REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS OF PARTIALLY HYPERBOLIC DYNAMICS ON 3 TORUS

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1 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS OF PARTIALLY HYPERBOLIC DYNAMICS ON 3 TORUS F. MICENA AND A. TAHZIBI Abstract. In this work we study relations between regularity of invariant foliations and the Lyapunov exponents of partially hyperbolic diffeomorphisms. We suggest a new regularity condition for foliations in terms of disintegration of Lebesgue measure which can be considered as a criterium for rigidity of Lyapunov exponents.. Introduction In this paper we address the regularity of invariant foliations of partially hyperbolic dynamics and its relations to Lyapunov exponents and rigidity. We suggest a new regularity condition (Uniform Bounded Density property) for foliations, which is defined in terms of disintegration of Lebesgue measure along the leaves of the foliation. In principle it can be compared with the absolute continuity of foliations, however for (un)stable foliations of partially hyperbolic diffeomorphisms the works of Pesin-Sinai [], Ledrappier [9] shed light on the subject and it turns out that for these foliations our condition imposes a kind of regularity much stronger than absolute continuity. However, we believe that exploiting this regularity condition is a geometric measure theoretical criterium for the rigidity of partially hyperbolic dynamics. From now on, we shall consider a smooth measure m (Lebesgue measure) on T 3 and a C 2 diffeomorphism f : M M preserving m. f is called (absolute) partially hyperbolic if there exists a D f -invariant splitting of the tangent bundle TM = E s f Ec f Eu and constants ν f ν + < µ µ + < λ λ + and C > 0, satisfying C νn v D f n (x)v Cν n + v, v E s f (x), C µn v D f n (x)v Cµ n + v, v E c f (x), C λn v D f n (x)v Cλ n + v, v E u f (x). Date: September 24, 203.

2 2 F. MICENA AND A. TAHZIBI It is possible to choose a riemannian metric in M that makes C = in the above definition. In this paper all partially hyperbolic diffeomorphisms are defined on T 3. For simplicity we denote D f (x) E σ(x) by f Jσ f (x), σ {s, c, u}. The distributions E s and E u, respectively stable and unstable bundle, are uniquely f f integrable to foliations F s and F u (See [8]). In general case, E c is not integrable. However for absolute partially hyperbolic diffeomorphisms on T 3 the center bundle is integrable [2]. The foliation tangent to E c is denoted by F c. One special example is the case of Anosov diffeomorphisms on T 3 with splitting TM = E uu E u E s. In this case, the weak unstable bundle E u can be considered as the central bundle... Regularity of foliations. Roughly speaking, a foliation is an equivalence relation on a manifold such that the equivalence classes (the leaves) are connected immersed submanifolds. For dynamical invariant foliations, although typically the leaves enjoy a high degree of regularity they are not stacked up in a smooth fashion. To define the different regularity conditions we need foliated charts. For instance a codimension k foliation is C r if there exist a covering of the manifold by C r charts φ : U R n R k such that each plaque is sent into the hyperplane R n {φ(p)}. For a C r partially hyperbolic diffeomorphism the invariant foliations F s and F u typically are at most Hölder continuous with C r leaves. An important feature of stable and unstable foliations of partially hyperbolic diffeomorphism is their absolute continuity property. In smooth ergodic theory, absolute continuity of foliations has been used by Anosov to prove the ergodicity of Anosov diffeomorphisms. One of the weakest definitions (leafwise absolute continuity) is sufficient to prove the ergodicity of Anosov diffeomorphisms. See [3] for other definitions and state of art of absolute continuity of foliations. Consider F a foliation over M. Denote by m the riemannian measure over M; and λ Fx ; the riemannian measure over F x ; the leaf through x M. There is a unique disintegration [{m Fx }] of m along the leaves of the foliation. [{m Fx }] are equivalent class of measures up to scaling. In a foliation chart U M; denote m Fx ; the probability measure which comes from the Rokhlin disintegration of m restricted to U. In what follows we use the unique notation m Fx (B) to denote the disintegration of the plaque inside foliated box B, which is a probability measure. Definition. (leafwise Absolute Continuity). Let F be a foliation on M. We say that F is leafwise absolutely continuous, if it satisfies the following: A measurable set Z has zero Lebesgue measure if and only if for almost every p M, the leaf F p meets Z in a λ Fp zero measure set, that is, λ Fp (Z) = 0, for almost everywhere p M. Locally, it is equivalent to

