Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms

We consider symplectic cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the H\"older topology.


1.
Introduction. Positiveness of Lyapunov exponents was widely studied in the past years, some natural questions that are trying to be answered are: • Positive Lyapunov exponent are common?
• What is the generic behaviour, positive exponents or zero exponents?
• Are positive exponents stable? Avila [7] proved for several topologies, including C r topology for r ≥ 0, that there exists a dense, but not necessarily open or generic, set of cocycles taking values in SL(2, R) with non zero Lyapunov exponents. This was generalized to Sp(2d, R) cocycles by Xu [21].
A result stated by Mañé and proved by Bochi [11] expose that the generic behaviour in the C 0 topology is not positiveness of Lyapunov exponents unless you have a strong hyperbolicity: C 0 generically SL(2, R) cocycles are uniformly hyperbolic or have zero Lyapunov exponents.
In the case of more regular cocycles with some hyperbolic behaviour on the base map and bunching condition on the fibers, the generic behavior changes radically. Viana [16] proved that, when the base maps are non-uniformly hyperbolic and the cocycle take values in the group SL(d, R), the Lyapunov exponents are generically non-zero. This was generalized by Bessa-Bochi-Cambrainha-Matheus-Varandas-Xu [10] for any non-compact semi-simple Lie group (for example Sp(2d, R)).
These results were extended in the partially hyperbolic setting when the map is volume preserving, accessible and the cocycle takes values in SL(d, R) by Avila-Viana-Santamaria [4].
Here we deal with Sp(2d, R) cocycles over a different class of partially hyperbolic maps, not contained in the previous results: they are not volume preserving neither 1.1. Organization of the paper. In sections 2 and 3 we recall some definitions of linear cocycles, Lyapunov exponents, partial hyperbolicity and we present the two classes of partially hyperbolic that we consider.
In section 4 we give the precise statements of our results, in section 5 and 6 we introduce some technical tools, and the concept of holonomy invariant disintegrations.
We work separately with the disintegrations for partially hyperbolic maps with center leaves in section 7 and for time one maps in 8. In Section 9 we recall some technical results already known that we prove for completeness.
In section 10, section 11 and 12 we conclude the proofs of the theorems.
In the appendix A we give a proof of the closedness of s and u-states for general partially hyperbolic maps.
2. Linear cocycles. Let G be a linear subgroup of GL(d, R), for some d ∈ N, the linear cocycle defined by a measurable matrix-valued function A : M → G over an invertible measurable map f : M → M is the (invertible) map F A : M × R d → M × R d given by Its iterates are given by F n Sometimes we denote this cocycle by (f, A).
Let µ be an f -invariant probability measure on M and suppose that log A and log A −1 are integrable. By Kingman's sub-additive ergodic theorem, see [18], the limits exist for µ-almost every x ∈ M . When there is no risk of ambiguity we write just λ + (x) = λ + (A, µ, x). By Oseledets [14], at µ-almost every point x ∈ M there exist real numbers λ + (x) = λ 1 (x) > · · · > λ k (x) = λ − (x) and a decomposition R d = E 1 x ⊕ · · · ⊕ E k x into vector subspaces such that for every non-zero v ∈ E i x and 1 ≤ i ≤ k. Let Sp(2d, R) be the symplectic subgroup of GL(2d, R), this means that there exists some non-degenerated skew-symmetric bilinear form ω : R 2d × R 2d → R preserved by the group action, i.e: for every v, w ∈ R 2d , ω(Av, Aw) = ω(v, w), in particular for dimension 2, SL(2, R) is a symplectic group for the area form dx ∧ dy. Observe that any GL(2, R) cocycle defines a SL(2, R) cocycle by takinĝ A = (1/ det A)A, so positive exponent forÂ means λ + (A) > λ − (A).
Define L(A, µ) = λ + dµ, we say that A has positive exponent if L(A, µ) > 0. When µ is ergodic, as we are going to assume later, we have L(A, µ) = λ + (x) for µ-almost every x ∈ M .
