TOPOLOGICAL STRUCTURE OF PARTIALLY HYPERBOLIC ATTRACTORS. José F. Alves

Transcription

1 TOPOLOGICAL STRUCTURE OF PARTIALLY HYPERBOLIC ATTRACTORS by José F. Alves Contents Introduction Partially hyperbolic sets Sets with positive volume Hölder control of tangent direction Hyperbolic times and bounded distortion A local unstable disk inside the attractor Limit sets Hyperbolic sets with positive volume Transitive sets Conservative case Spectral decomposition Markov structures for partially hyperbolic sets Markov structures Returning disks Partition on a reference leaf Product structure Metric estimates References

2 2 JOSÉ F. ALVES Introduction Since the 60 s that hyperbolic sets play an important role in the development of the Theory of Dynamical Systems. A hyperbolic set is a compact invariant set over which the tangent bundle splits into two invariant subbundles, one of them contracting and the other one expanding. In this notes we are concerned with discrete time dynamical systems (smooth transformations of a compact manifold). In the last decades an increasing emphasis has been put on the dynamics of partially hyperbolic sets, i.e. compact invariant sets for which the tangent bundle has a dominated splitting into two invariant subbundles having contracting/expanding behavior in one direction. Precise definitions of all these objects will be given in the next section. In this context, a special role has been played by the horseshoes, which were introduced by Smale in the sixties, and as shown in [Sm1], always appear near a transverse homoclinic point associated to some hyperbolic periodic point of saddle type, i.e. a point whose orbit asymptotically approaches that saddle point, both in the past and in the future. Horseshoes can be used to show that transverse homoclinic points are accumulated by periodic points, but the usefulness of these objects goes far beyond this issue. Horseshoes are Cantor sets which are, in dynamical terms, topologically conjugated to full shifts. A special interest lies in the horseshoes that appear when one unfolds a homoclinic tangency. Knowing how fat these horseshoes are can have several implications in the dynamical behavior after the homoclinic bifurcation. In this setting we mention the thickness, which has been used by Newhouse [Ne] to prove the existence of infinitely many sinks, and the Hausdorff dimension, which has been used by Moreira, Palis, Takens, Yoccoz to study the prevalence of hyperbolicity after the unfolding of a homoclinic tangency; see [MY, PT, PY1, PY2]. One interesting subject is the Lebesgue measure (volume) of horseshoes. As shown by Bowen in [Bo1], there are C 1 diffeomorphisms with hyperbolic horseshoes of positive volume. On the other hand, Bowen has proved in [Bo2] that a basic set (locally maximal hyperbolic set with a dense orbit) of a C 2 diffeomorphism which attracts a set with positive volume, necessarily attracts a neighborhood of itself. In particular, the unstable manifolds through points of this set must be contained in it, and consequently C 2 diffeomorphisms have no horseshoes with positive volume. For diffeomorphisms whose differentiability is higher than one, Alves and Pinheiro prove in [AP1] the nonexistence of horseshoe-like sets with positive volume in a context of sets with some partially hyperbolic structure. Going back to context of hyperbolic sets, it is shown in [AP1] that the above mentioned result in [Bo2] still holds without the local maximality assumption, i.e. a transitive hyperbolic set which attracts a set with positive volume necessarily attracts a neighborhood of itself. It is also proven in [AP1] that there are no proper transitive hyperbolic sets with positive volume for diffeomorphisms whose differentiability is higher than one. Similar results for sets with nonempty interior had already been obtained by Abdenur, Bonatti, Díaz in [ABD] and Fisher in [Fi]. On the other hand, as described in [ABD], there exist (non-transitive) hyperbolic sets with positive volume which do not attract neighborhoods of themselves; see [Fi] for a detailed construction. The results in [AP1] give also a description of the limit set of almost every point in a hyperbolic set with positive volume: there is a finite number of basic sets for which the ω-limit set of Lebesgue almost every point is contained in one of these basic

3 PARTIALLY HYPERBOLIC ATTRACTORS 3 sets. In the partially hyperbolic context these ω-limit sets are contained in the closure of finitely many hyperbolic periodic points. In the conservative setting, Bochi and Viana show in [BV] that a hyperbolic set for a volume preserving C 2 diffeomorphism has either zero volume or coincides with the whole manifold. Markov partitions have played a fundamental role in the development of the theory of hyperbolic dynamical systems. For non-uniformly hyperbolic dynamical systems, Benedicks and Young introduced in [BY] some structures of Markov style in certain regions of the phase space with infinitely many branches and variable return times. This structures enabled them to obtain exponential decay of correlations and deduce the Central Limit Theorem for Hénon maps. Further developments by Young in [Yo2] lead to a uniform theory for some non-uniformly hyperbolic diffeomorphisms, including Hénon maps, billiards with convex scatterers and Axiom A attractors. This kind of approach has also been successfully implemented by Young in [Yo3] for studying the rates of mixing of non-invertible systems with some non-uniformly expanding behavior. These Markov structures play also a key role in [ACF] for proving the continuity of the SRB measure for Hénon maps of the Benedicks-Carleson type. The existence of this Markov structures for partially hyperbolic systems whose central direction is mostly expanding is part of content in [AP3]. These notes were prepared for a course lectured in the Smooth Ergodic Theory Workshop at the Morningside Center of Mathematics, Chinese Academy of Sciences, May Most of this presentation is based on the works [ABV, ALP, AP1, AP3], with the fist two sections corresponding to [ABV, AP1] and the last one corresponding to [ALP, AP3]. The example from [Bo2] is also presented in the first section. Acknowledgements. The author thanks partial financial support from the Morningside Center of Mathematics and from FCT through CMUP. The author acknowledges the hospitality at the Morningside Center of Mathematics, in particular to Yongluo Cao, Xiongping Dai, Gaofei Zhang, Lanyu Wang and, most specially, to Huyi Hu for making so pleasant the stay in Beijing.

