Dominated Splitting, Partial Hyperbolicity and Positive Entropy

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary. We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The conditions deal with the dynamical behaviour of the (non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue measure is $\delta$-recurrent then the entropy of $f$ is positive. We give counterexamples showing that these sufficient conditions are not necessary. Finally, in the case of partially hyperbolic diffeomorphisms, we give a positive lower bound for the entropy relating it with the dimension of the unstable and stable sub-bundles.


Introduction
It is known that every Anosov system f (or horseshoe) has positive entropy and by structural stability the system g close to f is topologically conjugated to f so that the entropy function restricted on Anosov systems is locally positive constant. The entropy being positive is mainly based on the uniform hyperbolicity. After people know that uniform hyperbolicity is not dense in the whole space of dynamical systems, researchers started to study some systems with weak hyperbolicity such as nonuniform hyperbolicity, partial hyperbolicity and dominated splitting, etc. It is known for non-uniform hyperbolicity. More precisely, if f is C 1+α and preserves a non-atomic ergodic hyperbolic measure, by classical C 1+α Pesin theory there is a horseshoe so that f has positive entropy. For any partially hyperbolic system, it is also true from a recent result of [8]. Let Diff 1 (M) denote the space of all C 1 diffeomorphisms. One natural question arises: Question 1.1. Has any C 1 diffeomorphism f ∈ Diff 1 (M) with dominated splitting positive entropy?
We conjecture that it is true. Along this paper we will state and prove several theorems that partial positive answers to Question 1.1. All these theorems give sufficient conditions for a diffeomorphism with dominated splitting have positive entropy. The results are classified in three groups: • The general dominated splitting case: Theorems 2.3, 2.4 and 2.6 hold for rather general C 1 diffeomorphisms f with dominated splitting, that do not necessarily preserve a measure absolutely continuous w.r.t. Lebesgue measure. Nevertheless we prove that the entropy is positive only if f satisfies certain other conditions, as for instance the so called recurrence property for the Lebesgue measure.
• The smooth invariant measure case: Theorems 2.7, 2.9, 2.11 and 2.12 hold when the C 1 diffeomorphism f preserves a measure µ that is absolutely continuous w.r.t. Lebesgue measure. It also includes diffeomorphisms that do not have a global uniformly dominated splitting, but exhibit just a weak form of measurable domination which is called almost dominated splitting.
• The partially hyperbolic case: Theorem 2.17 holds when the dominated splitting is partially hyperbolic. It strengthens a previous result of [8] about positive entropy of partially hyperbolic C 1 diffeomorphisms.
The paper is organized as follows: in Section 2 we recall some definitions, state the main results to be proved along the paper, and prove the immediate corollaries of the main theorems. In Section 3 we prove the theorems in the general case of dominated splitting under the assumption of recurrence of the Lebesgue measure or under other rather general assumptions. In Section 4 we provide a simple example to show that the recurrence condition for the Lebesgue measure is sufficient but not necessary to have positive entropy. In Section 5 we prove the theorems of positive entropy in the smooth measure preserving case. In Section 6 we prove the the theorems in the partial hyperbolic case. Finally, in Section 7 we add some comments, prove other complementary results, and show some arguments to support the conjecture that any C 1 -diffeomorphism with dominated splitting has positive entropy.

Definitions and statement of the results
Before stating the main results let us recall the following definitions: where m(A) denotes the minimal norm of linear map A.
Remark that the continuity of the splitting in the definition is not necessary because it can be naturally deduced from the required inequality in the dominated splitting(for example, see [2]). Remark that This means if T M M = E ⊕ F is a σ−dominated splitting of f for some σ > 1, then for any integer k ≥ 1, There is an equivalent statement of dominated splitting. T M = E ⊕ F is a dominated splitting if there exist C > 0 and 0 < λ < 1 such that ≤ Cλ n , ∀x ∈ M, n ≥ 1.
Remark that Gourmelon ([6]) proved that there always exists an adapted metric for which C = 1.

