy of f given η ∈ P u ε 0 with respect to µ is defined as And the unstable metric entropy of f with respect to µ is defined as The following theorems will be used in proving the main results.

The result in Theorem 2.2 is a version of Shannon-McMillan-Breiman theorem for the unstable metric entropy.
This section gives the proof of Theorem 1.1.
Lemma 3.1. Suppose µ is an ergodic measure of f . Let η ∈ P u ε 0 be given, and ξ ∈ P ε 0 with H µ (ξ|η) < ∞. Then there exists a set M 1 ⊂ M with µ(M 1 ) = 1, and for each for each y ∈ G x . Note that for each y ∈ G x , µ η x = µ η y . Therefore, for each x ∈ M 1 and y ∈ G x , we have Now, we are going to prove Theorem 1.1. Proof.
Fix x ∈ M 1 . For n ∈ N and γ > 0, set the set Y n contains at most exp((h µ (f, ξ|η) + γ)n) elements of ξ n−1 , so Y n can be covered by d u n -balls with radius ε of the same number. So Since γ can be taken arbitrarily small and by Theorem 2 for every x ∈ M 1 . Noting that µ(M 1 ) = 1, we have for µ-a.e. x ∈ M.
It can also be viewed as a semi-metric on M.
Furthermore, for y ∈ D l 2 , n ≥ l 2 , B u n (y, γ) is contained in the set of points z whose (ξ, n)-names are ε-close to the (ξ, n)-name of y, i.e.
By Stirling's formula, there exists a large number l 3 ∈ N and for any n ≥ l 3 , it can be shown that the total number K n of such (ξ, n)-names consisting of B d ξ n (y, ε) admits the following estimate: where For n ≥ max{l 2 , l 3 }, set According to (1) and ε Then we apply the relation (2), so we have Thus, It is clear that S n = V ∈Pn V and By Lemma 3.1 and Egorov Theorem, there exists a large number l 4 > max{l 2 , l 3 } such that, µ η x (T n ) ≥ 1−δ 4 for each n ≥ l 4 , where Write t n := #{ξ n−1 0 (y) : y ∈ T n }. Then Hence, we have Noting that by Theorem 2.1 h µ (f, ξ|η) = h u µ (f ), and using (3), we obtain lim inf n→∞ log N u µ η x (n, γ 2 , δ) n ≥ h µ (f, ξ|η) − ε − lim sup n→∞ log K n n + lim Let ε → 0. Since γ < ε, lim In this section, we will prove Theorem 1.2. Proof.