LOCAL UNSTABLE ENTROPY AND LOCAL UNSTABLE PRESSURE FOR RANDOM PARTIALLY HYPERBOLIC DYNAMICAL SYSTEMS

. Let F be a random partially hyperbolic dynamical system generated by random compositions of a set of C 2 -diﬀeomorphisms. For the unstable foliation, the corresponding local unstable measure-theoretic entropy, local un- stable topological entropy and local unstable pressure via the dynamics of F along the unstable foliation are introduced and investigated. And variational principles for local unstable entropy and local unstable pressure are obtained respectively.

1. Introduction. It is well known that among the concepts in differentiable dynamical systems, entropy including measure-theoretic entropy and topological entropy, pressure, and Lyapunov exponents play important roles for both deterministic and random cases, which describe the complexity of a given dynamical system from different points of view.
Plentiful results on entropies have been obtained, since measure-theoretic entropy was introduced by Kolmogorov and topological entropy was introduced by Adler, Konheim and McAndrew respectively. One important result is the so-called variational principle relating measure-theoretic entropy and topological entropy, which says that for a given dynamical system, its topological entropy is equal to the supremum of measure-theoretic entropies over all invariant measures. Another momentous result is the famous Pesin's entropy formula relating measure-theoretic entropy and Lyapunov exponents for a C 2 diffeomorphism with an SRB measure, which was first introduced by Pesin and was explored in depth by Ledrappier and Young in [4]. Moreover, as a generalization of Pesin's entropy formula, for a C 2 diffeomorphism, Ledrappier and Young gave the dimension formula for all invariant measures in [5]. All above results are also generalized to random cases under different settings, and the reader can refer to [7] for a review.
In order to describe the complexity of a dynamical system more finely, various forms of entropies were introduced. One useful entropy is the so-called local entropy including local measure-theoretic entropy and local topological entropy, and a local version of variational principle was also obtained. For a topological dynamical system (X, T ) and an open cover U of X, Romagnoli [12] introduced two notions of local measure-theoretic entropies with respect to U, h µ (T, U) and h + µ (T, U) with h µ (T, U) ≤ h + µ (T, U), then a variational principle was obtained as follows: where h top (T, U) is the local topological entropy with respect to U. For random case, Ma and Chen [9] introduced random entropies h An interesting question is that can we introduce unstable entropies including both topological and measure-theoretic versions such that a similar variational principle can be obtained? Further more, can we obtain similar results for local entropies? Recently, Hu, Hua and Wu in [2] and Wu in [15] gave positive answers to the two questions for deterministic case respectively. In [2], for a C 1 -partially hyperbolic diffeomorphism on a closed Riemannian manifold, Hu, Hua and Wu introduced the so-called unstable measure-theoretic entropy and unstable topological entropy, and a variational principle was obtained relating them; in [15], in the same settings, the so-called local unstable measure-theoretic entropy and local unstable topological entropy were introduced by Wu, and a variational principle relating them was also established. For random case, a positive answer to the first question mentioned above under the condition of partial hyperbolicity is also obtained in [14] by Wang, Wu and Zhu. For a random dynamical system F with an invariant measure µ, they introduce the definitions of h u top (F) and h u µ (F) and following the line in [2] they establish the corresponding variational principle as follows sup{h u µ (F) : µ is F-invariant} = h u top (F). In this paper, we want to give a positive answer to the second question mentioned above for random case. Given a C 2 random partially hyperbolic dynamical system F (refer to Section 2 for its definition), an invariant measure µ respect to F, and an open cover U (refer to Section 2 for its