3. Partial hyperbolicity. A diffeomorphism f : M → M of a compact C k , k > 1, manifold M is said to be partially hyperbolic if there exists a non-trivial splitting of the tangent bundle invariant under the derivative Df , a Riemannian metric · on M , and positive continuous functions ν,ν, γ,γ with ν,ν < 1 and ν < γ <γ −1 <ν −1 such that, for any unit vector v ∈ T p M , All three sub-bundles E s , E c , E u are assumed to have positive dimension. From now on, we take M to be endowed with the distance dist : M × M → R associated to such a Riemannian structure. Suppose that f : M → M is a partially hyperbolic diffeomorphism. The stable and unstable bundles E s and E u are uniquely integrable and their integral manifolds form two transverse continuous foliations W s and W u , whose leaves are immersed sub-manifolds of the same class of differentiability as f . These foliations are referred to as the strong-stable and strong-unstable foliations. They are invariant under f , in the sense that where W s (x) and W s (x) denote the leaves of W s and W u , respectively, passing through any x ∈ M .
A partially hyperbolic diffeomorphism f : M → M is called dynamically coherent if there exist invariant foliations W cs and W cu with smooth leaves tangent to E c ⊕E s and E c ⊕ E u , respectively. Intersecting the leaves of W cs and W cu one obtains a center foliation W c whose leaves are tangent to the center sub-bundle E c at every point.
3.1. Class A: Compact center leaves. The first class to be consider is when the center leaves are compact manifolds. LetM = M/W c be the quotient of M by the center foliation and π : M →M be the quotient map. We say that the center leaves form a fiber bundle if for any smooth along the verticals {d} × W c (x) and mapping each vertical onto the corresponding center leaf d.
Remark 1. If f is a volume preserving, partially hyperbolic, dynamically coherent diffeomorphism in dimension 3 whose center foliation is absolutely continuous and whose generic center leaves are circles, then, according to Avila, Viana and Wilkinson [6], all center leaves are circles and they form a fiber bundle up to a finite cover.
We define the center Lyapunov exponents of f : M → M as and Again by the Oseledets theorem, this limits exist for µ-almost every x ∈ M . We say that f : M → M has zero center Lyapunov exponent if λ c (f ) Remark 2. When all the center Lyapunov exponents of f are non-zero, the problem falls in the hypothesis of Viana [16]. In this work we deal with the case of zero center Lyapunov exponents, where the previous techniques fails.
Let µ be an f -invariant ergodic measure, defineμ = π * µ, we say that µ has projective product structure if there exist measures µ s on W s (x) and µ u on W u (x) such that locallyμ ∼ µ s × µ u (see definition 6.1), we also assume supp(μ) =M .
From now on we refer to these partially hyperbolic maps as class A.

3.2.
Class B: Time one map of Anosov flows. The second class of partially hyperbolic maps that we consider are the time one maps of Anosov flows. We say that a flow φ t : M → M is an Anosov flow if there exists a splitting where E s is contracted and E u is expanded by the derivative of φ t .
Let f : M → M be the time one map of a C 2 Anosov flow (i.e: f = φ 1 ). This map is partially hyperbolic with center bundle E c = X.
In this particular case the center bundle E c = X is integrable and C 1 (observe that W c (p) = φ R (p)). The bundles E c + E u and E s + E c are integrable and absolutely continuous (See [1]). We call them center stable W cs and center unstable W cu foliations.
We say that a measure µ is an SRB measure for the flow if the disintegration of the measure along the center unstable leaves is in the Lebesgue class along these sub-manifolds (for more details see [1]). Observe that we are asking for the measure to be invariant for the flow not just for the time one map.
Recall that by the spectral decomposition theorem the non-wandering set of the flow can be decomposed into basic sets Ω(φ t ) = ∆ 1 ∪ · · · ∪ ∆ n , where the ∆ i are invariant and the restriction φ t is transitive in these sets, in particular the support of every ergodic measure is supported in one of these sets.