3 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS 3 λ Fp m Fp, m almost everywhere p U. In general setting it is not easy to understand the disintegration [{m Fx }] of m. In the case of leafwise absolutely continuous foliations the Radon-Nikodym derivative dm Fp dλ Fp is an interesting object to be studied. This motivated us to introduce new regularity condition. We show that, if we assume m Fx is universally proportional to λ Fx, for almost everywhere x M, independent of the size of F x U then many rigidity results hold. To begin, we need to work with long foliated boxes. Definition.2 (Long Foliated Box). Let F be a one dimensional foliation of M n. A set B M is called a foliated box by F of size greater than or equal to R > 0, if: () B is homeomorphic to D n (0, ) where D n is (n ) dimensional ball representing transversal to the plaques in B; (2) for each x B, the length of the connected component of F x B containing x is greater than or equal to R > 0 in the intrinsic riemannian metric of F x. For any foliated box B we denote by m B the normalized Lebesgue measure of B and for any plaque F x (B) the probability induced Lebesgue measure on the plaque is denoted by Leb Fx (B). In the cases where the box is fixed, we write just Leb Fx. Definition.3 (Uniform Bounded Density). Let F be an one dimensional foliation on M. We say that F has uniform bounded density (U.B.D) property, if there is K > such that for every long foliated box of F in M we have K < dm F x < K dleb Fx independent of the size of the foliated box and x. For example if A is a linear partially hyperbolic automorphism of torus then the invariant foliations have U.B.D property. In fact this is the case for any f close to A and C conjugated to it. Another example of foliations with U.B.D property is the case of central foliation of ergodic partially hyperbolic diffeomorphisms on M 3 whenever it is absolutely continuous and the leaves are circles. Indeed, as the length of central leaves are uniformly bounded (See [5] and [6] for general statements.) the U.B.D property is equivalent to leafwise absolute continuity. A recent result of Avila- Viana-Wilkinson [] establishes that absolute continuity of central foliation in This condition is similar to bounded jacobian or global absolute continuity in []. However, we do not have any evidence that they are equivalent.

4 4 F. MICENA AND A. TAHZIBI this setting implies C regularity. We hope that U.B.D property of central foliations in general, may imply its differentiability. Lyapunov exponents are important constants and measure the assymptotic behaviour of dynamics in the tangent space level. Let f : M M be a measure preserving C diffeomorphism. Then by Oseledets Theorem, for almost every x M and any v T x (M) the following limit exists: lim n n log D f n (x)v and is equal to one of the Lyapunov exponents of the orbit of x. For a conservative partially hyperbolic diffeomorphism of T 3 which is the main object of the study in this paper, we get a full Lebesgue measure subset R such that for each x R : lim n n log D f n (x)v σ = λ σ ( f, x) where σ {s, c, u} and v σ E σ. Every diffeomorphism of the torus f : T n T n induces an automorphism of the fundamental group and there exists a unique linear diffeomorphism f which induces the same automorphism on π (T n ). f is called the linearization of f and in this paper we study the relations between Lyapunov exponents of f and its linearization in the partially hyperbolic setting. Acknowledgement. We are grateful to a referee for a carefull reading of the paper and several useful comments and corrections of the first version. 2. Statement of Results and Questions First we prove that the uniform bounded density is a criterium for the rigidity of Lyapunov exponents in the context of partially hyperbolic diffeomorphisms of T 3. Theorem 2.. Let f : T 3 T 3, be a conservative partially hyperbolic diffeomorphism. Denote by A = f and suppose that stable and unstable foliations have the uniform bounded density property, then λ σ ( f, ) = λ σ A, σ {s, c, u} for almost every x T3. Remark 2.2. In the above theorem if we just assume the U.B.D property of one of the foliations F s or F u, we conclude the rigidity of the corresponding Lyapunov exponent. In the above theorem the rigidity of central Lyapunov exponent is just a corollary of volume preserving property of f. However, the same rigidity for central exponent also holds if we just assume F c has U.B.D property. As we do not have a good description for the disintegration along the central leaves, the proof for the central exponent rigidity is different from the stable and unstable foliation cases and it appears in the proof of theorem 2.5.

