FORMULA OF ENTROPY ALONG UNSTABLE FOLIATIONS FOR C 1 DIFFEOMORPHISMS WITH DOMINATED SPLITTING

. Metric entropies along a hierarchy of unstable foliations are inves-tigated for C 1 diﬀeomorphisms with dominated splitting. The analogues of Ruelle’s inequality and Pesin’s formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.


Introduction
It is well known that among the major concepts of smooth ergodic theory are the notions of invariant measures, entropy and Lyapunov exponents. Entropies, including measure-theoretic entropy and topological entropy, play important roles in the study of the complexity of a dynamical system.
Intuitively, topological entropy measures the exponential growth rate in n of the number of orbits of length n up to a small error, measure-theoretic entropy gives the maximum average information with respect to some invariant measure one can get from a system. While Lyapunov exponents reflect the rate at which two nearby orbits separate from each other. What interests one is the relation between entropy and Lyapunov exponents. Let f be a C 1 diffeomorphism on a compact Riemannian manifold M without boundary. For any regular point in the sense of Oseledec [9] x ∈ M , let λ 1 (x) > λ 2 (x) > · · · > λ r(x) (x) denote its distinct Lyapunov exponents, and E 1 (x) ⊕ · · · ⊕ E r(x) (x) be the corresponding decomposition of its tangent space T x M . In 1970s, Ruelle [14] gave the following inequality h µ (f ) ≤ M λi(x)>0 denote its distinct Lyapunov exponents and let be the corresponding decomposition of its tangent space. Df (2) (Dominated splitting on an invariant set) Let ∆ be an f -invariant set and T ∆ M = E ⊕ F be a Df -invariant splitting on ∆. We call T ∆ M = E ⊕ F to be (N, i(y))-dominated splitting, if the dimension of F at y is i(y)(1 ≤ i(y) ≤ dim M − 1) and there exists a constant N ∈ + such that In the following, we consider two cases of the invariant measure µ.
Case 1 µ is ergodic. In this case, the functions x → r(x), λ i (x) and dim E i (x) are constant µa.e., denote them by r, λ i and m i respectively. Let u = max{i : λ i > 0}, u(i) = u − i + 1, and where d is the Riemannian metric on M . The following result ensures that W i (x) is an immersed C 1 -manifold under the assumption of dominated splitting.
Proposition 1 ([1, Proposition 8.9]). Let µ be an ergodic measure whose support admits a dominated splitting E ⊕ F , let λ + E < λ − F be the maximal Lyapunov exponent in E and the minimal Lyapunov exponent in F of the measure µ.
which is a stable manifold, and for any For x ∈ Γ and 1 ≤ i ≤ u, let .
Remark 1. It is obvious that any hyperbolic automorphism on two dimensional tori satisfies Assumption 1.
Under Assumption 1, we know that, by Proposition 1, and contains an open neighborhood of x in W i (x). An important property with respect to such a partition is that there is a canonical system of conditional measures {µ ξ i x }. The following lemma ensures the existence of such partitions.
Proof. For the proof, the reader can refer to [11].
For more details about measurable partitions and conditional measures the reader can refer to Section 0.1 -0.3 in [7] and Section 3 and 4 in [13].
Let ξ i be a measurable partition subordinate to W i with conditional measures h i µ (f, x) is well defined and is independent of the choice of ξ i or µ ξ i x , and it is easy to verify that Definition 2.3. We define the entropy of f along ith unstable foliation by Since µ is ergodic, we know that h i µ (f, x) = h i µ (f ), for µ-a.e.x ∈ M .
Case 2 µ is arbitrary. In this case, the functions x → r(x), λ i (x) and dim E i (x) are now measurable.
Let u(x) = max{i : λ i (x) > 0}, u(i, x) = u(x) − i + 1, and Γ i = {x ∈ Γ : u(i, x) > 0}. Then we can define W u(i,x) (x) as in (2.1) except that u(i) and λ u(i) should be replaced by u(i, x) and λ u(i,x) (x) respectively, and the choice of ε depends on x such that ε < λ u Similar to that in Proposition 1, under Assumption 2, and contains an open neighborhood of x in W i (x). For the existence of such ξ i , one can simply disintegrate µ into its ergodic components and note that the entire leaf W i (x) is contained in the ergodic component of x(cf. [6]). There is a canonical system of conditional measures