Definition 2.2. (Measurable recurrence)
Let f : M → M be an homeomorphism. We call a measurable set B ⊂ M recurrent to the future if there exists n j → +∞ such that f n j (B) ∩ B = ∅ for all j ≥ 0.
Let ρ be a (non necessarily f -invariant) probability measure on M. We say that ρ is an f -recurrent measure if there exists δ > 0 such that any measurable set B ⊂ M satisfying ρ(B) > 1 − δ is recurrent.
It is immediate to check that if ρ is f -invariant then ρ is recurrent, satisfying the above definition for any 0 < δ ≤ 1. In fact, if B is a measurable set such that ρ(B) > 0, then, after Poincaré Recurrence Lemma, f n (B) ∩ B = ∅ for arbitrarily large values of n.
In the following, we consider the Lebesgue measure on the compact manifold M after a re-scaling to make it a probability.

2.1
The general dominated splitting case. Theorem 2.3. Let f : M → M be a C 1 diffeomorphism on a compact Riemanian manifold M exhibiting a σ−dominated splitting T M = E ⊕ F (with σ > 1). Assume that the Lebesgue measure on M is f -recurrent. Then the topological entropy of f is positive.
We will prove Theorem 2.3 in Section 3. We remark that the hypothesis of Theorem 2.3 which assumes that the Lebesgue measure is f -recurrent is not necessarily satisfied for all the diffeomorphisms having a dominated splitting and positive topological entropy. In fact, in Section 4 we provide an example.
For such cases of diffeomorphisms for which the Lebesgue measure is not recurrent, we can apply the following set of theorems. They give other sufficient conditions for a diffeomorphism with dominated splitting have positive entropy.
Assume that there exists a set A with positive Lebesgue measure such that for all x ∈ A at least one of the following inequalities holds: Then the topological entropy of f is positive.
We will prove Theorem 2.4 in Section 3.
Now we give a condition which is stronger than the condition of Theorem 2.4 and easier to check for applications.
Assume that at least one of the following inequalities holds: (2.6) Then the topological entropy of f is positive.
Then the topological entropy of f is positive.
We will prove Theorem 2.6 in Section 3.

The smooth-invariant measure case
In the particular case that f preserves a smooth measure µ (i.e., µ is absolutely continuous w.r.t. Lebesgue measure), we obtain the following result without additional hypothesis: We note that the proof of the particular case stated in Theorem 2.7 can be obtained as a corollary of Theorem 2.3. In fact, we denote by Leb the Lebesgue probability measure on M. If µ is f -invariant, then µ is f -recurrent for any 0 < δ < 1. But if µ ≪ Leb, and B is the measurable set where the density of µ is positive, then µ(B) = 1. So α = Leb(B) > 0. Since any measurable set A with Leb(A) > 1 − α will intersect B on a set A ∩ B with positive Lebesgue measure and where the density of µ is positive, then µ(A ∩ B) > 0. We deduce that A ∩ B is an f -recurrent set (because µ is f invariant). We conclude that Leb is an f -recurrent measure. Finally Theorem 2.3 implies that f has positive entropy. So Theorem 2.7 is a corollary of Theorem 2.3.
Also, the proof of the particular case stated in Theorem 2.7 can be easily and independently deduced from the following already known theorem: If f ∈ Diff 1 (M) has a σ-dominated splitting, with σ > 1, and if µ is a smooth f -invariant probability measure, then: where χ 1 ≥ χ 2 ≥ . . . ≥ χ dim(M) are the Lyapunov exponents.
Proof of Theorem 2.7: Since f −1 has also a dominated splitting, and µ ≪ Leb. is also f −1 -invariant, we can apply Inequality (2.7) to f −1 : χ j dµ > 0, and so by (2.7) the entropy is positive, or From the dominated splitting condition we obtain Thus, we can bound from above the integral at right in Inequality (2.8) as follows: So, Inequality (2.8) gives h µ (f ) > 0, ending the proof of Theorem 2.7.
Moreover, we point out that the latter proof is adaptable to systems with almost dominated splitting, according to the following definition: Definition 2.8. (Almost dominated splitting) Fix a point x ∈ M and denote by orb(x) its orbit. A splitting Let µ be an f −invariant measure µ and let N(·) : M → N be an f -invariant measurable function. We say µ has an almost dominated splitting, if for µ a. e. x ∈ M, there is an at x. We say µ has a non-trivial almost dominated splitting, if there is an almost dominated splitting of µ a.e. x and the set for which the following inequality holds has µ−positive measure: Now we state the main result in the case of smooth invariant measure: Suppose that f preserves a smooth measure µ which has a non-trivial almost dominated splitting. Then µ has positive entropy and thus f has positive entropy.
We will prove Theorem 2.9 in Section 5.
In particular, Theorem 2.9 immediately implies the following corollary for volume-preserving diffeomorphisms. Let Leb denote the volume measure (i.e. the Lebesgue measure) on M and Diff 1 Leb (M) denote the space of all C 1 volume-preserving diffeomorphisms.
. If Leb has a non-trivial almost dominated splitting, then f has positive entropy.
From this corollary, we obtain the following: such that for every f ∈ R, either for Leb−a.e.x ∈ M the Oseledec splitting of f is trivial (i.e., all Lyapunov exponents are zero), or Leb has positive entropy and thus f has positive entropy.
The proof of Theorem 2.11 is straightforward from Corollary 2.10 and the following known result: such that for every f ∈ R and for Leb−a.e.x ∈ M the Oseledec splitting of f is either trivial (i.e., all Lyapunov exponents are zero) or dominated at x.
It is known (see [11]) that for any C 1 diffeomorphism f far away from homoclinic tangencies and for any f -ergodic measure ν, the stable, center and unstable bundles of the Oseledec splitting are dominated on supp(ν) (the support of ν), and besides the center bundle is at most one dimensional.
Then, one can use the Ergodic Decomposition Theorem (see for instance [10]) to obtain that for any f -invariant measure µ, µ−a.e. x has an Oseledec splitting that is dominated at x. So, we deduce: M be a C 1 diffeomorphism far from tangencies that preserves a smooth invariant measure µ. Then f has positive entropy.
In particular, for volume preserving diffeomorphisms far from tangencies we deduce: If f is far from tangency, then f has positive entropy.