5 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS 5 The above result shows that U.B.D property imposes restrictions on the dynamics in the level of Lyapunov exponents. The above theorem assumes U.B.D property and conclude some rigidity of Lyapunov exponents. We should mention that even leafwise absolute continuity imposes some restrictions on the Lyapunov exponents, as we see in the following theorem. Recall that stable and unstable foliation of any C 2 partially hyperbolic diffeomorphism are leafwise absolutely continuous ([3]). Theorem 2.3. Let f be a C 2 conservative partially hyperbolic diffeomorphism on the 3 torus and A its linearization then λ u ( f, x) λ u (A) and λ s ( f, x) λ s (A) for almost everywhere x T 3. Similar to the statement of the above theorem appears in [4] and proved in [5] for f C -close to A. In [5], the authors need unique homological data for the strong unstable foliation and they prove that it is the case when f is closed to its linearization. Corollary 2.4. Any conservative linear partially hyperbolic diffeomorphism is a local maximum point for f λ u ( f )dm. Analogously any conservative linear partially hyperbolic diffeomorphism is a local minimum point for f λ s ( f )dm. Problem. Classify the local maximum points of unstable Lyapunov exponent. Are these diffeomorphisms C conjugated to linear? Problem 2. In the context of the above theorem, suppose that λ c ( f ) > 0 and F c is upper leafwise absolutely continuous then λ c ( f ) λ c (A). Another interesting issue in the setting of partially hyperbolic diffeomorphisms is the characterization of topological type of central leaves. It is clear that for a general partially hyperbolic diffeomorhism (general 3 manifolds) with one dimensional central bundle, the leaves of central foliation may be circles, line or both of them (consider suspension of an Anosov diffeomorphism of T 2 ). However by Hammerlindl s result [7], central leaves of a partially hyperbolic diffeomorphism on T 3 are homeomorphic to central leaves of its linearization and consequently all the leaves have the same topological type. A very natural question is that Question. Suppose f is volume preserving (absolute) partially hyperbolic on T 3 and central Lyapunov exponent vanishes almost everywhere. Is it true that all center leaves are compact?

6 6 F. MICENA AND A. TAHZIBI In general setting, this question has been answered negatively in [2]. We would like to mention that by a recent result of Hammerlindl and Ures, a non-ergodic derived from Anosov diffeomorphism on T 3, if exists, will have zero central Lyapunov exponent and non-compact central leaves. It is interesting to know whether exists example of such partially hyperbolic non-ergodic diffeomorphisms on torus. Assuming U.B.D property of central foliation we get the following theorem which gives an affirmative answer to the above question. Theorem 2.5. Let f : T 3 T 3, be a conservative partially hyperbolic diffeomorphism. Suppose that F c has the uniform bounded density property, then λ c ( f, x) = λ c ( f ), almost every where. In particular if λ c ( f, x) = 0 for a.e. x T 3, then all center leaves are circles. We do not know whether one can prove that all the central leaves are homeomorphic to S just under the assumption of absolute continuity of central foliation and vanishing central Lyapunov exponent. 3. Preliminaries In this section we review some definitions and known results about partially hyperbolic diffeomorphisms on T Partially hyperbolic diffeomorphisms on T 3. In the rest of the preliminaries section we will recall some nice topological properties of invariant foliations of partially hyperbolic diffeomorphisms on 3-torus. One of the key properties of the invariant foliations of partially hyperbolic diffeomorphisms in 3-torus is their quasi-isometric property. Quasi isometric foliation W of R d means that the leaves do not fold back on themselves much. Definition 3.. A foliation W is quasi-isometric if there exist positive constant Q such that for all x, y in a common leaf of W we have d W (x, y) Q x y. Here d W denotes the riemannian metric on W and x y is the distance on the ambient manifold of the foliation. In the partially hyperbolic case, we denote by d σ (, ), the riemannian metric on F σ, σ {s, c, u}. We define d c (, ) in the dynamical coherent case. The foliation F σ is called quasi isometric if its lift to the universal covering (R 3 ) is quasi isometric. Let us recall some recent results concerning geometric and topological properties of the invariant foliations of partially hyperbolic diffeomorphisms on T 3.

