Measure theoretic pressure and dimension formula for non-ergodic measures

This paper first studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carath\'{e}odory-Pesin structure described in \cite{Pes97}, and show that this quantity is equal to the essential supremum of the free energy of the measures in an ergodic decomposition. To the best of our knowledge, this formula is new even for entropy. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the free energy if the measure is ergodic. Particularly, if the dynamical system satisfies the uniform separation condition and the ergodic measures are entropy dense, this quantity is still equal to the the free energy even if the measure is non-ergodic. As an application of the main result, we find that the Hausdorff dimension of an invariant measure supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and volume-preserving, the Hausdorff dimension of any invariant measure on the hyperbolic set is equal to the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions.


Introduction
Let (X, f ) be a topological dynamical system, i.e., f is a continuous transformation on a compact metric space X. Given a continuous function ϕ on X and an f -invariant measure µ, the following quantity h µ (f ) + X ϕdµ is called the free energy of (f, ϕ) with respect to µ, where h µ (f ) is the Kolmogorov-Sinai entropy of f defined via measurable partitions (see Walters' book [18] for more details).
The free energy is an important quantity in the study of dynamical system, e.g., it appears as rate function in the study of large deviations in dynamical system (see [23]), and the well-known variational principle which relates the topological pressure with free energy (see [17]). To give other viewpoints for free energy, using ideas of Katok's entropy formula [9], in [8] He et al. showed that the free energy can be regarded as the growth rate of the minimal value of a potential on an (n, ǫ, δ)spanning set ( the union of these n-th Bowen balls centered at the points of this set has measure at least 1 − δ). In [13], using the theory of Carathéodory-Pesin structure, Pesin gave an explanation for free energy from the dimensional viewpoint. The authors in [6] extended the works in [8] and [13] for a larger class of potentials, e.g., the sub-additive and super-additive potentials, it is useful to estimate the dimension of an ergodic measure supported on non-conformal repellers (see [6] and [19]). In [24], the third author of this paper extended the works in [8] and [13] to amenable group actions.
We would like to stress that all the results mentioned above require that the measure to be ergodic. However, we know nothing if the measure is non-ergodic. In [2], Barreira and Wolf studied the Hausdorff dimension and point-wise dimension of invariant measures that are not necessarily ergodic. Namely, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces, they established formulas for the point-wise dimension of an invariant measure in terms of the local entropy and of the Lyapunov exponents. Furthermore, they proved that the Hausdorff dimension of an invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition.
The results described above indicate that a given global quantity has behind it or in certain case can even be build with the help of a local quantity. Another example is that the Kolmogorov-Sinai entropy of an invariant measure can be obtained by integrating the Brin-Katok's local entropy (see (2.3)). Motivated by Barreira and Wolf's results in dimension theory, following the approach described in [13], we define the measure theoretic pressure from the dimensional viewpoint. Furthermore, we define the point-wise measure theoretic pressure by combining Brin-Katok's local entropy formula and Birkhoff's ergodic theorem. In this paper, we prove that the measure theoretic pressure of any invariant measure is the essential supremum of the point-wise measure theoretic pressure (see Proposition 4.1), then we show that the measure theoretic pressure of any invariant measure can be regarded as the essential supremum of the free energies of the measures in an ergodic decomposition (see Theorem 4.1). We would like to point out that these results are new even for entropy.