The partially hyperbolic case
Usually, partial hyperbolicity means that there exists a splitting in three subbundles such that one (which is called the unstable bundle) is uniformly expanding, one (which is called the stable bundle) is uniformly contracting and the other one (which is called the center bundle) may have no hyperbolicity but is dominated by the unstable bundle and dominates the stable one. Here we adopt a more general notion of partial hyperbolicity, by using a splitting into two subbundles: Definition 2.14. (Partial hyperbolicity) We call T M = E ⊕ F a partially hyperbolic splitting, if it is a dominated splitting and E is uniformly contracting, i.e. there exist C > 0 and 0 < λ < 1 such that or F is uniformly expanding, i.e. there exist C > 0 and 0 < λ < 1 such that

Note that
Uniform Hyperbolicity ⇒ Partial Hyperbolicity ⇒ Dominated Splitting. Proof: A partially hyperbolic diffeomorphism is a diffeomorphism with dominated splitting which besides satisfies the hypothesis of Theorem 2.4. So, Theorem 2.4 implies that f has positive entropy.
Remark 2.16. We have shown that the new result stated on Theorem 2.4 is a generalization of Theorem 2.15. This latter was firstly proved in [8]. To prove positive entropy for partial hyperbolicity, the authors of [8] constructed a n-separated set on one unstable (or stable) manifold, and prove that the cardinality of this n-separated set has positive exponential growth with n. This method may be not adaptable to the case of dominated splitting without partial hyperbolity, because we may not have unstable-stable bundles. So, the proof of the more general Theorem 2.4 must be different from the proof of Theorem 2.15 in [8]. We mainly base it on some recent advances about Pesin's entropy formula for the so called SRB-like measures of C 1 diffeomorphisms with dominated splitting ( [4]).
Moreover, we will also prove the following result, which strengthens the theorem of Saghin-Sun-Vargas [8] by providing an explicit positive lower bound of the entropy: We will prove Theorem 2.17 in Section 6.