precise definition), we introduce the definitions of h u µ (F, U), h u,+ µ (F, U), and h u top (F, U). And as a generalization of the topological entropy, a local version of pressure is also introduced. In order to establish a variational principle relating h u µ (F, U), h u,+ µ (F, U), and h u top (F, U), some relations are obtained between unstable entropies and their corresponding local versions; furthermore, the unstable topological conditional entropy and unstable tail entropy are introduced. In fact, both of them vanish in our settings, which implies us that the principle obtained in [14] can be applied in our proofs. In the end, as a generalization of the entropy, we establish the variational principle for local pressure.
This paper is organised as follows. In Section 2, we give some basic knowledge necessary for our goal and state our main results. In Section 3, we give the definitions of two kinds of local unstable measure-theoretic entropies, and some properties of these two local entropies and relations between them are also obtained. In Section 4, we give the definition of local unstable topological entropy with some important properties of them. In Section 5, we give the definitions of unstable topological conditional entropy and unstable tail entropy and their relations with local unstable entropies and unstable entropies, which are crucial to the proof of our variational principles. In the last section, i.e. Section 6, we give the proofs of the variational principles for both local entropies and local pressure.
2. Preliminaries and main results. Throughout this paper, let M be a C ∞ compact Riemannian manifold without boundary. Denote by B(M ) the Borel σalgebra of M . Let (Ω, F , P) be a Polish probability space and θ be an invertible and ergodic measure-preserving transformation on Ω.
which has the following properties: (i) F is measurable; (ii) the maps F(n, ω) : M → M form a cocycle over θ, i.e. they satisfy for all n, m ∈ Z and ω ∈ Ω; (iii) the maps F(n, ω) : M → M are C 2 for all n ∈ Z and ω ∈ Ω.
For each ω ∈ Ω, we define Associated with Ω × M , there is a skew product Θ induced by F, i.e., Definition 2.2 (Invariant measure). A measure µ on Ω × M is said to be an F-invariant measure, if it is Θ-invariant and has marginal measure P on Ω. In particular, an F-invariant measure µ is said to be ergodic, if it is ergodic with respect to Θ.
We denote by M P (F) the set of all F-invariant measures. In order to apply Rokhlin's results for Lebesgue space, in the following part of this paper, for µ ∈ M P (F) we always consider the µ-completion of F × B(M ), which is still denoted by F × B(M ) for simplicity.
Throughout this paper, we always assume that the Probability P on Ω satisfies where |f | C 2 denotes the usual C 2 norm of f ∈ Diff 2 (M ), and log + a = max{log a, 0}. Similar to the deterministic case, we can define the Lyapunov exponents for F. Let Λ be the set of all regular points (ω, x) ∈ Ω × M in the sense of Oseledec. For (ω, x) ∈ Λ, Let λ 1 (ω, x) > · · · > λ r(ω,x) (ω, x) be its distinct Lyapunov exponents of F with multiplicities m j (ω, x) (1 ≤ j ≤ r(ω, x)).
Let µ ∈ M P (F), and denote by · the norm of vectors in the tangent space of M . By the Oseledec Multiplicative Ergodic Theorem, we know that Λ is Θ-invariant and µ(Λ) = 1. For each (ω, x) ∈ Λ, there is a splitting of T x M as follows such that for i = 1, . . . , r(ω, x), dimE i (ω, x) = m i (ω, x) and (By (1) we can choose Λ such that all the above Lyapunov exponents are finite.) Let u(ω, x) = max{j : λ j (ω, x) > 0}. For (ω, x) ∈ Λ, we define the set where d(·, ·) is the metric on M induced by its Riemannian structure. Let . It is clear that both E u (ω, x) and F u (ω, x) are invariant under the tangent map, i.e. for n ∈ Z, and D x f n ω F u (ω, x) = F u (θ n ω, f n ω x). The following proposition from [1] ensures that W u (ω, x) is an immersed submanifold of M .
When (ω, x) ∈ Ω × M \ Λ, we let W u (ω, x) = {x}, making the definitions of unstable topological entropy in subsequent sections more transparent. We call the where I is an index set. Denote by U(ω, x) the element of U containing (ω, x). It is clear that It is clear that a measurable partition α of Ω × M can be regarded as a Borel cover of Ω × M , and α ω is a measurable partition of M , for all ω ∈ Ω.