As an application of Theorem 4.1, we show that the Hausdorff dimension of an invariant measure (not necessarily ergodic) supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure (see Theorem 4.5), which can be regarded as Bowen's equation in the measure theoretic sense. To show this result, we first show that the Hausdorff dimension of an invariant measure on average conformal repellers of C 1 maps is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition (see Theorem 4.4), which extends Barreira and Wolf's Hausdorff dimension formula (see [2,Theorem 10]) to C 1 maps and non-conformal case. Following the same approach, we can show that the Hausdorff dimension of an invariant measure supported on average conformal hyperbolic sets of a C 1 volume-preserving diffeomorphism is given by the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions (see Theorem 4.7). This paper also defines the measure theoretic pressure of invariant measures (not necessary ergodic) by using (n, ǫ)-separated sets and (δ, n, ǫ)-separated sets (see Section 2.3 for details). For each ergodic measure, we show that the measure theoretic pressure defined in this way is exactly the free energy, which means that this definition of measure theoretic pressure is respectively equivalent to the ones in [8] and [13]. Furthermore, if the dynamical system has the uniform separation property (see Definition 2.4) and the ergodic measures are entropy dense (see Definition 2.3), then this kind of measure theoretic pressure is still equal to the free energy even if the measure is non-ergodic. This paper is organized as follows. In Section 2, we introduce the concepts of point-wise measure theoretic pressure and measure theoretic pressure for any invariant measure. We also define the measure theoretic pressure via separated sets. In Section 3, we give some auxiliary results that are useful in the proof of main results in this paper. Section 4 contains the statements and detailed proofs of our main results. We first prove that the measure theoretic pressure is the essential supremum of the sum of Brin-Katok's local entropy and the limit of Birkhoff's sum of a continuous function, which helps us to show that the measure theoretic pressure is the essential supremum of the sum of Kolmogorov-Sinai entropy and the integral of the potential with respect to the measures in an ergodic decomposition. Meanwhile, we show that the measure theoretic pressure of ergodic measures defined by separated sets is equal to the free energy. In particular, the measure theoretic pressure defined via separated sets is still equal to the free energy even if the measure is non-ergodic, provided the system has the uniform separation property and the ergodic measures are entropy dense. Finally, we show that the zero of measure theoretic pressure is exactly the Hausdorff dimension of an invariant measure on average conformal repellers of a C 1 map; the Hausdorff dimension of an invariant measure supported on hyperbolic sets is the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions, provided that the diffeomorphism is average conformal and volume-preserving.

Notation and Preliminaries
We give the definitions and notation in this section. Throughout this section, let (X, f ) be a topological dynamical system (TDS for short), i.e., f : X → X is a continuous transformation on a compact metric space X equipped with metric d. Let M(X) denote the space of all Borel probability measures on X, and let M(X, f ) and E(X, f ) denote the set of all f −invariant respectively, ergodic f −invariant Borel probability measures on X.
2.1. Point-wise measure theoretic pressure. For x, y ∈ X, define a dynamical metric as d n (x, y) = max{d(f i x, f i y) : 0 ≤ i < n}, and let B n (x, ǫ) = {y ∈ X : d n (x, y) < ǫ} denote the dynamical ball centered at x of radius ǫ and length n.
Given a continuous function ϕ : X → R and an invariant measure µ ∈ M(X, f ), the following quantity is called the point-wise measure theoretic pressure of ϕ at the point x (w.r.t. the measure µ), where S n ϕ(x) := n−1 i=0 ϕ(f i x). It follows from the Brin-Katok's local entropy formula (see [4]) and Birkhoff's ergodic theorem (e.g., see [18]) that the following limits exist µ-almost everywhere. Hence, Particularly, the two limits in (2.1) satisfy that where h µ (f ) is the Kolmogorov-Sinai entropy of f (see [18] for more details).
Definition 2.1. We call the quantity the topological pressure of (f, ϕ) on the set Z.
The above definition is equivalent to the one given by Pesin and Pitskel' [12] (see also [13] ), see [7] for the detailed proof.
Definition 2.2. We call the quantities the lower and upper topological pressures of (f, ϕ) on the set Z.
Given an f -invariant measure µ, let and then we call the following quantity the measure theoretic pressure of (f, ϕ) with respect to the measure µ. Let further We call the following quantities the lower and upper measure theoretic pressures of (f, ϕ) with respect to the measure µ. It is proved in [13] that for any f -invariant ergodic measure µ, see [6, Theorem A] for a generalization of the previous formula to a larger class of potentials.

2.3.
Measure theoretic pressure defined via separated sets. Given ǫ > 0 and δ > 0, recall that a subset E ⊂ X is (n, ǫ)-separated, if for any distinct points x, y ∈ E we have d n (x, y) > ǫ. Furthermore, a subset A ⊂ X is called (δ, n, ǫ)separated, if any distinct points x, y ∈ A satisfy that Given x ∈ X and n ∈ N, consider the empiric measure at point x ∈ X as follows: For each neighborhood F ⊂ M(X), put Given a continuous function ϕ : X → R, ǫ > 0 and ν ∈ M(X). Let F ⊂ M(X) be a neighborhood of ν, put and where the infimum is taken over any base of neighborhoods of ν and , we denote this common value as SP ν (f, ϕ). If we consider (δ, n, ǫ)-separated set in (2.5), we write the corresponding quantities as respectively. In Section 4, for a given TDS (X, f ) and a continuous function ϕ on X, we will prove that both SP ν (f, ϕ) and SP ′ ν (f, ϕ) exist and equal to the free energy of (f, ϕ) provided that ν is ergodic. Furthermore, if the system (X, f ) has uniform separation property and the ergodic measures of (X, f ) are entropy dense (which we will recall in below), then both SP ν (f, ϕ) and SP ′ ν (f, ϕ) exist and equal to the free energy of (f, ϕ) for any invariant measure ν (not necessarily ergodic).