4 4 JOSÉ F. ALVES 1. Partially hyperbolic sets Let f : M M be a diffeomorphism of a compact connected Riemannian manifold M. We say that f is C 1+ if f is C 1 and Df is Hölder continuous. We use Leb to denote a normalized volume form extended to the Borel sets of M that we call Lebesgue measure. Given a submanifold γ M we use Leb γ to denote the measure on γ induced by the restriction of the Riemannian structure to γ. A set Λ M is said to be invariant if f(λ) = Λ, and forward invariant if f(λ) Λ. Definition 1.1. Given a forward invariant compact set K, we define Λ = n 0 f n (K). Suppose that there is a continuous splitting T K M = E cs E cu of the tangent bundle restricted to K, and assume that this splitting is Df-invariant over Λ. 1. This is a dominated splitting (over Λ) if there is a constant 0 < λ < 1 such that for some choice of a Riemannian metric on M Df E cs x Df 1 E cu f(x) λ, for every x Λ. We call E cs the centre-stable bundle and E cu the centre-unstable bundle. 2. We say that Λ is partially hyperbolic, if additionally E cs is uniformly contracting or E cu is uniformly expanding, meaning that there is 0 < λ < 1 such that, respectively, or Df E cs x λ, for every x Λ, Df 1 E cu f(x) λ, for every x Λ. 3. We say that f is non-uniformly expanding along the centre-unstable direction for a point x K if 1 n lim inf log Df 1 E cu n + f n j (x) < c. (NUE) j=1 Condition NUE means that the derivative has expanding behavior in the centre-unstable direction in average over the orbit of x for an infinite number of times. Let us mention that if condition NUE holds for every point in a compact invariant set Λ, then E cu is uniformly expanding in the centre-unstable direction in Λ; see [AAS, Ca]. Example 1.2. Take a linear Anosov diffeomorphism f 0 on the d-dimensional torus M = T d with d 3. We write T M = E u E s the corresponding hyperbolic decomposition. Let V be a small closed domain in M such that there exist unit open cubes K 0 and K 1 in R d such that V π(k 0 ) and f 0 (V ) π(k 1 ), where π : R d T d is the canonical projection. Now, let f be a diffeomorphism on T d such that (a) f admits invariant cone fields C cu and C cs, with small width α > 0 and containing, respectively, the unstable bundle E u and the stable bundle E s of the Anosov diffeomorphism f 0 ;

5 PARTIALLY HYPERBOLIC ATTRACTORS 5 (b) there is σ 1 > 1 so that det(df T x D cu ) > σ 1 for any x M and any disk D cu through x tangent to C cu. (c) there exists σ 2 < 1 satisfying (Df T x D cu ) 1 < σ 2 for any x M \ V and any disk D cu through x tangent to C cu. (d) there exists some small δ 0 > 0 satisfying (Df T x D cu ) 1 < (1 + δ 0 ) for any x V and any disks D cu tangent to C cu. For instance, if f 1 is a torus diffeomorphism satisfying (a), (b), (d), and coinciding with f 0 outside V, then any map f in a C 1 neighbourhood of f 1 satisfies all the previous conditions. We argue that any f satisfying (a) (d) is non-uniformly expanding along its centreunstable direction in a strong sense: there is c > 0 such that lim sup n + 1 n n j=1 log Df 1 E cu f j (x) < c. (1) on a full Lebesgue measure set of points x M. To explain this, let B 1,..., B p, B p+1 = V be any partition of T d into small domains, in the sense that there exist open unit cubes K 0 i and K 1 i in R d such that B i π(k 0 i ) and f(b i ) π(k 1 i ). (2) Let F0 u be the unstable foliation of f 0, and F j = f j (F0 u ) for every j 0. By (a), each F j is a foliation of T d tangent to the centre-unstable cone field C cu. For any subset E of a leaf of F j with j 0, we denote Leb j (E) the Lebesgue measure of E inside that leaf. Claim 1. There is C 0 > 0 such that for any small disk D 0 contained in a leaf of F 0 Leb 0 ({x D 0 : f j (x) B ij for every n 1 and every i 0,..., i n 1 in {1,..., p, p + 1} for 0 j < n}) C 0 σ n 1 Proof. Let F j be the lift to R d of F j, for j 0. Using (2) one can easily conclude, by induction on j, that f j ([i]) is contained in the image π(k 1 j 1 F j ) of the intersection of K 1 j 1 with some leaf Fj of Fj, for every 0 j n. So, using (b) and the fact that (π K 1 n 1) is a diffeomorphism and an isometry onto its image, Leb 0 ([i]) σ n 1 Leb n (f n ([i])) σ n 1 Leb n (F n K 1 n 1). (3) Recall that we took f 0 linear, so that its unstable foliation F u 0 lifts to a foliation F u 0 of R d by affine hyperplanes. The leaves of every F n are C 1 submanifolds of R d transverse to these hyperplanes, with angles uniformly bounded away from zero at every intersection point. Consequently, the intersection of a leaf of F n with any unit cube in R d has Lebesgue measure (inside the leaf) bounded by some uniform constant C 0. In particular, the last factor in (3) is bounded by C 0.

6 6 JOSÉ F. ALVES Claim 2. There exists θ > 0 such that the orbit of Lebesgue almost every point x D 0 spends a fraction θ of the time in B 1 B p, that is, for every large n. #{0 j < n : f j (x) B 1 B p } θ n Proof. Let n be fixed. Given a sequence i = (i 0, i 1,..., i n 1 ) in {1,..., p+1}, we denote [i] = B i0 f 1 (B i1 ) f n+1 (B in 1 ). Moreover, we define g(i) to be the number of values of 0 j n 1 for which i j p. We begin by noting that, given any θ > 0, the total number of sequences i for which g(i) < θ n is bounded by k<θ n ( n k ) p k k θ n ( n k ) p θ n A standard application of Stirling s formula (see e.g. [BV, Section 6.3]) gives that the last expression is bounded by e γn p θ n, where γ depends only on θ and goes to zero when θ goes to zero. On the other hand, by Claim 1 we have Leb([i]) C 0 σ1 n. Then the measure of the union I n of all the sets [i] with g(i) < θ n is less than C 0 σ (1 θ)n 1 e γn p θ n. Since σ 1 > 1, we may fix θ small so that e γ p θ < σ1 1 θ. This means that the Lebesgue measure of I n goes to zero exponentially fast as n. Thus, by Borel-Cantelli lemma, Lebesgue almost every point x D 0 belongs in only finitely many sets I n. Clearly, any such point x satisfies the conclusion of the lemma. Let θ > 0 be the constant given by Claim 2, and fix δ > 0 small enough so that σ0(1 θ + δ) e c for some c > 0. Let x be any point satisfying the conclusion of the lemma. Then n 1 j=0 Df 1 E cu f j (x) σθ n 0 (1 + δ) (1 θ)n e cn for every large enough n. This implies that 1 n lim sup log Df 1 E cu f n + n j (x) < c j=1 for Leb 0 -almost every point x D 0. Since D 0 was an arbitrary disk inside a leaf of F s 0, and the latter is an absolutely continuous foliation, we conclude that f is non-uniformly expanding along E cu, Lebesgue almost everywhere in M = T d Sets with positive volume. In this section we present some results on the topological structure of partially hyperbolic sets attracting a positive volume set of orbits with non-uniform expansion, whose proofs we leave to the next sections. Definition 1.3. We say that an embedded disk γ M is an unstable manifold, or an unstable disk, if dist(f n (x), f n (y)) 0 exponentially fast as n, for every x, y γ. Similarly, γ is called a stable manifold, or a stable disk, if dist(f n (x), f n (y)) 0 exponentially fast as n, for every x, y γ.