Proofs in the general case of dominated splitting
Route of the proofs: To prove Theorems 2.3, 2.4 and 2.6, we will use similar ideas to those in Subsection 2.2 for C 1 diffeomorphisms with dominated splitting that preserve a smooth measure. But now, smooth invariant measures may not exist. So, we do not have from the very beginning an adequate invariant measure that satisfies simultaneously inequalities (2.7) and (2.8). Anyway, we will construct two or more invariant probabilities, some satisfying inequality (2.7) and the other ones satisfying inequality (2.8). Finally we will prove that at least one of those measures has positive entropy.
First, the construction of such adequate probabilities will be based on the theory of SRB-like measures for C 1 maps introduced in [5]. Second, we will apply a result in [4] which provides a generalization to all the SRB-like measures of the Pesin's formula of the entropy previously proved in [9] for C 1 diffeomorphisms with dominated splitting. Third and finally, we will deduce that under the hypothesis of Theorems 2.3 or 2.4 or 2.6, Pesin's formula of the entropy implies that at least one SRB-like measure, either for f or for f −1 , has positive entropy.
Remark 3.1. As roughly explained above, we will prove that for some SRB-like measure the dominated splitting assumption plus the assumptions of Theorems 2.3 or 2.4 or 2.6 imply positive entropy. However, the sufficient conditions to obtain positive entropy in the case of general SRB-like measures must be stronger than the hypothesis of Theorem 2.9 for the case of smooth measures. In fact, for general SRB-like measures the hypothesis of non-trivial almost dominated splitting is not enough to get positive entropy. For example -as said in Remark 1.7 of [4] -the eight-figure diffeomorphism (see [1] - Figure 10.1) has an invariant Dirac measure supported on a fixed hyperbolic point that is SRB-like. By hyperbolicity it has a non-trivial almost dominated splitting, but the entropy of the diffeomorphism is zero.
To start the proofs, we are going to show that Theorem 2.3 is indeed a corollary of Theorem 2.4: If the Lebesgue measure is f -recurrent, then at least one of inequalities of (2.1) and (2.2) of Theorem 2.4 holds for the points in a set with positive Lebesgue measure.
More generally, for any a > 0 there exists a set A with positive Lebesgue measure such that for all x ∈ A at least one of the following inequalities holds: Proof: Assume by contradiction that for Lebesgue almost all x ∈ M the following two inequalities hold simultaneously: Denote by Leb the Lebesgue measure normalized to make Leb(M) = 1. By continuity of the D x f every set A N is closed, and so it is measurable.
where 0 < δ < 1 is the number of Definition 2.2, which exists by the hypothesis of recurrence of the Lebesgue measure. Therefore A N is a recurrent set. We assert that In fact, fix n ≥ N. Since |det(Df n (x))|, |det(Df −n (x))| ≤ e −nα for all x ∈ A N , we deduce the following inequalities, for any measurable set B ⊆ A N : (3.11) In particular, if we take B 0 = A N in the first inequality above, we obtain Therefore: By induction we deduce and so Leb(A N ∩ f n (A N )) = 0. Since n ≥ N is arbitrary we have proved that Finally, construct contradicting the hypothesis of recurrence of the Lebesgue measure.