In [14, Theorem 2.1], Pfister and Sullivan proved that the ergodic measures of a TDS (X, f ) are entropy dense if the system (X, f ) has the g-almost product property, e.g., all β-shift have the g-almost product property, and hence the ergodic measures of all β-shifts are entropy dense.
In [15, Theorem 3.1], Pfister and Sullivan proved that a TDS (X, f ) has the uniform separation property if the system (X, f ) is expansive or is asymptotically h-expansive.

Dimension of invariant measures.
In this subsection, we will recall the definition of dimension of a measure. Particularly, we will recall some formula for dimension of measures supported on average conformal repellers and average conformal hyperbolic sets. We say that f is expanding on J and that J is a repeller of f if v ∈ T x M and n ≥ 1, where · is the norm induced by the Remannian metric on M .
For x ∈ M and v ∈ T x M , the Lyapunov exponent of v at x is the limit if the limit exists. Given an f -invariant measure µ, by the Oseledec multiplicative ergodic theorem [11], for µ-almost every point x, every vector v ∈ T x M has a Lyapunov exponent, and they can be denoted by See [2] for the formula of point-wise dimension in the case of conformal repellers of C 1+α maps. Next, we briefly recall the Hausdorff dimension of a probability measure. Given a set Z ⊂ M , its Hausdorff dimension is defined by where U is an open cover of Z and diam U = sup{diam U : U ∈ U}. If ν is a probability measure on M , then the Hausdorff dimension of the measure ν is given by In fact, this estimation is valid for other dimensions of a measure, e.g., box dimension, information dimension etc, see [22] for details. This implies that for any ergodic measure ν supported on average conformal repellers, one has there exist an open neighborhood U such that J = n∈Z f n (U ), and a continuous splitting of the tangent bundle T J M = E s ⊕ E u , and constants c > 0 and κ ∈ (0, 1) such that for each x ∈ J the following properties hold: That is, the map has only two different Lyapunov exponents λ s (ν) < 0 and λ u (ν) > 0 with respect to each ergodic measure ν on J. Let and µ an f -invariant measure on J. Since f is average conformal on J, for µ-almost every x ∈ J the following limits are well-defined See Lemma 2.1 in [21] for the proof. The numbers λ s (x) and λ u (x) are respectively the negative and positive values of the Lyapunov exponent at x. Furthermore, if µ is ergodic then they are constant almost everywhere. Assume that f is a C 1 diffeomorphism with a compact f -invariant locally maximal hyperbolic set J on which f is average conformal. Then the following properties hold: (1) for any ergodic measure ν on J, one has with the essential supremum taken with respect to µ, see [20, Corollary 2] for details.

Auxiliary results
In this section, we will provide some preliminary results about the separated sets and free energy.
Let (X, f ) be a TDS, and let ξ be a finite measurable partition of X and µ ∈ M(X, f ). Then for any Borel set A ⊂ X with 0 < µ(A) < 1, we have Proof. See [18,Theorem 8.1] for the proof.
Next we recall some notations that would be used in the following. Given a finite set A, let #A denote the cardinality of A and let Λ n := {0, 1, · · · , n − 1}. For Proof. See [14, Lemma 2.1] for the proof.
Utilizing the method in [14], we prove the following proposition.