7 PARTIALLY HYPERBOLIC ATTRACTORS 7 It is well-known that every point in a hyperbolic set possesses a local stable manifold u (x) and a local unstable manifold Wloc (x) which are disks tangent to Es x and Ex u at x W s loc respectively. Theorem 1.4. Let f : M M be a C 1+ diffeomorphism and let K M be a forward invariant compact set with a continuous splitting T K M = E cs E cu dominated over Λ = n 0 f n (K). If NUE holds for a positive Lebesgue set of points x K, then Λ contains some local unstable manifold. The next result is a direct consequence of Theorem 1.4, whenever E cu is uniformly expanding. If, on the other hand, E cs is uniformly contracting, then we just have to apply Theorem 1.4 to f 1. Corollary 1.5. Let f : M M be a C 1+ diffeomorphism and let Λ M be a compact invariant set with a dominated splitting and Leb(Λ) > If E cs is uniformly contracting, then Λ contains some local stable manifold. 2. If E cu is uniformly expanding, then Λ contains some local unstable manifold. The same conclusions hold for partially hyperbolic sets intersecting a local stable manifold or a local unstable manifold in a positive Lebesgue measure subset, as Corollary 1.7 below shows. Theorem 1.6. Let f : M M be a C 1+ diffeomorphism and let K M be a forward invariant compact set with a continuous splitting T K M = E cs E cu dominated over Λ = n 0 f n (K). Assume that there is a local unstable manifold γ such that NUE holds for every x in a positive Leb γ subset of γ K. Then Λ contains some local unstable manifold. The next result is an immediate consequence of Theorem 1.6, in the case that E cu is uniformly expanding, and a consequence of the same theorem applied to f 1 when E cs is uniformly contracting. Actually, we shall prove a stronger version of this result in Theorem Corollary 1.7. Let f : M M be a C 1+ diffeomorphism and let Λ M with Leb(Λ) > 0 be a compact invariant set with a dominated splitting. 1. If E cs is uniformly contracting and there exists a local stable manifold γ such that Leb γ (γ Λ) > 0, then Λ contains some local stable manifold. 2. If E cu is uniformly expanding and there exists a local unstable manifold γ such that Leb γ (γ Λ) > 0, then Λ contains some local unstable manifold. The previous results give in particular that there cannot be partially hyperbolic horseshoes with positive volume, provided the dynamics is C 1+. If the diffeomorphism is just C 1 then the previous conclusions do not hold. The following example was presented by Bowen in [Bo1] and makes this point clear. Example 1.8. We shall construct a C 1 diffeomorphism which has a compact invariant hyperbolic Cantor set (horseshoe) with positive Lebesgue measure. First we consider some notation for Cantor sets. Let I be a closed interval and (α n ) n a squence of positive numbers

8 8 JOSÉ F. ALVES with n 0 α n length(i). Let a = a 1 a 2... a n denote a sequence of 0 s and l s of length n = n(a); we allow the empty sequence a = with n( ) = 0. Define [ a + b I = I = [a, b], I = α 0 2 2, a + b + α ] and Ia I a recursively as follows: let I a0 and I a1 be the left and right intervals remaining when the interior of Ia is removed from I a ; let Iak (k=0, 1) be the closed interval of length α n(ak) 2 n(ak) and having the same center as I ak. The Cantor set K I is given as K I = I a. m 0 n(a)=m This is the standard construction of the Cantor set except that we permit some flexibility in the lengths of intervals removed. The measure of K I is m(k I ) = length(i) n 0 α n Assume now we have another interval J and β n > 0 with n 0 β n length(i). One can then construct Ja J a and K J as above. Suppose that β n αn 1 γ 0 as n. Pick a sequence δ n 0 and for each a let g : Ia Ja be a C 1 orientation preserving diffeomorphism such that: (i) g (x) = γ for x and endpoint of I a; (ii) g (Ia) is contained in the interval spanned by γ ± δ n and β n ± δ n. α n Then g extends from a Ia by continuity to a homeomorphism g : I J which is C 1 with g (x) = γ at each point x K I. We will now construct a horseshoe with positive measure. Choose β n > 0 with Let Then n 0 n 0 β n < 2 and β n+1 β n 1 J = [ 1, 1], I = [ ] β0 2, 1 ( e.g. β n = 1 (n + 100) 2 and α n = β n+1 2. α n < length(i) and γ = lim β n α n = lim 2β n β n+1 = 2. So we get a C 1 diffeomorphism g : I J as above. One defines a diffeomorphism f of the square S = J J into R 2 by (i) f(x, y) = (g(x), g 1 (y) for (x, y) I J; (ii) f(x, y) = (g( x), g 1 (y) for (x, ( y) ( I) J; (iii) f(t ) (J J) =, where T = β 0 2, β ) 0 J. 2 Then Λ = + n= f n (S) = K J K J ).

9 PARTIALLY HYPERBOLIC ATTRACTORS 9 has Lebesgue measure m(λ) = m(k J ) 2 = ( 2 n 0 β n ) 2 > 0. The mapping f extends to a diffeomorphism of S 2 exactly as in [Sm2] Hölder control of tangent direction. In this section we present some results from [ABV] concerning the Hölder control of the tangent direction of certain submanifolds. Let K be a forward invariant compact set for which there is a continuous splitting T K M = E cs E cu of the tangent bundle restricted to K which is Df-invariant over Λ = n 0 f n (K). We fix continuous extensions of the two bundles E cs and E cu to some compact neighborhood U of Λ, that we still denote by E cs and E cu. Replacing K by a forward iterate of it, if necessary, we may assume that K U. Definition 1.9. Given 0 < a < 1, we define the centre-unstable cone field (C cu a (x)) x U of width a by Ca cu (x) = { v 1 + v 2 Ex cs Ex cu such that v 1 a v 2 }. (4) We define the centre-stable cone field (C cs a (x)) x U of width a in a similar way, just reversing the roles of the subbundles in (4). We fix a > 0 and U small enough so that, up to slightly increasing λ < 1, the domination condition remains valid for any pair of vectors in the two cone fields, i.e. Df(x)v cs Df 1 (f(x))v cu λ v cs v cu, for every v cs C cs a (x), v cu C cu a (f(x)), and any x U f 1 (U). Note that the centreunstable cone field is forward invariant: Df(x)C cu a (x) C cu a (f(x)), whenever x, f(x) U. Indeed, the domination property together with the invariance of E cu over Λ imply that Df(x)C cu a (x) C cu λa(f(x)), for every x Λ. (5) This extends to any x U f 1 (U) just by continuity, slightly increasing λ < 1, if necessary. Definition We say that an embedded C 1 submanifold N U is tangent to the centre-unstable cone field if the tangent subspace to N at each point x N is contained in the corresponding cone Ca cu (x). Then f(n) is also tangent to the centre-unstable cone field, if it is contained in U, by the domination property. We choose δ 0 > 0 small enough so that the inverse of the exponential map exp x is defined on the δ 0 neighbourhood of every point x in U. From now on we identify this neighbourhood of x with the corresponding neighbourhood U x of the origin in T x N, through the local chart is contained in the a (y) with defined by exp 1 x. Reducing δ 0, if necessary, we may suppose that Ex cs centre-stable cone Ca cs (y) of every y U x. In particular, the intersection of C cu