7 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS 7 Theorem 3.2 ([4], [7]). If f : T 3 T 3 is partially hyperbolic diffeomorphism, then F σ, σ {s, c, u} are quasi isometric foliations. Proposition 3.3 ([7]). Let f : T 3 T 3 be a partially hyperbolic diffeomorphism and A : T 3 T 3 the linearization of f. Then y x lim y x + y x = Eσ A, and the convergence is uniform. y F σ x, σ {s, c, u} Proposition 3.4 ([7]). Let f : T 3 T 3 be a partially hyperbolic diffeomorphism and A : T 3 T 3 the linearization of f then for each k Z and C > there is an M > 0 such that for x, y, x y > M C < f k (x) f k (y) A k (x) A k (y) < C. More general, for each k Z, C >, and linear map π : R d R d there is an M > 0 such that for x, y R d, π(x y) > M C < π( f k (x) f k (y)) π(a k (x) A k (y)) < C. Theorem 3.5 ([7]). Every partially hyperbolic diffeomorphism of the 3-torus is leaf conjugated to its linearization, by a homeomorphism h. Furthermore h restricted to each center leaf is bi-lipschitz and denoting h a lift of h in R 3 one has that is bounded. h Id R 3 The above theorem and propositions has the following corollaries which are useful in the rest of the paper. Lemma 3.6 ([0]). Let f : T 3 T 3 be a partially hyperbolic diffeomorphism and A : T 3 T 3 the linearization of f. For all n Z and ɛ > 0 there exists M such that for x, y with y Fx σ and x y > M then ( ε)e nλσ A y x A n (x) A n (y) ( + ε)e nλσ A y x where λ σ is the Lyapunov exponent of A corresponding to E σ and σ {s, c, u}. Proof. Let us fix σ and denote by E A the eigenspace corresponding to λ σ, µ := A. A eλσ Let N Z and Choose x, y F σ (x), such that x y > M. By proposition 3.3, f we have x y x y = v + e M,

8 8 F. MICENA AND A. TAHZIBI where the vector v = v EA is a unit eigenvector of A, in the E A direction and e M is a correction vector that converges to zero uniformly as M goes to infinity. So, considering µ the eigenvalue of A in the E A direction ( ) ( ) x y x y A N = µ N v + A N e M = µ N µ N e M + A N e M x y x y It implies that x y (µ N µ N e M A N e M ) A N (x y) x y (µ N + µ N e M + A N e M ). Since N is fixed, we can choose M > 0, such that µ N e M + A N e M εµ N. and the lemma is proved. 4. Technical Rigidity Results and proof of Theorem 2. In this section we prove some technical rigidity results for Lyapunov exponents which will be used in the proofs of the main theorems. In particular we prove Theorem 2.. Let us concentrate on volume preserving partially hyperbolic diffeomorphisms of T 3. One important result which appears in the works of Pesin-Sinai and Ledrappier (see [9] and []) is the exact formula for the disintegration of the Lebesgue measure along unstable manifolds (even in the Pesin theory setting): Take ξ be a measurable partition subordinated to the unstable foliation. For y ξ(x) define J u f ( f i (x)) u (x, y) := J u f ( f i (y)). (4.) After normalizing ρ(y) := u (x, y) L(x) i= where L(x) = u (x, y)dleb x and ρ( ) is ξ(x) the Radon-Nikodym derivative dm x dleb x. We emphasize that such a clear formula for the disintegration along a genral leafwise absolutely continuous foliation (for instance for central foliation whenever it is absolutely continuous) is not available. We use this formula in the proof of Theorem 2.. Here we observe some elementary properties of u. First of all note that C 2 regularity of f and Hölder continuous dependence of E u with the base point give us: Lemma 4.. For any ɛ > there exists δ > 0 such that if y W u δ,x F u x ɛ u (x, y) + ɛ. then