If h ν (f ) = 0, we choose Γ n = {x}, where x ∈ K n , then the desired result follows immediately. If In the same way to define φ : X → {1, · · · , k} N , x → (ω 0 ω 1 · · · ω n−1 · · · ). Let Since h ν (f, ξ) > h ′ , there exists n ξ such that whenever n ≥ n ξ , Since ν is regular, for a small number δ > 0, there exists compact subsets B j ⊂ A j (j = 1, · · · , k) so that Let F ⊂ M(X) be a neighborhood of ν, and let where I B denotes the indicator function of B. Since B is closed, I B is upper semicontinuous. By the Birkhoff Ergodic theorem, we have that lim n→∞ ν(X B n,F ) = 1. Hence, for all sufficiently large n. We define ν n,δ so that for each w ∈ Y n , Note that for any µ ∈ M(X), we have that H(µ, ξ n ) ≤ n log k. Therefore, there exists n F,δ ≥ n ξ such that for each n ≥ n F,δ we have where the second inequality follows from Lemma 3.1. Put For any w ∈ Y n , take a point x n,w ∈ φ −1 n (w) such that x n,w ∈ X B n,F . Let Ξ n denote the set which consists of all of these points x n,w chosen in this way. Obviously, Ξ n ⊂ X B n,F . By (3.2) and the construction of Ξ n , we have that Let Γ n ⊂ Ξ n be a set of maximal cardinality satisfying that Since {B j : j = 1, 2, · · · , k} are mutually disjoint compact subsets, there exists ǫ δ > 0 such that Using (3.4) and (3.5), for x = x ′ ∈ Γ n we have that This implies that Γ n is (δ, n, ǫ δ )-separated. For each x ∈ Ξ n , by the maximality of Γ n , there exists x ′ ∈ Γ n such that d H n (φ n (x), φ n (x ′ )) ≤ 3δn. Hence, for n ≥ n F,δ , by Lemma 3.2 and (3.3) we have that In Proposition 3.1, the numbers δ * and ǫ * in the (δ * , n, ǫ * )-separated may depend on the ergodic measure ν. However, if the dynamical system (X, f ) has uniform separation property, δ * and ǫ * can be chosen independently of the ergodic measure.
The following result is one of the key ingredient in proving the variational principle of topological pressure, see [18] for details.

Main results
In this section, we will give the statements and proofs of the main results in this paper. Our first result shows that the measure theoretic pressure of an invariant measure is the essential supremum of point-wise measure theoretic pressure. Using the first result, one can further show that the measure theoretic pressure of an invariant measure is the essential supremum of the free energies of the measures in an ergodic decomposition. Meanwhile, we show that the measure theoretic pressure of ergodic measures defined via separated set is equal to the free energy. Furthermore, if the dynamical system has the uniform separation property and the ergodic measures are entropy dense, then the measure theoretic pressure of invariant measures (not necessarily ergodic) defined via separated set is still equal to the free energy. Finally, we show that the Hausdorff dimension of an invariant measure supported on a C 1 average conformal repeller is exactly the zero of the measure theoretic pressure of the same measure. Similarly, we show that the Hausdorff dimension of an invariant measure on hyperbolic sets is the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions, provided that the diffeomorphism is volume-preserving and average conformal.
4.1. Point-wise measure theoretic pressure and measure theoretic pressure. We first recall a useful property relating point-wise measure theoretic pressure and topological pressure on arbitrary subsets: [16, Theorem A] for proofs. See [10] for the original version of Bowen entropy. Let (X, f ) be a TDS, and ϕ : X → R a continuous function on X and µ ∈ M(X, f ), the following properties hold: Proof. See [16,Theorem A] for the proof of the second statement. To prove the first statement, let Z = {x ∈ X : P µ (f, ϕ, x) ≥ α}. It follows that µ(Z) = 1, and by (1) we have that By the definition of measure theoretic pressure, we have P µ (f, ϕ) ≤ α.
Let (X, f ) be a TDS, and ϕ : X → R a continuous function on X and µ ∈ M(X, f ), then with the essential supremum taken with respect to µ.
Note that the functions h µ (·) and ϕ * (·) in (2.1) are f -invariant almost everywhere, using(2.2) and Proposition 4.1, we obtain the following formula for the measure theoretic pressure of non-ergodic measures. Let (X, f ) be a TDS, and ϕ : X → R a continuous function on X and µ ∈ M(X, f ), then with the essential supremum taken with respect to µ. If, in addition, µ is ergodic then Consequently, we get the following formula for entropy. with the essential supremum taken with respect to µ. If, in addition, µ is ergodic then for µ-almost every x ∈ X.

4.2.
Measure theoretic pressure and ergodic decompositions. This subsection is devoted to discuss the relation between the measure theoretic pressure of an invariant measure with its ergodic decompositions. Recall that a probability Borel measure τ on M(X, f ) (with the weak-star topology) is an ergodic decomposition of a measure µ ∈ M(X, f ) if τ (E(X, f )) = 1 and for every ϕ ∈ L 1 (µ).