10 10 JOSÉ F. ALVES Ex cs reduces to the zero vector. Then, the tangent space to N at y is parallel to the graph of a unique linear map A x (y) : T x N E cs x. Definition Given constants C > 0 and 0 < ζ 1, we say that the tangent bundle to N is (C, ζ)-hölder if for every y N U x and x U A x (y) Cd x (y) ζ, (6) where d x (y) denotes the distance from x to y along N U x, defined as the length of the shortest curve connecting x to y inside N U x. Recall that we have chosen the neighbourhood U and the cone width a sufficiently small so that the domination property remains valid for vectors in the cones Ca cs (z), Ca cu (z), and for any point z in U. Then, there exist λ 1 (λ, 1) and ζ (0, 1] such that Df(z)v cs Df 1 (f(z))v cu 1+ζ λ 1 < 1 (7) for every norm 1 vectors v cs C cs a (z) and v cu C cu a (z), at any z U. Then, up to reducing δ 0 > 0 and slightly increasing λ 1 < 1, condition (7) remains true if we replace z by any y U x, with x U (taking to mean the Riemannian metric in the corresponding local chart). We fix ζ and λ 1 as above, choosing ζ with the additional property that f is of class C 1+ζ. Given a C 1 submanifold N U, we define κ(n) = inf{c > 0 : the tangent bundle of N is (C, ζ)-hölder}. (8) Proposition There exist λ 0 < 1 and C 0 > 0 so that if N U f 1 (U) is any C 1 submanifold tangent to the centre-unstable cone field, then κ(f(n)) λ 0 κ(n) + C 0. Proof. We only need to consider the case when κ(n) is finite, that is, the tangent bundle of N is (C, ζ)-hölder for some C > 0. Let x N be fixed. We use (u, s) T x N Ex cs and (u 1, s 1 ) T f(x) f(n) Ef(x) cs, respectively, to represent the local coordinates in U x and U f(x) introduced above. We write the expression of our map in these local coordinates as f(u, s) = (u 1 (u, s), s 1 (u, s)). Observe that if x K then the partial derivatives of u 1 and s 1 at the origin 0 T x N are u u 1 (0) = Df T x N, s u 1 (0) = 0, u s 1 (0) = 0, s s 1 (0) = Df E cs x. This is because Ex cs = Ex cs is mapped to Ef(x) cs = Ecs f(x) under Df(x) and, similarly, T xn is mapped to T f(x) N. Then, given any small ε 0 > 0 we have that u u 1 (y) Df T x N, s u 1 (y), u s 1 (y), s s 1 (y) Df E cs x, (9) are all less than ε 0 for every x U and y U x, as long as δ 0 and U are small. Taking the cone width a also small, we get Df T y N Df E cu x ε 0 and Df 1 T f(y) f(n) Df 1 E cu f(x) ε 0, (10) for every x U and y U x. Since f is C 2, there is also some constant K 2 > 0 such that s u 1 (y) K 2 d x (y) ζ and u s 1 (y) K 2 d x (y) ζ. (11)

11 PARTIALLY HYPERBOLIC ATTRACTORS 11 For y 1 in U f(x), let A f(x) (y 1 ) be the linear map from T f(x) f(n) to Ef(x) cs whose graph is parallel to T y1 f(n). We are going to prove that, fixing ε 0 sufficiently small, then A f(x) (y 1 ) satisfies (6) for any C > λ 0 κ(n) + C 0, with convenient λ 0 and C 0. Let us begin by noting that A f(x) (y 1 ) is bounded by some uniform constant K 1 > 0, since f(n) is tangent to the centre-unstable cone field. We will choose the constant C 0 K 1 /(δ 0 / Df 1 ) ζ, so that (6) is immediate when d f(x) (y 1 ) δ 0 / Df 1 : A f(x) (y 1 ) K 1 C 0 (δ 0 / Df 1 ) ζ C 0 d f(x) (y 1 ) ζ. Here Df 1 is the supremum of all Df 1 (z) with z U w, w U, where the norms are taken with respect to the Riemannian metrics in the local charts. This permits us to restrict to the case when d f(x) (y 1 ) < δ 0 / Df 1 in all that follows. Let Γ 1 be any curve on f(n) U f(x) joining f(x) to y 1 and whose length approximates d f(x) (y 1 ). Then Γ = f 1 (Γ 1 ) is a curve in N U x joining x to y = f 1 (y 1 ), with length less than δ 0. In fact, cf. (10), d x (y) length(γ) ( Df 1 E cu f(x) + ε 0 ) length(γ 1 ). This shows that d x (y) ( Df 1 E cu f(x) + ε 0)d f(x) (y 1 ). Now we observe that A f(x) (y 1 ) = [ u s 1 (y) + s s 1 (y) A x (y) ] [ u u 1 (y) + s u 1 (y) A x (y) ] 1. On the one hand, by (9) and (11), u s 1 (y) + s s 1 (y) A x (y) K 2 d x (y) ζ + ( Df E cs x + ε 0 ) κ(n)dx (y) ζ ( K 2 + ( Df E cs x + ε 0 ) κ(n) ) dx (y) ζ. On the other hand, s u 1 (y) A x (y) ε 0 K 1, which can be made much smaller than 1/ ( u u 1 (y) 1. As a consequence, recall (10) and (11), [ u u 1 (y) + s s 1 (y) A x (y) ] 1 Df 1 E cu f(x) + ε 1, where ε 1 can be made arbitrarily small by reducing ε 0. Putting these bounds together, we conclude that A f(x) (y 1 ) d f(x) (y 1 ) ζ is less than ( Df E cs x + ε 0 )( Df 1 E cu ( Df 1 E cu f(x) + ε 0) ζ f(x) + ε 1) κ(n) + K 1 2( Df Ecu f(x) + ε 1) ( Df 1 Ef(x) cu + ε 0). ζ Hence, choosing δ 0, U, a sufficiently small, we can make ε 0, ε 1 sufficiently close to zero so that the factor multiplying κ(n) is less than some λ 0 (λ 1, 1); recall (7). Moreover, the second term in the expression above is bounded by some constant that depends only on f. We take C 0 larger than this constant. Corollary There exists C 1, ζ > 0 such that, given any C 1 submanifold N U tangent to the centre-unstable cone field, there is n 0 1 such that: 1. κ(f n (N)) C 1 for every n n 0 such that f k (N) U for all 0 k n; 2. if κ(n) C 1, then κ(f n (N)) C 1 for n 1 such that f k (N) U for all 0 k n; 3. if N and n are as in 2, then the functions J k : f k (N) x log det ( Df T x f k (N) ), 0 k n, are (L, ζ)-hölder continuous with L > 0 depending only on C 1 and f.