After Lemma 3.2, it is enough to prove Theorems 2.4 and 2.6. We restate these two theorems together as follows: lim sup Now, to end the proofs of all the results of Subsection 2.1 it is enough to prove only Theorem 3.3. Recall that T M = E ⊕ F is a σ−dominated splitting for some σ > 1. To prove Theorem 3.3, we need the following property of domination: The inequality is also true for lim inf n→+∞ .
Proof. From definition, Thus, Similarly, one can show the other inequalities.
We recall some previous known results of Ergodic Theory: By Variational Principle from [10], the topological entropy is the supremum of the metric entropy of all invariant measures. That is, for any dynamical system f : So if the metric entropy of an invariant measure (e.g., SRB measure or SRB-like measure) has positive entropy, then the system has positive topological entropy. Thus, we only need to show that there exists some SRB-like measure with positive entropy.
Let's recall some basic definitions and results related with SRB-like measures. Let P denote the space of all probability measures, and P f ⊂ P denote the space of f -invariant probability measures. For a point x ∈ M we consider the following sequence where δ y is the Dirac probability measure supported at y ∈ M. We define the nonempty and compact set pω(x) of probability measures: It is standard to check that pω f (x) ⊂ P f .
We denote with O f the set of all SRB-like measures for f : M → M. It is easy to see that every SRB-like measure for f is f -invariant.
We call basin of attraction A(K) of any nonempty weak * compact subset K of probabilities, to We need a following theorem, which is a reformulation of the main results of [5]: The set O f of all SRB-like measures for f is the minimal weak * compact subset of P whose basin of attraction has total Lebesgue measure.
In other words: O f is nonempty and weak * compact, and the minimal nonempty weak * compact set that contains, for Lebegue almost all the initial states x ∈ M, the limits of the convergent subsequences of Then we can take a subsequence {n j } ↑ ∞ such that Moreover, take a subsequence {n j k } ↑ ∞ of {n j } such that the limit exists in weak * topology. If denote this limit measure by µ, then µ ∈ pw f (x). Since dominated splitting is always continuous(for example, see [2]), then the function m(Df | F (y) ) is continuous on M. Thus, by (3.19) log m(Df | F (y) )dµ = lim Let χ 1 (y) ≥ χ 2 (y) · · · ≥ χ dim(M ) (y) denote the Lyapunov exponents of (µ, f ) for µ a.e. y ∈ M. Then it is easy to see and therefore by sub-additional property of minimal norm, These are deduced from Oseledec theorem and Birkhorff Theorem and remark that the above two kind of limits always exist for µ a.e. y ∈ M.
To obtain positive entropy, we recall another recent result on Pesin's entropy formula for systems with dominated splitting [4]: To end the proof of Lemma 3.7 we apply Theorem 3.8. In fact:  Then, we can take a subsequence {n j } ↑ ∞ such that Moreover, take a subsequence {n j k } ↑ ∞ of {n j } such that the limit exists in weak * topology. Denoting by µ this limit measure, we have µ ∈ pw f (x). Since f is of C 1 , then the function log |detDf | y | is continuous on M. Thus, by (3.19) log |detDf | y |dµ = lim Let χ 1 (y) ≥ χ 2 (y) · · · ≥ χ dim(M) (y) denote the Lyapunov exponents of (µ, f ) for µ a.e. y ∈ M. On the one hand, from Theorem 3.8 and from Oseledec Theorem, we obtain: On the other hand, similarly to formula (3.21) of Case 1, one has So, from Lemma 3.4: So h µ (f ) > 0 as wanted.