The following theorem gives a formula for the measure theoretic pressure of an invariant measure in terms of its ergodic decomposition. with the essential supremum taken with respect to τ . Consequently, the following property hold: Proof. The second equality in the statement is clear since P ν (f, ϕ) = h ν (f )+ X ϕdν for each ν ∈ E(X, f ). Therefore, to complete the proof of the theorem, it suffices to prove the first equality. Take a subset Z ⊂ X with µ(Z) = 1, then ν(Z) = 1 for τ -almost every ν ∈ E(X, f ). This implies that P Z (f, ϕ) ≥ P ν (f, ϕ) for τ -almost every ν ∈ E(X, f ), which yields that Since Z is chosen arbitrarily, by the definition of measure theoretic pressure we have that P µ (f, ϕ) ≥ ess sup{P ν (f, ϕ) : ν ∈ E(X, f )}. To prove the reverse inequality, let By Brin-Katok's local entropy formula and Birkhoff's ergodic theorem, the subset X 0 is f -invariant and of full µ-measure. Given a small number ǫ > 0 and x ∈ X 0 , define Clearly, X 0 = x∈X0 Υ(x). We can choose points y i ∈ X 0 for i = 1, 2, · · · such that each Υ i := Υ(y i ) has positive µ-measure and µ( i≥1 Υ i ) = 1. Note that each Υ i is f -invariant, since the functions h µ (x) and ϕ * (x) are f -invariant. Fix i and let µ i denote the normalised restriction of µ to Υ i . It follows from (2.3) and the definition of Υ i that there is a one-to-one correspondence between the f -invariant ergodic probability measure on Υ i and the measures in E i (X, f ) := M i (X, f )∩E(X, f ). Using standard arguments, one can show that τ (E i (X, f )) > 0 and that the normalization of τ i of τ |M i (X, f ) provides and ergodic decomposition of µ i . Since Hence, for each ν ∈ A i we have Since τ (A i ) > 0 and µ(Υ i ) > 0, it follows from Corollary 4.2 that Since ǫ is arbitrary, the desired result immediately follows.
To complete the proof of the theorem, we notice that This together with the first result yield that Consequently, we have the following formula.
with the essential supremum taken with respect to τ . Consequently, we have that Remark 4.2. We provide an example that the inequality in Corollary 4.3 may be strict. In fact, choose two ergodic measures µ 1 , µ 2 such that h µ1 (f ) > h µ2 (f ), let µ = 1 2 µ 1 + 1 2 µ 2 , it is clear µ is f -invariant and satisfies that

4.3.
Measure theoretic pressure and separated sets. In this subsection, we will show that the measure theoretic pressure of ergodic measures defined via separated sets are exactly the free energy without any additional condition on the dynamical system (X, f ). The same phenomena can be observed in the non-ergodic case provided that the system (X, f ) has the uniform separation property and the ergodic measures are entropy dense.
Let (X, f ) be a TDS, and ϕ : X → R a continuous function on X. For any µ ∈ E(X, f ) we have that Proof. We divide the proof into two small steps: Step 1: SP µ (f, ϕ) ≤ h µ (f ) + ϕdµ for any µ ∈ M(X, f ).
Next, we will consider the measure theoretic pressure of non-ergodic measures defined via separated sets. Then for any µ ∈ M(X, f ), Proof. Since P (F ; ϕ, δ, n, ǫ) ≤ P (F ; ϕ, n, ǫ) for any δ ∈ (0, 1) and any ǫ > 0, we have that From Corollary 3.1, we have that This completes the proof of the theorem.  Proof. Given an f -invariant measure µ, by Proposition 4 in [2] we have that On the other hand, we follow the proof of Theorem 7 in [2] to prove the reverse inequality. By (2.6) and Proposition 4 in [2] we have that it yields that Let X = {x ∈ J : λ(x) and h µ (x) are well-defined }, by Brin-Katok's local entropy formula and Birkhoff's ergodic theorem, the subset X is f -invariant and of full µ-measure. Given a small number ǫ > 0 and x ∈ X, define Υ(x) = y ∈ X : |h µ (y) − h µ (x)| < ǫ and |λ(y) − λ(x)| < ǫ .