12 12 JOSÉ F. ALVES Proof. It suffices to take any C 1 C 0 /(1 λ 0 ) Hyperbolic times and bounded distortion. Let K M be a forward invariant compact set and let Λ K U be as in Section 1.2. The following notion will allow us to derive uniform behaviour (expansion, distortion) from the non-uniform expansion. Definition Given σ < 1, we say that n is a σ-hyperbolic time for x K if n Df 1 E cu f j (x) σk, for all 1 k n. j=n k+1 In particular, if n is a σ-hyperbolic time for x, then Df k Ef cu n (x) every 1 k n: n Df k E cu f n (x) j=n k+1 is a contraction for Df 1 E cu f j (x) σk. (12) If a > 0 is taken sufficiently small in the definition of our cone fields, and we choose δ 1 > 0 also small so that the δ 1 -neighborhood of K should be contained in U, then by continuity Df 1 (f(y))v 1 Df 1 E cu σ f(x) v, (13) whenever x K, dist(f(x), f(y)) δ 1 and v C cu a (f(y)). Given any disk M, we use dist (x, y) to denote the distance between x, y measured along. The distance from a point x to the boundary of is dist (x, ) = inf y dist (x, y). Lemma Take any C 1 disk U of radius δ, with 0 < δ < δ 1, tangent to the centre-unstable cone field. There is n 0 1 such that for x K with dist (x, ) δ/2 and n n 0 a σ-hyperbolic time for x, then there is a neighborhood V n of x in such that: 1. f n maps V n diffeomorphically onto a disk of radius δ 1 around f n (x) tangent to the centre-unstable cone field; 2. for every 1 k n and y, z V n, dist f n k (V n )(f n k (y), f n k (z)) σ k/2 dist f n (V n )(f n (y), f n (z)); 3. for every 1 k n and y V n, n Df 1 E cu f j (y) σk/2. j=n k+1 Proof. First we show that f n ( ) contains some disk of radius δ 1 around f n (x), as long as n > 2 log(δ/(2δ 1)). (14) log(σ) Define 1 as the connected component of f( ) U containing f(x). For k 1, we inductively define k+1 f k+1 ( ) as the connected component of f( k ) U containing

13 PARTIALLY HYPERBOLIC ATTRACTORS 13 f k+1 (x). We shall prove that n contains some disk of radius δ 1 around f n (x) for n as in (14). Observe that since j U, the invariance (5) gives that for every j 1 T w j C cu λ j a (w), for every w j. (15) Let η 0 be a curve of minimal length in n connecting f n (x) to f n (y) n for which dist n (f n (x), f n (y)) < δ 1. For 0 k n, writing η k = f k (η 0 ) we have η k n k. We prove by induction that length(η k ) < σ k/2 δ 1, for 0 k n. Let 1 k n and assume that length(η j ) < σ j/2 δ 1, for 0 j k 1. Denote by η 0 (w) the tangent vector to the curve η 0 at the point w. Using the fact that η k n k and (15) we have Df j (w) η 0 (w) C cu λ n j a (f j (w)) C cu a (f j (w)). Then, by the choice of δ 1 in (13) and the definition of σ-hyperbolic time, n Df k (w) η 0 (w) σ k/2 η 0 (w) Df 1 E cu f j (x) σk/2 η 0 (w). Hence, j=n k+1 length(η k ) σ k/2 length(η 0 ) < σ k/2 δ 1. This completes our induction. In particular we have length(η n ) < σ n/2 δ 1. The k preimage of the ball of radius δ 1 in n centered at f n (x) is contained in U for each 1 k n. If η n is a curve in connecting x to y, then we must have n < 2 log(δ/(2δ 1)). log(σ) Moreover, f n ( ) contains some disk of radius δ 1 around f n (x) for n as in (14). Let now D 1 be the disk of radius δ 1 around f n (x) in f n ( ) and let V n = f n (D 1 ), for n as in (14). Take any y, z V n and let η 0 be a curve of minimal length in D 1 connecting f n (y) to f n (z). Defining η k = f n+k (η 0 ), for 1 k n, and arguing as before we inductively prove that for 1 k n length(η k ) σ k/2 length(η 0 ) = σ k/2 dist f n (V n )(f n (y), f n (z)), which implies that for 1 k n dist f n k (V n)(f n k (y), f n k (z)) σ k/2 dist f n (V n )(f n (y), f n (z)). This completes the proof of the first two items of the lemma. Given y V n we have dist(f j (x), f j (y)) δ 1 for every 1 j n, which together with (13) gives n j=n k+1 Df 1 E cu f j (y) σ k/2 n j=n k+1 Df 1 E cu f j (x) σk/2. Recall that f j (x) K for every j, and n is a σ-hyperbolic time for x. We shall sometimes refer to the sets V n as hyperbolic pre-balls and to their images f n (V n ) as hyperbolic balls. Notice that the latter are indeed balls of radius δ 1.

14 14 JOSÉ F. ALVES Corollary 1.16 (Bounded Distortion). There exists C 2 > 1 such that given as in Lemma 1.15 with κ( ) C 1, and given any hyperbolic pre-ball V n with n n 0, then for all y, z V n 1 det Df n T y C 2 det Df n T z C 2. Proof. For 0 i < n and y, we denote J i (y) = det Df T f i (y)f i ( ). Then, log det Df n n 1 T y det Df n T z = ( log Ji (y) log J i (z) ). i=0 By Corollary 1.13, log J i is (L, ζ)-hölder continuous, for some uniform constant L > 0. Moreover, by Lemma 1.15, the sum of all dist f j ( )(f j (y), f j (z)) ζ over 0 j n is bounded by 2δ 1 /(1 σ ζ/2 ). Then it suffices to take C 2 = exp(2δ 1 L/(1 σ ζ/2 )) A local unstable disk inside the attractor. Now we are able to prove Theorem 1.4 and Theorem 1.6. Those results will be obtained as corollaries of the next slightly more general result. Take K M a forward invariant compact set and let Λ K U be as before. Theorem Let f : M M be a C 1+ diffeomorphism and let K M be a forward invariant compact set with a continuous splitting T K M = E cs E cu dominated over Λ = n 0 f n (K). Assume that there is a disk tangent to the centre-unstable cone field intersecting K in a positive Leb set of points where NUE holds. Then Λ contains some local unstable disk Consequences of Theorem Let us show that Theorem 1.17 implies Theorem 1.4. Assume that NUE holds for Lebesgue almost every x K with Leb(K) > 0. Choosing a Leb density point of K, we laminate a neighborhood of that point into disks tangent to the centre-unstable cone field contained in U. Since the relative Lebesgue measure of the intersections of these disks with K cannot be all equal to zero, we obtain some disk as in the assumption of Theorem For showing that Theorem 1.17 implies Theorem 1.6, we just have to observe that local unstable manifolds are tangent to the centre-unstable subspaces Proof of Theorem Let H K be the set of points where NUE holds. It is easy to check that f(h) H. Let be a disk of radius δ > 0 tangent to the centreunstable cone field intersecting H in a positive Leb. Since NUE remains valid under positive iteration, by Corollary 1.13 we may assume that κ( ) < C 1. It is no restriction to assume that H intersects the sub-disk of of radius δ/2, for some 0 < δ < δ 1, in a positive Leb subset, and we do so. Lemma 1.18 (Pliss). Given A > c 2 > c 1 > 0 let θ = (c 2 c 1 )/(A c 1 ). Given real numbers a 1,..., a N satisfying a j A for every 1 j N and N a j c 2 N, j=1