Case 4.
Lemma 3.11. If there is a set H 3 ⊆ M with positive Lebesgue measure such that for any then f has positive entropy.
Proof. Consider f −1 in the role of f and apply Lemma 3.10 of Case 3. Then we can take a subsequence {n j } ↑ ∞ such that Moreover, take a subsequence {n j k } ↑ ∞ of {n j } such that the limit exists in weak * topology. If denote this limit measure by µ, then µ ∈ pw f (x). Since the bundles of dominated splitting are always continuous(see [2]), then the function log | det Df | F (y) | is continuous on M. Thus, by (3.19) By Theorem 3.8, Proof. Consider f −1 in the role of f and apply Lemma 3.12 of Case 5.

Proof of Theorem 3.3:
Any map f with dominated splitting is either in one of the above Cases 1 to 6, and so it has positive entropy, or otherwise it must satisfy all the conditions of the following case: and lim sup We ended the proof of Theorem 3.3, and so also the proofs of Theorems 2.3, 2.4 and 2.6.

Example
In this section we construct a simple example to show that the hypothesis of recurrence of the Lebesgue measure in Theorem 2.3 is sufficient but not necessary for a diffeomorphism with dominated splitting have positive entropy.
Consider the circle S 1 and a Morse-Smale order preserving diffeomorphism f 1 : S 1 → S 1 having exactly two fixed points: a hyperbolic sink x 1 and a hyperbolic source x 2 and such Consider the torus T 2 and an area-preserving linear Anosov diffeomorphism f 2 : T 2 → T 2 with expanding eigenvalue σ 2 > 1 and contracting eigenvalue λ 2 = σ −1 2 . We can choose such a diffeomorphism so σ 2 > σ 1 . Such a choice does exist, since from any linear Anosov diffeomorphism g on the torus, one can take f 2 = g N for N ≥ 1 large enough. Denote by the hyperbolic splitting for f 2 , where S and U are the stable and unstable sub-bundles respectively. Construct Therefore f has a σ-dominated splitting T T 3 = E ⊕ F , where E = T S 1 ⊕ S, F = U, and σ = σ 2 /σ 1 > 1.
Besides, f has positive entropy, since f is the product map f 1 × f 2 , and so: Also note that f is not transitive, since it is a product map f 1 ×f 2 and f 1 is not transitive. So, this example shows that non transitive diffeomorphisms with dominated splitting may have positive entropy.
It is standard to check that the SRB-like measure µ for f is unique and coincides with δ x 1 × Leb T 2 . This is because, in this example δ x 1 × Leb T 2 is physical with basin of statistical attraction of full Lebesgue measure on T 3 .
Analogously, the SRB-like measure ν for f −1 is unique, and coincides δ x 2 × Leb T 2 . It is physical for f −1 with basin of statistical attraction of full Lebesgue measure on T 3 . Moreover, in this example µ and ν are ergodic measures for f and where χ 1 (x) ≥ χ 2 (x) ≥ · · · ≥ χ dim M (x) are the Lyapunov exponents at µ a.e. x.
Note that µ is also a smooth invariant measure for f −1 . So, Theorem 5.1 implies the following: Theorem 5.2. Let f : M → M be a C 1 diffeomorphism on a compact Riemannian manifold. Suppose that f preserves a smooth measure µ and µ has an almost dominated splitting. Then where χ 1 (x) ≥ χ 2 (x) ≥ · · · ≥ χ dim M (x) are the Lyapunov exponents at µ−a.e. x ∈ M.
Proof of Theorem 2.9: By assumption, there is a set B with µ positive measure such that for any point x ∈ B there exists a N(x)−dominated splitting where Notice that B L is an f −invariant set, and besides N(x)|S for all x ∈ B L , Then, for any point Thus g has a non trivial dominated splitting for all x ∈ B L . Now, we need the following lemma: The inequality is also true for lim inf n→+∞ .
To prove Lemma 5.3 repeat the proof of Lemma 3.4 after replacing f by g, σ by 2, and the whole manifold M by B L . Now, we continue the proof of Theorem 2.9. Define ν := µ| B L . Then ν is a g-invariant smooth measure. From Theorem 5.1 applied to g in the role of f , we obtain log m(Dg| F (g i (x)) )dν; and from Theorem 5.2 Then . So f has positive entropy, as wanted.

6
Proof in the partially hyperbolic case The purpose of this section is to prove Theorem 2.17 which explicits a positive lower bound for the entropy of C 1 diffeomorphisms with a partially hyperbolic splitting (see Subsection 2.2).
To prove assertion (a) of Theorem 2.17, we assume that F is an expanding subbundle. Thus, the Lyapunov exponents of any invariant measure w.r.t. F are positive. Precisely: Lemma 6.1. Let f : M → M be a C 1 diffeomorphism on a compact Riemanian manifold M with a continuous bundle F . Assume that there exist C > 0 and 0 < λ < 1 such that Df −n | F (x) ≤ Cλ n , ∀x ∈ M, n ≥ 1. Then every invariant measure ν satisfies the following inequalities for ν a.e. x ∈ M: Proof. By assumption, for any x ∈ M we have This implies that for any x ∈ M, Let ν be an invariant measure. Then by Oseledec theorem, for ν a.e. x ∈ M, where χ 1 (x) ≥ χ 2 (x) ≥ · · · ≥ χ dim M (x) denote the Lyapunov exponents at ν a.e. x. This ends the proof of Lemma 6.1.
Proof of Theorem 2.17: To prove assertion (a) of Theorem 2.17 we take a SRB-like measure µ. Joining Theorem 3.8 with Lemma 6.1, we obtain proving assertion (a). Finally, to prove assertion (b) we just apply assertion (a) replacing f by f −1 , ending the proof of Theorem 2.17.