Also these basic sets coincide with the closure of the homoclinic class of its periodic points, as the support of an SRB measures is saturated by the center unstable foliation this implies that supp(µ) = ∆ i for some i ∈ {1, . . . , n}, moreover this set is an attractor. Conversely, every non trivial attractor of an hyperbolic flow admits some SRB measure.
From now on we refer to these partially hyperbolic maps as class B.

4.
Statements. Let us denote by H α (M ) the space of α-Hölder continuous maps A : M → Sp(2d, R) endowed with the α-Hölder topology which is generated by norm A cocycle generated by an α-Hölder function A : M → Sp(2d, R) is fiber-bunched if there exists C 3 > 0 and θ < 1 such that for every x ∈ M and n ≥ 0. Our first result, for partially hyperbolic maps of class A is In the theorem above, for the restriction of the cocycle to the periodic center leaf we take the natural invariant measure: • given by the disintegration in center leaves for class A, • the invariant measure equivalent to Lebesgue in the orbits of the flow for class B, this will be explained in more detail in section 6. We also study sub-sets of continuity of the Lyapunov exponents for G = SL(2, R).
Definition 4.1. Given an invertible measurable map g : N → N an invariant measure η (not necessarily ergodic) and an integrable cocycle A : N → SL(2, R) we say that A is a continuity point for the Lyapunov exponents if for every {A k } k converging to A we have that λ + A k : N → R converges in measure to λ + A : N → R. Observe that as A k → A this implies that sup k A k is bounded and consequently λ + A k is also bounded. Thus, since we are dealing with probability measures, convergence in measure is equivalent to convergence in L 1 η . When the measure is ergodic this definition coincide with the classical definition of continuity, namely L(A k , µ) → L(A, µ) whenever A k → A. An interesting question is if in this context the Lyapunov exponents are continuous.
In general this is not true. For example, take a volume preserving Anosov diffeomorphism g : T 2 → T 2 , θ ∈ R irrational and define f : this is a volume preserving partially hyperbolic diffeomorphism. By Wang-You [20] there exist discontinuity points for the Lyapunov exponents for monotonic cocycles in the smooth topology, let A : T 1 → T 1 be one of this with t → t + θ. Then it is easy to see that the cocycleÂ : With an additional condition in a center leave we can find open sets of continuity.
Theorem D. Let f : M → M be a partially hyperbolic of class A or B. Let A : M → SL(2, R) be a fiber bunched cocycle whose restriction to some compact periodic center leaf, intersecting the support of µ, has positive Lyapunov exponent and is a continuity point for the Lyapunov exponents. Then A is accumulated by open sets restricted to which the Lyapunov exponents vary continuously.
An interesting question is whether these hypotheses are open or generic in some topology.
Following Avila-Krikorian [2], given > 0, we say that a function f : This definition extends to functions defined on S 1 = R/Z and taking values on S 1 by considering the standard lift. We say that A : [2] that the Lyapunov exponents are continuous functions of -monotonic cocycles. Moreover, the set of -monotonic cocycles is an open subset of H 1 (S 1 ).
Thus as corollary we have: is -monotonic. Then the Lyapunov exponents vary continuously in an open and dense subset of H p .
We also have an equivalent result for time one maps: For volume preserving and accessible partially hyperbolic maps Liang, Marin and Yang [13], proved that there exists an open and dense set of points of continuity for the Lyapunov exponents.
5. Invariant holonomies. At this section we introduce the key notion that we are going to use in the proof of Theorem C, namely invariant holonomies.
Suppose that f : M → M is partially hyperbolic. The stable and unstable foliations are, usually, not transversely smooth: the holonomy maps between any pair of cross-sections are not even Lipschitz continuous, in general, although they are always γ-Hölder continuous for some γ > 0. Moreover, if f is C 2 then these foliations are absolutely continuous, in the following sense.