The following proposition ensures the existence of a class of useful partitions.
Proposition 2 (Cf. [1]). Let µ ∈ M P (F). Then there exists a measurable partition ξ u of Ω × M satisfying the following properties: where ε is the partition of Ω × M into points, i.e. ξ u is a generator; (iv) B( A partition satisfying Proposition 2 is called an increasing partition subordinate to unstable manifolds, and denote by Q u (Ω × M ) the set of all such partitions. Now we give the definition of random partially hyperbolic dynamical systems. A random variable s : [13] for deterministic case). Notice that the uniform expansion in (ii) is a crucial property for our proofs of the main results. We think that the similar results should hold for C 2 RDSs with u-domination, but more complicated techniques involving Pesin theory must be applied.
Basic assumption. In the remaining of this paper, we always assume that F is a random partially hyperbolic dynamical system.
Example. Let f be a C 2 partially hyperbolic diffeomorphism. Combining the techniques in [6] and [3], we can obtain a random partially hyperbolic dynamical system satisfying the above assumption via small C 2 random perturbations of f .
Choose and fix a variable λ 0 such that Denote by P u (Ω × M ) the set of all partitions constructed by the above method.
Remark 3. It is easy to check that if for P-a.e. ω, µ ω ( A∈αω ∂A) = 0, then a measurable partition described as above is a partition subordinate to the W ufoliation.
Given U ∈ C o Ω×M , as generalizations of local unstable measure-theoretical entropies and local unstable topological entropy in [15], we introduce corresponding notions for random case, denote them by h u . Now we can give our main results as follows.
Theorem A. Let F be a random partially hyperbolic dynamical system, and U ∈ C o Ω×M with small enough diameter. Then for any µ ∈ M P (F) and is the unstable topological entropy (for more details, we refer the reader to the paper [14]).
As a corollary of Theorem A and Theorem D in [14], we have the following theorem.
Theorem B. Let F be a random partially hyperbolic dynamical system, and U ∈ C o Ω×M with small enough diameter. Then for any We can generalize above results to the unstable pressure for random case. Firstly, we have the following theorem. Denote the set Theorem C. Let F be a random partially hyperbolic dynamical system, and U ∈ C o Ω×M with small enough diameter. Then for any φ ∈ L 1 (Ω, C(M )), we have Applying Theorem C in [14], we can obtain a variational principle as follows.
Corollary D. Let F be a random partially hyperbolic dynamical system, and U ∈ C o Ω×M with small enough diameter. Then for any φ ∈ L 1 (Ω, C(M )), we have 3. Local unstable measure-theoretic entropy. In this section, we give the precise definition of local unstable measure-theoretic entropy for a random partially hyperbolic dynamical system. Firstly we give some knowledge on the information function, which is slightly modified in our context. Definition 3.1. Let µ be an invariant measure of (Ω × M, Θ), α and η be two measurable partitions of Ω × M . The information function of α with respect to µ is defined as and the entropy of α with respect to µ is defined as The conditional information function of α with respect to η is defined as ∈Ω×M is the canonical system of conditional measures of µ with respect to η. Then the conditional entropy of α with respect to η is defined as For simplicity, sometimes we will use the following notations. For ω ∈ Ω, denote It is clear that The following definition comes from [14].
The conditional entropy of F with respect to ζ is defined as and the conditional entropy of F along W u -foliation is defined as In the following, we give some useful conclusions from [14]. For deterministic case, see [2,16].
Lemma 3.7. Let α ∈ P(Ω × M ) with diameter smaller than 0 /λ 0 , and η ∈ P u (Ω× M ), then for any n ∈ N, we have For the choice of y, we have f As 0 is small enough, we have Then, we can obtain by induction. Recall that F is uniformly expanding on W u , so we have d u θ n ω (x, y) < C 0 (θ n−1 ω) 0 (θ n−1 ω) <¯ 0 (θ n ω),