15 PARTIALLY HYPERBOLIC ATTRACTORS 15 there are l > θn and 1 < n 1 < < n l N so that n i j=n+1 for every 0 n < n i and i = 1,..., l. Proof. Define for each 1 n N, S n = a j c 1 (n i n) n (a j c 1 ), and also S 0 = 0. j=1 Then define 1 < n 1 < < n l N to be the maximal sequence such that S ni S n for every 0 n < n i and i = 1,..., l. Note that l cannot be zero, since S N > S 0. Moreover, the definition means that n i j=n+1 a j c 1 (n i n), for 0 n < n i and i = 1,..., l. So, we only have to check that l > θ 0 N. Observe that, by definition, for every 1 < i l. Moreover, S ni 1 < S ni 1 and so S ni < S ni 1 + (A c 1 ) S n1 (A c 1 ) and S nl S N N(c 2 c 1 ). This gives, N(c 2 c 1 ) S nl = which completes the proof. l ( ) Sni S ni 1 + Sn1 < l(a c 1 ), i=2 Corollary There is 0 < σ < 1 (depending only on f and λ) such that every x H has infinitely many σ-hyperbolic times. Proof. Given x H, by NUE we have infinitely many positive integers N for which N j=1 log Df 1 E cu f j (x) cn. Take c 1 = c/2, c 2 = c, A = sup log Df 1 E cu, and aj = log Df 1 E cu f j (x) in the previous lemma. Note that under assumption NUE we are unable to prove the existence of positive frequency of hyperbolic times at infinity. This will be possible in Section 3 where we shall use a stronger form of non-uniform expansion replacing lim inf by lim sup in the definition of NUE. The existence of infinitely many hyperbolic times is enough for the present situation.

16 16 JOSÉ F. ALVES Lemma Let O be an open set in such that Leb (O H) > 0. Given any small ρ > 0 there is a hyperbolic time n, a hyperbolic pre-ball V O and W V such that n = f n (W ) is a disk of radius δ 1 /4 tangent to the centre-unstable cone field and Leb n (f n (H)) Leb n ( n ) 1 ρ. Proof. Take a small number ɛ > 0. By regularity of Leb measure, there is a compact set C contained in O H and A an open neighborhood of O H in such that Leb (A \ C) < ɛleb (C). It follows from Corollary 1.19 and Lemma 1.15 that we can choose for each x C a σ- hyperbolic time n(x) and a hyperbolic pre-ball V x such that V x A. Recall that V x is the neighborhood of x which is mapped diffeomorphically by f n(x) onto the ball B δ1 (f n(x) (x)) of radius δ 1 around f n(x) (x), tangent to the centre-unstable cone field. Let W x V x be the pre-image of the ball B δ1 /4(f n(x) (x)) of radius δ 1 /4 under this diffeomorphism. By compactness there are x 1,..., x m C such that C W x1... W xm. Assume that {n(x 1 ),..., n(x m )} = {n 1,..., n s }, with n 1 < n 2 <... < n s. (16) Let I 1 N be a maximal set of {1,..., m} such that if i I 1 then n(x i ) = n 1 and W xi W xj = for all j I 1 with j i. Inductively we define I k for 2 k s as follows: Supposing that I k 1 has already been defined, let I k N be a maximal set of {1,..., m} such that if i I k, then n(x i ) = n k and W xi W xj = for all j I k with j i, and also W xi W xj = for all j I 1... I k 1. Let I = I 1 I s. By maximality, each W xj, for 1 j m, intersects some W xi with i I and n(x j ) n(x i ). Thus, given any 1 j m, taking i I such that W xj W xi and n(x j ) n(x i ), we get Lemma 1.15 assures that and so f n(x i) (W xj ) B δ1 /4(f n(x i) (x i )). diam(f n(x i) (W xj )) δ 1 2 σ(n(x j) n(x i ))/2 δ 1 2, f n(x i) (W xj ) B δ1 (f n(x i) (x i )). This implies that W xj V xi. Hence {V xi } i I is a covering of C. It follows from Corollary 1.16 that there is a uniform constant γ > 0 such that Hence Leb (W xi ) Leb (V xi ) γ, for every i I. ( ) Leb i I W xi = Leb (W xi ) i I i I γ Leb (V xi ) γ Leb ( i I V xi ) γ Leb (C).

17 PARTIALLY HYPERBOLIC ATTRACTORS 17 Let { } Leb (W xi \ C) τ = min : i I. Leb (W xi ) We essentially want to prove that τ can be made arbitrarily small. This will achieved by taking ɛ > 0 small. We have ε Leb (C) > Leb (A \ C) Leb ( i I W xi \ C ) = Leb (W xi \ C) i I ( ) τ Leb i I W xi τγ Leb (C). This implies that τ < ε/γ. Since ε > 0 can be taken arbitrarily small, we may choose W xi with the relative Lebesgue measure of C in W xi arbitrarily close to 1. Then, by bounded distortion, the relative Lebesgue measure of f n(xi) (H) f n(xi) (C) in f n(xi) (W xi ) can also be made arbitrarily close to 1. Recalling that f n(xi) (W xi ) is a disk of radius δ 1 /4 around f n(xi) (x i ) tangent to centre-unstable cone field, we just have to take V = V xi, W = W xi and n = n(x i ). Remark Observe that we did not use the forward invariance of H in the proof of Lemma We have just used the fact that points in H have infinitely many hyperbolic times. Corollary There are sets W 1 W 2 and integers 1 n 1 n 2 such that: 1. W k is contained in some hyperbolic pre-ball with hyperbolic time n k ; 2. k = f n k (Wk ) is a disk of radius δ 1 /4, centered at some point x k, tangent to the centre-unstable cone field; 3. f n k (Wk+1 ) is contained in the disk of radius δ 1 /8 centered at x k ; Leb k (f nk (H)) 4. lim = 1. k Leb k ( k ) Proof. Take a constant 0 < ρ < 1 such that for any disk D of radius δ 1 /4 centered at a point x tangent to the centre-unstable cone field the following holds: If Leb D (A) (1 ρ)leb D (D) for some A D, then we must have Leb D (A) > 0 for the disk D D of radius δ 1 /8 centered at the same point x. Note that it is possible to make a choice of ρ in these conditions only depending on the radius of the disk and the dimension of the disk. Surely, once we have chosen some ρ satisfying the required property, then any smaller number still has that property. We shall use Lemma 1.20 repeatedly in order to define the sequence of sets (W k ) k and hyperbolic times (n k ) k inductively. Let us start with O = and 0 < ρ < 1 with the property mentioned above. By Lemma 1.20 there are n 1 1 and W 1 V 1 O, where V 1 is a hyperbolic pre-ball with hyperbolic time n 1, such that 1 = f n 1 (W 1 ) is a disk of