9 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS 9 Proof. Taking logarithm, as α Hölder continuity of unstable bundle and J u f implies that log u (x, y) = log J u f ( f i (x)) log J u f ( f i (y) i= C d( f i (x), f i (y)) α (C i= i= λ iα )d α (x, y) where λ comes from the definition of partial hyperbolicity. This completes the proof of the lemma. Lemma 4.2. Suppose that F u has bounded density property. There exists K > such that for almost every x T 3 and every y, y 2 F u x : Moreover, for any n N : K 2 K u (y, y 2 ) K. (4.2) n i=0 J u f ( f i (y )) J u f ( f i (y 2 )) K2. (4.3) Proof. By definition of uniform bounded density (.3) it comes out that ρ(y 2) ρ(y ) [K 2, K 2 ]. Abusing the notations for simplicity, we substitute K 2 by K and conclude the first claim of the lemma. We can suppose that the points x satisfying (4.2) belong to an invariant set. So changing x to f n (x) we have K u ( f n (y ), f n (y 2 )) K. (4.4) Dividing equation (4.4) by (4.2) we conclude the proof of the second claim of lemma. In stable case, we take f and apply (4.) in the E s = E u direction. Similarly f f for y Fx s we define J s f ( f i (x)) s (x, y) := J s f ( f i (y)). (4.5) i=0 From now on we use the notation σ ( f n (y ), f n (y 2 )) = O() to denote that σ ( f n (y ), f n (y 2 )) is bounded from below and above by constants just depending on f. Now we state a technical proposition which guarantees the constancy of unstable periodic data and rigidity of Lyapunov exponents.

10 0 F. MICENA AND A. TAHZIBI Proposition 4.3. Let f be a partially hyperbolic diffeomorphism of T 3. Suppose that for any x in an invariant full measure set and any y, y 2 F σ x : σ ( f n (y ), f n (y 2 )) = O(), then λ σ ( f, x) = λ σ (A) where A is the linearization of f and σ {s, u}. 4.. Proof of Theorem 2.. As we mentioned above (Lemma 4.2) the U.B.D. property of unstable foliation implies the desired boundedness condition and using proposition 4.3 we conclude that λ u ( f, x) = λ u. Similarly, taking the A inverse f, U.B.D. property implies that λ u ( f, x) = λ u, it means λ s ( f, x) = λ s A A and as f is conservative λ c ( f, x) = λ c. A 4.2. Proof of Proposition 4.3. Take any σ {s, u} and suppose that Z = {x T 3 λ σ ( f, x) > λ σ A } has positive volume. Let ε > 0 be a small number and define A n = {x Z J σ f m (x) > e m(λσ A +ε) for all m n}. Take n large enough such that Qe n(λσ A +ε) m(a n ) > 0 and > 2. 2K 2 e nλσ A where Q be as in definition (3.) of quasi-isometric foliations (We know that stable, unstable and central foliations of partially hyperbolic diffeomorphisms in T 3 are quasi-isometric.) and the constant K is such that Similar to (4.3) we get K σ ( f n (y ), f n (y 2 )) K. K 2 n i=0 J σ f ( f i (y )) J σ f ( f i (y 2 )) K2. (4.6) for any n N. Using proposition 3.4 and lemma 3.6, choose M > 0 such that for any y Fx σ and d σ (x, y) M 2 < f n x f n y < 2, A n x A n y (4.7) A n x A n y 2 enλσ A < < 2e nλσ A. x y (4.8) Take any regular point x A n. By definition we have J σ f n (x) > e n(λσ A +ɛ) and by (4.6) we get J σ f n (y) K 2 en(λσ A +ε)

11 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS for any y F σ x. Now f n x f n y A n x A n y Qdσ ( f n x, f n y) A n x A n y = Q D σ f n dλ F σ x F σ (B) A n x A n y (4.9) Qen(λσ A +ε) x y 2K 2 e nλσ A x y Qen(λσ A +ε) 2K 2 e nλσ A which gives a contradiction. Thus {x T 3 λ σ ( f, x) > λ σ } has zero volume. In A the same way, considering f, it comes out that > 2. m({x T 3 λ σ ( f, x) > λ σ }) = m({x T 3 λ σ ( f, x) < λ σ A A }) = Proof of theorem Local maxima for Lyapunov exponents Proof. We prove the statement on λ u ( f, ). Suppose by contradiction that there is a positive volume set Z T 3, such that, for every x Z we have λ u ( f, x) > λ u A. Since f is C 2, the unstable foliation F u for f is upper absolutely continuous, in particular there is a positive volume set B such that for every point x B we have λ F u x (Fx u Z) > 0. (5.) Choose a p B satisfying (5.) and ε > 0 a small number. Now consider a segment [x, y] u Fp u satisfying λ F u p ([x, y] u Z) > 0 such that the length of [x, y] u is bigger than M as required in lemma 3.6 and proposition 3.4. We can choose M such that and Ax Ay < ( + ε)e λu A y x f x f y Ax Ay < + ε. whenever d u (x, y) M. The above equation implies that f x f y < ( + ε) 2 e λu A y x. Inductively, we assume that for n we have f n x f n y < ( + ε) 2n e nλu A y x. (5.2)