Then we have
On the other hand, we have where in the last inequality, Lemma 3.7 is applied. Then by Lemma 3.9 we have = ρ + h u µ (F, α|η). Since ρ > 0 is arbitrary, we have h u µ (F, β|η) ≤ h u µ (F, α|η); then by the arbitrariness of β, we complete the proof.
The following lemma gives the relationship between h u µ (F, β|η) and h u µ (F, β|ξ), whose proof is similar to that in [15], so we omit its proof. We also need the following theorem from [14].
Theorem 3.12 (Theorem B in [14]). Suppose µ is an ergodic measure of F, and η ∈ P u (Ω × M ). Then for any measurable partition α of Ω × M with H µ (α|η) < ∞, we have lim Now we give the following proposition, which plays an important role in this paper.
Proof. We prove Proposition 3 in two cases. Case 1 For ζ ∈ P u (Ω × M ). This is the results of Lemma 3.9 and 3.10. Case 2 For ζ ∈ Q u (Ω × M ).
The following corollary can be obtained easily from Proposition 3.

LOCAL UNSTABLE ENTROPY AND PRESSURE FOR RPHDS 93
The following proposition gives the relation between h u µ (F, U|ζ) and h u,+ µ (F, U|ζ), whose proof is similar to that of Proposition 3.16 in [15] and omitted here.
In the definition of h u µ (F, U|ζ), we use "lim sup"; in fact, we can show that for any η ∈ P u (Ω × M ), it can be replaced by "lim". To prove this, we need some lemmas.
in the third and last inequality Lemma 3.14 and Lemma 3.8 are applied respectively. Because of the arbitrariness of α and γ, we have As in the proof of Lemma 3.9, we have showed that {H µ (U n−1 0 |η) + H µ (β)} is a subadditive sequence, which implies what we need.
For the proof, we refer the reader to Proposition 3.19 in [15].
The following lemmas are useful for the proof of Theorem A.
Proof. The proof is similar to those of Lemma 3.7 and Lemma 3.17, and we refer the reader to the proof of Lemma 3.20 in [15]. Proof. In the proof, Lemma 3.17, Lemma 3.8 and Proposition 4 are used, which is completely parallel to that of Lemma 3.21 in [15], so we omit it here. Proof. Choose arbitrary α ≥ U n−1 0 , as in Lemma 3.10, we can show that Then follow the line of proof of Lemma 3.22 in [15], for any β ∈ P(Ω × M ) and ρ > 0 we can show that h u µ (F n , β|η) ≤ ρ + h u µ (F n , α|η). Then by the arbitrariness of β, ρ and α, we have And it is clear that nh u µ (F|η) ≥ h u,+ µ (F n , U n−1 0 |η).

4.
Local unstable topological entropy and pressure. In this section, we give the definition of local unstable topological entropy of F with respect to a Borel cover U ∈ C Ω×M . Let K ⊂ Ω × M . For any U ∈ C Ω×M , denote min{the cardinality of V : V ⊂ U, V ∈V V ⊃ K} by N (K, U), and denote log N (K, U) by H(U|K). |W u (ω, x, δ)), For any ρ > 0 and each ω ∈ Ω, there exists y ω such that Then choose 0 < δ 1 < δ small enough such that There exists a positive number N = N (ω) which depends on δ, δ 1 and the Riemannian structure on W u (ω, y ω , δ) such that for some y j ∈ W u (ω, y ω , δ), j = 1, 2, · · · , N . Then we have

XINSHENG WANG, WEISHENG WU AND YUJUN ZHU
Integrating both sides of the above inequality, we get Thus, by (2) we have Since ρ is arbitrary, we have , which completes the proof of Lemma 4.2.
As a generalization of local unstable topological entropy, we can give the definition of local unstable pressure of F.
Then the local unstable pressure of F with respect to φ is defined as Now we give the relation between local unstable measure-theoretic entropy and local unstable topological entropy.
, which completes the proof of Proposition 5.

5.
Unstable topological conditional entropy and unstable tail entropy. In this section, we give the definitions of unstable topological conditional entropy and unstable tail entropy, which are useful in the proof of Theorem A, for the case when The following proposition is a collection of properties for H u , whose proof is simple.
)} is subadditive, hence we complete the proof.
Because of Lemma 5.2, we have the following definition.  The following proposition is a collection of properties of unstable conditional entropy.
, then by Proposition 6, we obtained what we need.
The following proposition is important. In the next, we begin to define the unstable tail entropy of F in the sense of Bowen.
In fact, in our setting, both unstable conditional entropy and unstable tail entropy vanish. So we have the following theorem.
6. Variational principles for local unstable entropy and unstable pressure. In this section, we prove Theorem A, then variational principles for local unstable entropy and unstable pressure are obtained. First of all, we give two propositions as follows.

Then we have
Thus Since β ≥ U is arbitrary, we complete the proof.
Proof of Theorem A. We divide the proof into two cases. Case 1 for η ∈ P u (P × M ).
Let U ∈ C Ω×M with diam(U) 0 . By Corollary 1, we know that h u,+ µ (F, U|ζ) = h u µ (F|ζ) = h u µ (F) Then by (5), (6) and Proposition 5, we know that h u µ (F) = h u µ (F, U|ζ) ≤ h u top (F, U). By the variational principle for unstable entropy of F we can obtain And it is easy to see that h u top (F, U) ≤ h u top (F), then we have h u top (F, U) = h u top (F). This ends the proof of Theorem A for η ∈ P u (Ω × M ). Case 2 for ξ ∈ Q u (P × M ).
Let U ∈ C Ω×M . By Theorem 5.5 and Proposition 11 we have h u µ (F|ζ) = h u µ (F, U|ζ). By Corollary 1, we know that As an application of Theorem A, following the line of Proposition 3.25 in [15], we have the following proposition.
Proposition 12. For U ∈ C o Ω×M , the local unstable entropy map µ → h + µ (F, U|η) is upper semi-continuous for η ∈ P u (Ω × M ). Now we begin to discuss the variational principle for local pressure. Firstly, we need the following lemma from [10], which is adapted in our context. For V ∈ C Ω×M , let α be the Borel partition generated by V; given ω ∈ Ω, let P * ω (V) ={β ∈ P(M ) : β ≥ V ω and each atom of β is the union of some atoms of α ω }.