18 18 JOSÉ F. ALVES radius δ 1 /4 centered at some point x 1, tangent to the centre-unstable cone field, such that Leb 1 (f n 1 (H)) Leb 1 ( 1 ) 1 ρ. Considering 1 1 the disk of radius δ 1 /8 centered at x 1, then by the choice of ρ we have Leb 1 (H) > 0. Let O 1 W 1 be the part of W 1 which is sent by f n 1 diffeomorphically onto 1. We have Leb (O 1 H) > 0. Next we apply Lemma 1.20 to O = O 1 and ρ/2 in the place of ρ. Then we find a hyperbolic time n 2 and W 2 O 1 such that 2 = f n 2 (W 2 ) satisfies Leb 2 (f n 2 (H))) Leb 2 ( 2 ) 1 ρ 2. Observe that W 2 O 1 W 1. Then we take O 2 W 2 as that part of W 2 which is sent by f n 2 diffeomorphically onto the disk 2 of radius δ 1 /8 and proceed inductively. The next result gives the conclusion of Theorem Lemma The sequence ( k ) k has a subsequence converging to a local unstable disk of radius δ 1 /4 inside Λ. Proof. Let ( k ) k be the sequence of disks given by Corollary 1.22 and (x k ) k be the sequence of points at which these disks are centered. Up to taking subsequences, we may assume that the centers of the disks converge to some point x. Using Ascoli-Arzela, a subsequence of the disks converge to some disk centered at x, which must necessarily be contained in Λ. Note that each k is contained in the n k -iterate of, which is a disk tangent to the centre-unstable cone field. The domination property implies that the angle between k and E cu goes uniformly to 0 as n. In particular, is tangent to E cu at every point in Λ. By Lemma 1.15, given any n 1, then f n is a σ n/2 -contraction on k for every large k. Passing to the limit, we get that f n is a σ n/2 -contraction in the E cu direction over for every n 1. The fact that the Df-invariant splitting T Λ M = E cs E cu is dominated implies that any expansion Df may exhibit along the complementary direction E cs is weaker than the expansion in the E cu direction. Then there exists a unique unstable manifold Wloc u (x) tangent to Ecu and which is contracted by the negative iterates of f; see [Pe]. Since is contracted by every f n, and all its negative iterates are tangent to centre-unstable cone field, then is contained in Wloc u (x) Limit sets. Using the previous results we are able to give a description of the ω-limit of Lebesgue almost every point in a partially hyperbolic set whose center-unstable direction displays non-uniform expansion in a subset with positive volume. Recall that the ω-limit of a point in M is the set of accumulation points of its orbit. Theorem Let f : M M be a C 1+ diffeomorphism and let K M with Leb(K) > 0 be a forward invariant compact set with a continuous splitting T K M = E cs E cu which is dominated over Λ = n 0 f n (K). Assume that E cs is uniformly contracting and NUE holds for Lebesgue almost every x K. Then there are hyperbolic periodic points p 1,..., p k Λ such that:

19 PARTIALLY HYPERBOLIC ATTRACTORS W u (p i ) Λ for each 1 i k; 2. for Leb almost every x K there is 1 i k for which ω(x) W u (p i ). Moreover, if E cu has dimension one, then for each 1 i k 3. W u (p i ) attracts an open neighborhood of itself. The last conclusion also holds whenever E cu is uniformly expanding. Actually, more can be said in the case of uniformly hyperbolic sets with positive volume, as we shall see in the next subsection. Open problem. Can we obtain the conclusion of the third item for higher dimensional centre-unstable direction? Is there any counter-example? In the remaining of this section we prove Theorem By Corollary 1.22 there exist a sequence of sets W 1 W 2 contained in and a sequence of positive integers n 1 n 2 such that: 1. W k is contained in some hyperbolic pre-ball with hyperbolic time n k ; 2. k = f n k (Wk ) is a disk of radius δ 1 /4, centered at some point x k, tangent to the centre-unstable cone field; 3. f n k (Wk+1 ) is contained in the disk k of radius δ 1/8 centered at x k. Taking a subsequence, if necessary, we have by Lemma 1.23 that the sequence of disks ( k ) k accumulates on a local unstable disk of radius δ 1 /4 which is contained in Λ. Our aim now is to prove that Λ contains the unstable manifold of some periodic point. We choose δ > 0 small so that i) Wδ s (z) is defined for every z Λ, ii) the 2δ-neighborhood of Λ is contained in U, and iii) Df 1 (f(y))v σ 1/4 Df 1 E cu f(x) v, (17) whenever x U, dist(f(x), f(y)) 2δ, and v C cu a (f(y)). Lemma Given K 1 K with Leb(K 1 ) > 0, there exist a hyperbolic periodic point p Λ and δ 2 > 0 (not depending on p) such that: 1. W u (p) Λ; 2. the size of Wloc u (p) is at least δ 2; 3. Leb W u loc (p) almost every point in Wloc u (p) belongs to H; 4. there is x K 1 with ω(x) W u (p). Proof. Let x denote the center of the accumulation disk. cylinder C δ = Wδ s (y), y and the projection along local stable manifolds π : C δ. Let us consider the Slightly diminishing the radius of the disk, if necessary, we may assume that there is a positive integer k 0 such that for every k k 0 π( k C δ ) = and k C δ. (18)