Remark 2.
It is easy to check that when µ is ergodic, Γ i = Γ for 1 ≤ i ≤ u. So when µ is ergodic, the entropy along ith unstable foliation on Γ i coincides with the entropy along ith unstable foliation.
So we call the entropy defined as above the entropy of f along ith unstable foliation.
Standing hypotheses for the remaining of this paper: When µ is ergodic, we set Assumption 1, and when µ is arbitrary, we set Assumption 2. Now we are ready to state our main results of this paper: Theorem A. Let µ be an invariant measure. Then we have the following inequality Moreover, if µ satisfies some additional conditions, we have the following theorem.
Theorem B. Let µ be an invariant measure satisfying that for µ-a.e. x ∈ M and every measurable x is the corresponding Riemannian measure on W i (x). Then we have the following entropy formula In particular, if µ is ergodic, then we have Remark 3. We only need to prove the ergodic versions of Theorem A and Theorem B respectively, and the nonergodic versions of them follow immediately from the ergodic versions by decomposing µ into ergodic components (just as that has been done in [6]). So in the following two sections, we always assume that µ is ergodic.
In the following, we relate the entropy h i µ (f ) along the unstable foliation W i with the supremum of certain conditional entropy of finite partitions with respect to a measurable partition subordinate to W i . This idea derives from [4].
The following proposition gives an equivalent definition of h i µ (f ). [4], where ξ is a partition subordinate to the unstable foliation W u . We omit the details.

Proposition 2. Let µ be an ergodic measure, then we have
Remark 4. In fact, the partitions used in Definition 2.4 and Proposition 2 can be replaced by some more natural partitions. Roughly speaking, such partition is constructed via the intersection of a finite partition and the local unstable manifolds. For more details, the reader can refer to [4].
As the classical measure-theoretic entropy and the topological entropy, the entropy along ith unstable foliation also has the so-called power rule.

Proof of Theorem A
Now we complete the proof of Theorem A. Firstly, we need the following definition from [4].
Definition 3.1. Pick 0 < δ < r, where r is as in the proof of Lemma 2.2, and ε > 0 small enough.
we call S an (n, ε) i-separated set of W i (x, δ). Let N i (f, ε, n, x, δ) denote the largest cardinality of any (n, ε) i-separated set in W i (x, δ).
we call R an (n, ε) i-spanning set of W i (x, δ). Let S i (f, ε, n, x, δ) denote the smallest cardinality of any (n, ε) i-spanning set in W i (x, δ).
The following lemma gives us a relation between N i (f, ε, n, x, δ) and S i (f, ε, n, x, δ).  Proof. cf. the proof of Lemma 3.5 in [4].
The estimation of h i µ (f ) from above is based on the following lemma. Proof. Let ξ i = ξ be any measurable partition subordinate to ith unstable foliation as in Lemma 2.2.
Since µ is ergodic, then we can pick x ∈ Σ with the following property: there exists a set B ⊂ ξ(x) It is clear that µ ξ x (B) = 1 and B satisfies the property above. The property above implies that for any ρ > 0 and y ∈ B, there exists ε 0 (y), such that if 0 < ε < ε 0 (y), then . Then there exists a set R n with cardinality no more than S i (f, ε 2 , n, x, δ), such that and V i (f, z, n, ε 2 ) ∩ B = ∅. Choose an arbitrary point in V i (f, z, n, ε 2 ) ∩ B and denote it by y(z). Then we have And hence S i (f, ε 2 , n, x, δ) ≥ e n(h i µ (f,ξ)−ρ) . Thus we have Now we begin the proof of Theorem A. Let S n ⊂ W i (x, δ) be an (n, ε) i-separated set with the largest cardinality. When n is large enough, we can pick y n ∈ S n such that where exp x is the exponential map at x,Ṽ x,δ = Vol(exp −1 W i (x, δ)), and Vol(·) denotes the volume function.
Because of the compactness of M , we can choose δ > 0 small enough such that exp x is a diffeomor- In order to avoid a cumbersome computation, for every x ∈ M , we treat the tangent space T x M as it were Ê n . We denote the Jacobian determinant For any ε > 0, we can choose 0 < δ(ε) < δ 2 such that ||J x (y 1 )| − |J x (y 2 )|| < ε, for any x ∈ M and y 1 , y 2 ∈ π x B(0 x , δ(ε)), where π x : T x M → F (x) is the projection and 0 x is the null vector in T x M . Let ε 0 : = 1 2 inf{|J x (y)| : x ∈ M, y ∈ π x B(0 x , δ 2 )}. Then for any x ∈ M , we have for any y 1 , Then we have |Jỹ n (f n−1 (y))||J fỹn (f n−2 (y))| · · · |J f n−1ỹ n ((y))|dλ, wheref j (y) = exp −1 f n−j (ỹn) f −j exp f n (ỹn) , j = 1, 2, · · · , n − 1 and λ is the Lebesgue measure on B i n . Notice the definition of B i n , so we havẽ for j = 1, 2, · · · , n − 1 and y ∈ B i n . Hence by (3.3) we have So we have The last equality follows that D exp x | 0x is an identity.
Let R m,ε ′ be the set of x ∈ M such that for any n ≥ m and v ∈ E i (x), we have D x f −n v ≥ e n(−λi−ε ′ ) . By Oseledec's Theorem, we know that lim m→∞ µ(R m , ε ′ ) = 1.
It is easy to check that A is an f -invariant set and µ(A) = 0. So for ε, there exists N > 0 such that for any n ≥ N , And for every n ≥ N we can choose an appropriate x n ∈ M such that So when n is large enough such that we can pickỹ n from the set B(y n , ε 2 ) ∩ W i (f n (x n )) ∩ R n,ε ′ . Hence when n is large enough, we have where the constant C only related to f .
)), and notice that lim sup n→∞ 1 n log V x,δ = 0, so let ε ′ → 0, using Lemma 3.2 and Lemma 3.3, we obtain Let ρ → 0, we obtain For N > 0, let g = f N , then we have Let N → ∞, we obtain Now we have completed the proof of Theorem A.