12 2 F. MICENA AND A. TAHZIBI Since f expands uniformly on the u direction we have d u ( f n x, f n y) > M, it leads f ( f n x) f ( f n y) < ( + ε) A( f n x) A( f n y) < ( + ε) 2 e λu A f n x f n y < ( + ε) 2(n+) e (n+)λu A. For each n > 0, let A n Z be the following set A n = {x Z: D u f k > ( + 2ε) 2k e kλu A for any k n}. We have m(z) > 0 and A n Z. Consider a big n and α n > 0 such that Leb F u p ([x, y] u A n ) = α n Leb F u p ([x, y] u ). Note that when n increases to infinity the proportion α n converges to Leb F u p ([x, y] u Z). We can assume with lost generality α n > α 0 > 0 for any n >. Then f n x f n y > Q D f n (z) dλ F u p (z) > [x,y] u A n (5.3) > Q( + 2ε) 2n e nλu A λf u p ([x, y] u A n ) (5.4) > α 0 Q( + 2ε) 2n e nλu A x y. (5.5) The inequalities (5.2) and (5.5) give a contradiction. We conclude that λ u ( f, x) λ u, for almost everywhere x A T3. Considering the inverse f we conclude that λ s A λs ( f, x) for almost every x T 3. It would be interesting to prove a result similar to Theorem 2.3 for central Lyapunov exponent of partially hperbolic diffeomorphisms (See problem 2). However, the above arguments can be used to prove the same result for the special case of Anosov diffeomorphisms. In what follows we consider Anosov diffeomorphism as partially hyperbolic system and by central bundle we refer the weak unstable bundle of the invariant decomposition. Theorem 5.. Consider f : T 3 T 3 a C 2 volume preserving Anosov diffeomorphism with decomposition E uu E u E s. Let A : T 3 T 3, the linearization of f. Suppose that F c is quasi isometric, (upper) absolutely continuous foliation then λ c ( f, ) λ c A, a.e. x T 3 Proof. By [7] we have A is partially hyperbolic and dim E c = dim E c. Since f is f A Anosov with partially hyperbolic structure, then by propositions 3.3 and 3.4, we have that A is Anosov and λ c > 0. for a.e. Furthermore there is µ >, A such that D c f (x) > µ, for any x T 3. In the other words, f has uniform

13 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS 3 expansion in the central direction. Since F c is quasi isometric, we can apply f the same argument of the previous theorem, and we conclude λ c ( f, ) λ c, a.e. A x T d. 6. Topology of central leaves To prove theorem 2.5 we show that U.B.D property of central foliation implies that for almost every x we have λ c (x) = λ c. Consequently under the assumption A of λ c ( f, x) = 0, a.e, A has compact central leaves and by leaf conjugacy between f and A the same is true for f. The idea of the proof is similar to that of theorem 2.. However a main difficulty here is that we do not have a formula for the density of the disintegration along central foliation. Let and ε > 0 be a small number. Define Z = {x T 3 λ c ( f, x) > λ c A } A n = {x Z D f m (x) E c x > e m(λc A +ε) for all m n}. Take n large enough such that m(a n ) > α and Qαe n(λc A +ε) 4Ke nλσ A > 2. (6.) for some positive α > 0 which will be fixed. We choose M satisfying 4.7 and 4.8 for σ = c. Now the idea is to find a central plaque Fx c of size M such that Leb x (A n ) α/2k. Of course, if we could provide a measurable partition of M into plaques of size M by Rokhlin decomposition we would get a plaque such that m x (A n ) α and by definition Leb x (A n ) α/k. As we ignore the existence of such partition we construct a partition covering a large measurable subset of T 3 in the next subsection. Let us complete the proof of the theorem assuming the existence of such plaque. The idea is to get the same contradiction as in the proof of theorem 2.. More precisely, we get similar to (5.5) Q D f n E c dλ f n x f n F c x y A n x A n y Fx c (B) (6.2) A n x A n y Qαen(λc A +ε) x y Qαen(λc A +ε) > 2. 4Ke nλc A x y 4Ke nλc A To find such a plaque we need to construct a measurable partition by plaques as follows.