20 20 JOSÉ F. ALVES For each k k 0 let π k : k be the projection along the local stable manifolds. Notice that these projections are continuous and π π k = id. Take a positive integer k 1 > k 0 sufficiently large so that π( k1 C δ/2 ) = and λ n k 1 n k (19) We have k1 = f n k 1 (Wk1 ) f n k 1 n k0 (f n k0(wk0 +1)) f n k 1 n k0( k0 ), which together with (18) and (19) implies that there is some disk 0 such that π f n k 1 n k0 πk0 ( 0 ) =. Thus there must be some z 0 which is a fixed point for the continuous map π f n k 1 n k0 π k0. This means that there are z k0, z k1 W s δ (z) with z k 0 k0 and z k1 k1 such that f n k 1 n k0 (z k0 ) = z k1. Letting γ = W s δ (z), we have dist γ(w, z k1 ) 2δ for every w γ. This implies that dist γ (f n k 1 n k0 (w), zk1 ) = dist γ (f n k 1 n k0 (w), f n k1 n k0 (zk0 )) 2δλ n k 1 n k0, which together with (19) gives dist γ (f n k 1 n k0 (w), z) distγ (f n k 1 n k0 (w), zk1 ) + dist γ (z k1, z) δ. We conclude that f n k 1 n k0 (Wδ s(z)) W δ s(z). Since W δ s (z) is a topological disk, this implies that Wδ s(z) must necessarily contain some periodic point p of period m = n k 1 n k0. As z and p Wδ s (z) it follows that p Λ, by closeness of Λ. Let us now prove that p is a hyperbolic point. As p Wδ s (z), it is enough to show that Df m Ef cu m (p) < 1, where m = n k 1 n k0. Let q = Wδ s(z) f n k 0 (W k1 ). Observe that since p Λ Wδ s (z), then q belongs to the 2δ-neighborhood of Λ, which is contained in U. Since W k1 is contained in some hyperbolic pre-ball with hyperbolic time n k1, it follows from Lemma 1.15 and the choice of δ in (17) that Df m E cu f m (p) Thus we have proved the hyperbolicity of p. m j=1 σ m/4 m Df 1 E cu f j (p) (20) j=1 Df 1 E cu f j (q) σ m/4. (21) Now since p is a hyperbolic periodic point, there is Wloc u (p) a local unstable manifold through p tangent to the center unstable bundle. As cuts transversely the local stable manifold through p, then using the inclination lemma we deduce that the positive iterates of accumulate on the unstable manifold through p. Since these iterates are all contained in the closed set Λ, we must have W u (p) Λ, which then implies that W u (p) Λ. Thus we have proved the first part of the result. By (20) and (21) we deduce that every multiple of m is a σ 1/4 -hyperbolic time for p. Then we choose δ 2 > 0 such that an inequality as in (13) holds with δ 2 in the place of δ 1

21 PARTIALLY HYPERBOLIC ATTRACTORS 21 and σ 1/8 in the place of σ 1/2. Using Lemma 1.15 with Wloc u (p) in the place of and taking a sufficiently large σ 1/4 -hyperbolic time for p we deduce that there is a hyperbolic pre-ball inside Wloc u (p). This imples that its image by the hypebolic time, which is a disk of radius δ 2 around p, is contained in the local unstable manifold of p. This gives the second part of the result. Observe that as long as we take the local unstable manifold through p small enough, every point in Wloc u (p) belongs to the local stable manifold of some point in. By construction, is accumulated by the disks k = f n k (Wk ) which, by Corollary 1.22, satisfy Leb k (f n k (H)) lim = 1. (22) k Leb k ( k ) Since H is forward invariant, we have lim k Leb k (H) Leb k ( k ) = 1. Let now ϕ: Λ R be the continuous function given by ϕ(x) = log Df 1 E cu x. Since time averages of ϕ are constant for points on local stable manifolds and the local stable foliation is absolutely continuous, we deduce that Leb (H) Leb ( ) = 1. The same conclusion holds for the local unstable manifold of p in the place of by the same reason. Let us now prove the last item. Since H has full Lebesgue measure in K and K 1 K has positive Lebesgue measure, we may start our construction with the set H 1 = H K 1 in the place of H intersecting the disk in a positive Leb measure set of points. Although we have not invariance of H 1, by Corollary 1.22 we still have that the iterates of H 1 Λ 1 accumulate on the whole ; recall Remark Since the stable manifolds through points in Wloc u (p) intersect, there must be points in H 1 whose orbits accumulate on Wloc u (p). Let p 1 be a hyperbolic periodic point as in Lemma Let B 1 be the basin of W u (p 1 ), i.e. the set of points whose ω-limit is contained in W u (p 1 ). If Leb(K \ B 1 ) = 0, then we have proved the theorem. Otherwise, let K 1 = K \ B 1. Using again Lemma 1.25 we obtain a point p 2 Λ such that the basin B 2 of W u (p 2 ) attracts the orbit of some point in K 1. By definition of K 1 we must have W u (p 1 ) W u (p 2 ). We proceed inductively, thus obtaining periodic points p 1,..., p n Λ with W u (p i ) W u (p j ) for every i j. This process must stop after a finite number of steps. Indeed, if there were infinitely many points as above, by compactness, choosing p i1, p i2 sufficiently close we would get W u (p i1 ) = W u (p i2 ) by the Inclination Lemma. We have proved the first two items of Theorem 1.24.

22 22 JOSÉ F. ALVES Dimension one centre-unstable direction. Assume now that E cu has dimension one. We want to show that each W u (p i ) attracts an open set containing W u (p i ). Given 1 i k, by Lemma 1.25 we can find at least one point in each connected component of W u (p i ) \ {p i } belonging to H. Since these points have infinitely many hyperbolic times, then each connected component of W u (p i ) \ {p i } necessarily has infinite arc length; recall Lemma This implies that each point x W u (p i ) has an unstable arc γ u (x) W u (p i ) of a fixed length passing through it. Let B(x) = Wδ s (y). y γ u (x) By domination, the angles of γ u (x) and the local stable manifolds W s δ (y) with y γu (x) are uniformly bounded away from zero. Thus, B(x) must contain some ball of uniform radius (not depending on x), and so the set x W u (p i ) B(x) is a neighborhood of W u (p i ). Since, for each x W u (p i ), the points in B(x) have their ω-limit set contained in W u (p i ), we are done. 2. Hyperbolic sets with positive volume A compact invariant set Λ is called hyperbolic if there is an invariant splitting T Λ M = E s E u of the tangent bundle restricted to Λ, and a constant 0 < λ < 1 such that for some choice of a Riemannian metric on M we have Df E s x < λ and Df 1 E u x < λ, for every x Λ. Observe that in this case the splitting is obviously a dominated splitting. In the next subsections we will derive several consequences for hyperbolic sets from the results obtained before in the partially hyperbolic context Transitive sets. We are able to prove that transitive hyperbolic sets with positive volume necessarily coincide with the whole manifold, i.e. the diffeomorphism is Anosov. The main reason why we cannot generalize the next result to the context of partially hyperbolic sets is that the length of local stable/unstable manifolds may shrink to zero when iterated back/forth, respectively. Theorem 2.1. Let f : M M be a C 1+ diffeomorphism and let Λ M be a transitive hyperbolic set. 1. If Λ has positive volume, then Λ = M. 2. If Λ attracts a set with positive volume, then Λ attracts a neighborhood of itself. If Λ has positive volume, it follows from Corollary 1.5 that Λ must contain some local unstable disk and some local stable disk. The first item of Theorem 2.1 is a consequence of the following easy lemma. Lemma 2.2. If Λ is a transitive hyperbolic set containing the local unstable manifold of some point, then Λ contains the local unstable manifolds of all its points.