The sets Υ(x) form a cover of X and we can choose points y i ∈ X for i = 1, 2, · · · such that each Υ i := Υ(y i ) has positive µ-measure and µ( i≥1 Υ i ) = 1. Note that each Υ i is f -invariant, since the functions h µ (x) and λ(x) are f -invariant. Fix i and let µ i denote the normalised restriction of µ to Υ i . It follows from (2.3) and the definition of Υ i that there is a one-to-one correspondence between the f -invariant ergodic probability measure on Υ i and the measures in Using standard arguments, one can show that τ (E i (f | J )) > 0 and that the normalization of τ i of τ |M i (f | J ) provides and ergodic decomposition of µ i . Since Note that X λ(x)dν is exactly the unique Lyapunov exponent λ(ν), since J is an average conformal repeller. Hence, for each ν ∈ A i we have The above observation together with (2.8) yield that Letting ǫ → 0, we have that This completes the proof of the theorem.
The formula in the previous theorem was first established by Barreira and Wolf for conformal repellers of a C 1+α map (see [2]). Here, we relax the smoothness to C 1 and extend their result for average conformal repellers which are indeed non-conformal (see [25] for an example).
The following theorem relates the Hausdorff dimension of an invariant measure to the zero of measure theoretic pressure, which can be regarded as an Bowen's equation in the measure theoretic case. Let τ be an ergodic decomposition of µ, it follows from Theorem 4.1 that Since ϕdν = λ(ν) > 0 for each ν ∈ E(f | J ), it yields that By Theorem 4.4, we have that To prove the reverse inequality, by (2.8) and Theorem 4.4 we have that This together with Corollary 4.1 yield that By Theorem 4.1, we have that This completes the proof of theorem.
We now consider hyperbolic diffeomorphisms and derive formula for the Hausdorff dimension of an invariant measure (not necessarily ergodic). The formulas are versions of the ones in Theorems 4.4 and 4.5. Our approach is similar to that in Theorems 4.4 and 4.5, although it is now necessary to deal simultaneously with the stable and unstable directions.
In the following, we will first provide a dimension formula similar to that in Theorem 4.4 in the case of average conformal hyperbolic diffeomorphisms. Recall that ϕ i (x) = 1 di log D x f |E i (x) for i = s, u. Theorem 4.6. Let f be a C 1 diffeomorphism with a compact f -invariant locally maximal hyperbolic set J on which f is average conformal, and let µ be an finvariant measure on J. For any ergodic decomposition τ of µ we have dim H µ = ess sup{dim H ν : ν ∈ E(f | J )} with the essential supremum taken with respect to τ .
Choose now points y i ∈ X for i = 1, 2, · · · such that the f -invariant sets Υ i = Υ(yi) satisfy µ(Υ i ) > 0 for each i, and µ( i≥1 Υ i ) = 1. Fix i and consider the normalized restriction µ i of µ to Υ i . Using the same arguments as in the proof of Theorem 4.4, one can show that there exists a set A i ⊂ E i (f | J ) of positive τ i -measure (here τ i is the normalization of τ |M i (f | J )) such that for each ν ∈ A i and x ∈ Υ i we have that h Consequently, we have that for each ν ∈ A i . Since τ (A i ) > 0, it follows from (2.10) and (2.11) that dim H µ ≤ ess sup{dim H ν : ν ∈ E(f | J )} + C(ǫ) where ǫ → C(ǫ) is a function (independent of i and ν) that tends to zero as ǫ → 0. This completes the proof of the theorem.
The above theorem extends Theorem 10 in [2] in twofold: first it relaxes the smoothness of the diffeomorphism to C 1 ; on the other hand, it considers the nonconformal case in the hyperbolic setting.
Theorem 4.7. Let f be a C 1 volume-preserving diffeomorphism with a compact f -invariant locally maximal hyperbolic set J on which f is average conformal, and let µ be an f -invariant measure on J. Then where t s and t u are respectively the unique root of the following equations P µ (f, tϕ s ) = 0 and P µ (f, −tϕ u ) = 0.
It follows from Theorem 4.6 that t s + t u ≥ dim H µ.
To prove the reverse inequality, by (2.10) and Theorem 4.6 we have that for τ -almost every ν ∈ E(f | J ). Since f is volume preserving, we have that ϕ u dν = − ϕ s dν, ∀ν ∈ E(f | J ). Hence, It follows from Theorem 4.1 that Hence, Similarly, one can show that t s ≤ 1 2 dim H µ.
Hence, t u + t s ≤ dim H µ. This completes the proof of the theorem.