Comments and complementary results.
In Theorem 3.3 we proved that, in the hypothetical case that a C 1 -diffeomorphism with dominated splitting had zero entropy, it should satisfy the six statements (3.12) to (3.17).
Besides, if such a diffeomorphism existed, it should also satisfy all the statements of the following Theorem 7.1, simultaneously with the six statements of Theorem 3.3. Thus, the requirements to construct -if it exists-a map f with dominated splitting and zero entropy are so many, that seem to impose too strong restrictions to hold simultaneously. .
If the answer to Question 7.2 were positive, then joining inequality (3.20) with (7.32), we would immediately obtain h µ (f ) > 0 and a complete positive answer to Question 1.1. We conjecture that the answer to Question 7.2 is "yes", and so also the answer to Question 1.1. We base this conjecture on the procedure that we will use to construct the measure µ 2 along the proof of Theorem 7.1. In fact, on the one hand to prove part (B) of Theorem 7.1 we will construct (in Lemma 7.3) the invariant probability measure µ 2 as an ergodic component of a weak * accumulation point of countably many non-invariant measures σ n . Each σ n will be supported on a finite number of points that one can choose from a fixed set A ⊂ M which has positive Lebegue measure. On the other hand, the SRB-like measures constructed in [5] are also the weak * accumulation points of a sequence of measures, each one supported on a finite number of points of a set A with positive Lebesgue measure. Precisely, this construction of the SRB-like measures, with the property that Leb(A) > 0, plays a key role in the proof of the entropy inequality (3.20) of Theorem 3.8 in [4]. So, we have a hope that some of the probabilities µ 2 of part (B) of Theorem 7.1 -which are not necessarily SRB-like but are constructed similarly to SRB-like measures -may also satisfy the entropy inequality (3.20).
To prove Theorem 7.1 we need the following lemma: for every ergodic f -invariant measure, then there is N > 0 such that where M e (f ) is the space of all ergodic measures. By assumption, (a, b) ⊆ V. By Ergodic Decomposition Theorem, for any invariant measure µ, Moreover, one should satisfy that [a, b] V.
Otherwise, without loss of generality, assume a ∈ R\V. Since the space of invariant measures is compact, then there exists an invariant measure µ such that ϕdµ = a. By the definition of a and Ergodic Decomposition Theorem, this implies there is an ergodic component of µ, denoted by ν, such that ϕdν = a. It contradicts to the assumption.
Take ǫ > 0 small enough such that (a − 2ǫ, b + 2ǫ) ⊆ V. Firstly, we are going to prove that for any x ∈ M, there is n = n(x) ≥ 1 such that By contradiction, there exists x ∈ X such that for any n ≥ 1, Without loss of generality, we can assume that there exists a sequence of {n k } such that for any k ≥ 1, 1 n k n k −1 j=0 ϕ(f j (x)) ≤ a − ǫ.
Define the probability measures where δ y denotes the Dirac-probability supported on y. Consider any convergent subsequence of {σ n k } k∈N in the space of all the Borel probability measures on X endowed with the weak * -topology. Remark that every such measure is invariant. Take one such measure µ = lim m σ n km . Thus, ϕdµ = lim m 1 n km n km −1 j=0 ϕ(f j (x)) ≤ a − ǫ < a, which contradicts ϕdµ ≥ a. Secondly, let us find a common integer N ≥ 1 such that for any x ∈ X, a − 2ǫ < 1 N N −1 j=0 ϕ(f j (x)) < b + 2ǫ.
By continuity of ϕ, from (7.36) for any x ∈ M there is an open neighborhood of x, U x , such that for any y ∈ U x , a − ǫ < 1 n(x) n(x)−1 j=0 ϕ(f j (y)) < b + ǫ.
By compactness of M, there is a finite cover of M by {U x i } m i=1 for some m ≥ 1. Then for any y ∈ M, set l(y) := min{n(x i )| y ∈ U x i , 1 ≤ i ≤ m}. Define N 0 (y) = 0 and N k (y) by induction: N k+1 (y) = N k (y) + l(f N k (y) (y)), k ≥ 0.
Take N > T ǫ max{C + |b| + ǫ, C + |a| + ǫ} which is an integer independent of y ∈ M, then for any y ∈ M, Since (a − 2ǫ, b + 2ǫ) ⊆ V, then we complete the proof.
Proof of Theorem 7.1: Equalities (7.34) and (7.35) of assertion (C) are immediatly deduced from Birkhorff Ergodic Theorem. In fact, for any f -invariant measure µ, and for any µ-integrable real function ϕ the following limits exist and coincide µ-a.e. x ∈ M: To prove assertion (A) we use ϕ(x) := log |detDf x |, take any ergodic measure µ and apply Birkhoff Ergodic Theorem and the definition of ergodicity. We deduce that the limit at left of inequality (7.30) exists for µ-a.e. x ∈ M and equals ϕ dµ. Assume by contradiction that for any ergodic measure µ inequality (7.30) fails, namely ϕ dµ < 0. To prove assertion (B), we denote ϕ := log |det(Df n | Fx ), take any ergodic measure µ and apply Birkhorff Ergodic Theorem and the definition of ergodicity. We obtain: lim n→+∞ 1 n log |det(Df n | Fx )| = ϕ dµ for µ − a.e.x ∈ M.