Proof of Theorem B
Now we start to prove Theorem B. By Theorem A, we only need to complete the estimation of h i µ (f ) from below. Firstly, we give the following lemma.
Set A be the σ-algebra generated by partitions α n , n ≥ 0. Letμ ξ x ,λ i x be two measures on M satisfyingμ It is easy to verify thatμ ξ x ≪λ i x . Let k : M → Ê be aλ i x -integrable function with respect to A such that Such a function exists because thatμ ξ x ≪λ i x . It follows from (4.2) that And hence we have .
Observe that for ε 0 > 0 small enough, we can take C > 0, such that for any x ∈ M , Notice the relationship betweenμ ξ x and µ ξ x ,λ i x and λ i x respectively, if ε ≤ ε 0 is small enough, we have Integrating both side of (4.4), we obtain By Fatou's Lemma we obtain lim sup Combining (4.5), (4.6), and (4.7), we obtain Let{ε k } k≥1 be a sequence such that ε k > 0 and ε → 0 as k → ∞. Then by the monotone convergence theorem, we have Therefore (4.1) follows from (4.8) and (4.9).
Before going into the proof of Theorem B, we need a technical lemma from [16]. In the statement of the lemma, we will use the following definition from [8].
The number sup{ ψ(x)−ψ(y) x−y |x = y ∈ U } is called the dispersion of G.
The following lemma about graph transform on dominated bundles is a generalization of Lemma 3 in [8] by Sun and Tian [16].
The following lemma is also useful for the proof of Theorem B. any (E x , F x )-graph G with dispersion ≤ c contained in the Bowen ball V (g, x, n, δ)(n ≥ 0), its image Proof. cf. the proof of Lemma 3.4 in [16].
Now we are ready to prove Theorem B. Fix any ε > 0. Take N 0 so large that the set Γ i,ε = {x ∈ Γ : D i (x) ≤ N 0 } has µ-measure larger than µ(Γ i ) − ε. Let N = N 0 ! and g = f N , then the splitting T Γi,ε M = E ⊕ F satisfies (1, i(x))-dominated with respect to g: Note that Γ i,ε is f -invariant and thus is g-invariant.
In what follows, in order to avoid a cumbersome and conceptually unnecessary use of coordinate charts, we shall treat M as if it were a Euclidean space, and let λ be the Lebesgue measure on M . The reader will observe that all our arguments can be easily formalized by a completely straightforward use of local coordinates.
Since dominated splitting can be extended on the closure of Γ i,ε , and dominated splitting is always continuous(see [2]), we can fix two constants c > 0 and a > 0 so small that if x ∈ Γ i,ε , y ∈ M and d(x, y) < a, then for every linear subspace E ⊆ T y M which is a (E(x), F (x))-graph with dispersion < c we have Thus | det(D y g)| E | ≥ det |(D x g)| F (x) | · e −ε . (4.10) By Lemma 4.4, there exists δ ∈ (0, a) such that for every x ∈ Γ i,ε and any (E x , F x )-graph G with dispersion ≤ c contained in the ball V (g, x, n, δ)(n ≥ 0). Its image g n (G) is a ((D x g n )E x , (D x g n )F x )graph with dispersion ≤ c.
The estimation of h i µ (f ) from below is based on the following fact.
If Λ n (y) is not empty, by Lemma 4.4, we have g n (Λ n (y)) is a (E(g n (x)), F (g n (x)))-graph with dispersion ≤ c.
Using the fact, we obtain Since ε is arbitrary, this completes the estimation of h i µ (f ) from below.