14 4 F. MICENA AND A. TAHZIBI Measurable partition by long plaques. It is more convenient to work in the universal covering π : R 3 T 3. We lift the foliations to R 3 and use the same notations Fx σ for the leaf passing through x in R 3. First we recall a nice property of central foliation proved by Hammerlind [7]: Proposition 6.. There is a constant R c such that for all x R 3, F c x U Rc (A c x) where U Rc (A c x) denote the neighbourhood of radius R c around the central leaf of A through x. For M large enough, we take D a ball centered at O R 3 of radius M, transverse to F c and in the su leaf of A. Now saturate D by central plaques of size M and let ˆD := z D F c z,m. Lemma 6.2. If M is large enough there exists a plaque F c such that z,m Leb z (π (A n )) α/2k. Proof. Recall that m(a n ) α. As M is large, ˆD will include a large number N(M) of fundamental domains (cubes) where π is invertible. That is C i ˆD where C i are unitary cubes for i =,, N(M). However ˆD may intersect partially Ñ(M) other fundamental cubes, i.e ˆD C i but C i ˆD for i =,, Ñ(M). By the above proposition we claim that So for large enough M we have m(π (A) n ) ˆD) Ñ(M) lim M N(M) = 0. αn(m) N(M) + Ñ(M) α/2. Now we disintegrate along the plaques in ˆD by Rokhlin we get plaques such that m z (π (A n )) α/2 and by definition.3 of uniform bounded density property, it yields that Leb z (π (A n )) α/2k. References [] A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents, and rigidity I: geodesic flows. Arxive: [2] M. Brin, D. Burago, S. Ivanov, On partially hyperbolic of 3 manifolds with commutative fundamental group. Modern Dynamical Systems and Applications, Boris Hasselblatt, M. Brin and Y. Pesin, eds, Cambridge Univ. Press, New York, , [3] M. Brin, Y. Pesin, Partially hyperbolic Dynamical systems., :77 22, 974. Izv. Acad. Nauk. SSSR, vol.3, No.,, [4] M. Brin, D. Burago, S. Ivanov, Dynamical Coherence of Partially Hyperbolic Diffeomorfism of the 3 Torus. Journal of Modern Dynamical Systems and Applications, vol.3, No.,, 2009.

15 REGULARITY OF FOLIATIONS AND LYAPUNOV EXPONENTS 5 [5] P. Carrasco, Compact Dynamical Foliations. Ph.D Thesis, University of Toronto, 200. [6] A. Gogolev, Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 5, no. 4, , 20. [7] A. Hammerlindl, Leaf Conjugacies on the Torus. Ergodic Theory and Dynamical Systems, to appear. [8] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds. Lecture Notes in Math., 583, Springer-Verlag, New York, 977. [9] F. Ledrappier. Proprietes ergodiques des mesures de Sinaï. Publ. Math. I.H.E.S., 59:6388, 984. [0] F. Micena. Avanços em dinâmica parcialmente hiperbólica e entropia para sistema iterado de funções. tese de doutorado, Universidade de São Paulo 20. [] Ya. Pesin, Ya. Sinai. Gibbs measures for partially hyperbolic attractors. Ergodic Theory Dynam. Systems, 2 (983), no. 3-4, [2] G. Ponce, A. Tahzibi, Central Lyapunov exponents of partially hyperbolic diffeomorphisms of T 3, To appear in Proceedings of American Math. Society, 203. [3] C. Pugh, M. Viana, and A. Wilkinson. Absolute continuity of foliations. In preparation. [4] F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics. Partially hyperbolic dynamics, laminations, and Teichüller flow Fields Inst. Commun., 5, 35 87, [5] R. Saghin, Zh. Xia. Geometric expansion, Lyapunov exponents and foliations. Ann. Inst. H. Poincaré 26, no. 2, , Departamento de Matemática, IM-UFAL Maceió-AL, Brazil. address: fpmicena@gmail.com Departamento de Matemática, ICMC-USP São Carlos-SP, Brazil. address: tahzibi@icmc.usp.br