Let d = dim M and F be a continuous foliation of M with k-dimensional smooth leaves, 0 < k < d. Let F(p) be the leaf through a point p ∈ M and F(p, R) ⊂ F(p) be the neighbourhood of radius R > 0 around p, relative to the distance defined by the Riemannian metric restricted to F(p). A foliation box for F at p is the image of an embedding Φ : such that Φ(·, 0) = id, every Φ(·, y) is a diffeomorphism from F(p, R) to some subset of a leaf of F (we call the image a horizontal slice), and these diffeomorphisms vary continuously with y ∈ R d−k . Foliation boxes exist at every p ∈ M , by definition of continuous foliation with smooth leaves. A cross-section to F is a smooth codimension-k disk inside a foliation box that intersects each horizontal slice exactly once, transversely and with angle uniformly bounded from zero. Then, for any pair of cross-sections Σ and Σ , there is a well defined holonomy map Σ → Σ , assigning to each x ∈ Σ the unique point of intersection of Σ with the horizontal slice through x. The foliation is absolutely continuous if all these homeomorphisms map zero Lebesgue measure sets to zero Lebesgue measure sets. That holds, in particular, for the strong-stable and strong-unstable foliations of partially hyperbolic C 2 diffeomorphisms and, in fact, the Jacobians of all holonomy maps are bounded by a uniform constant.
If f : M → M is of class B, his center stable and center unstable foliations are also absolutely continuous. If we take two center manifolds W c (p) and W c (z) in the same strong stable manifold, from every point t ∈ W c (p) we can define the stable holonomy locally h s p,z : I ⊂ W c (p) → W c (z) and the Jacobian of h s p,z vary continuously with the points p and z. Actually the Jacobian is given by where Jf n c (t) = det Df n | E c t (t). Analogously for the unstable holonomies. When f is of class A, we can extend the stable and unstable holonomies to be defined as a map from a entire center manifold to another: x,ỹ (t) the first intersection between W s (t) and W c (ỹ). Analogously, forz ∈ W u (x) we define the unstable holonomy h ũ x,z : W c (x) → W c (z) changing stable by unstable manifolds. 5.1. linear holonomies. We say that A admits strong stable holonomies if there exist, for every p and q ∈ M with p ∈ W s loc (q), linear transformations H s,A p,q : R 2d → R 2d with the following properties: These linear transformations are called strong stable holonomies.
Analogously if p ∈ W u loc (q) we have the strong unstable holonomies. When there is no ambiguity we write H s p,q instead of H s,A p,q . Remark 3. The fiber bunched condition (1) implies the existence of the strong stable and strong unstable holonomies (see [3]).
Let RP 2d−1 be the real projective space, i.e: the quotient space of R 2d \ {0} by the equivalence relation v ∼ u if there exist c ∈ R \ {0}, such that, v = cu. Every invertible linear transformation B induce a projective transformation, that we also denote by B, We also denote by F : M × RP 2d−1 → M × RP 2d−1 the induced projective cocycle.
6. Measure disintegration. Given a measurable partition P of M , by Rokhlin disintegration theorem (see [18]) there exists a family of measures {µ P } P ∈P such that for every measurable set X ∈ M Moreover, such disintegrantion is essentially unique [15]. In general the partition by the invariant foliations is not measurable, to overcome this problem we disintegrate our measure locally as we explain now.
Denote by s, u and c the dimensions of E s , E u and E c respectively. We call and π : C x → Σ x the natural projection given by π(Φ(x s , y u , z c )) = Φ(x s , y u , 0 c ), observe that Σ x has a product structure Σ s × Σ u given by Φ. Also, in this case the partition by center leaves is a measurable partition, so using Rokhlin disintegration we can define a family of measuresx → µ c x , such that µ c x is concentrated in W c (x) and Moreover, we are going to prove (see section 7) thatx → µ c x is continuous and defined everywhere. Definition 6.1. We say that µ has projective product structure if π * µ | C x is absolutely continuous with a product measure µ s × µ u , where µ * is a measure on Σ * for * = s, u.
Remark 5. When f is the time one map of an Anosov flow, µ is an SRB measure for the flow if and only if for every x the disintegration of µ | C x given by the partition {W cu Cx (y)} y∈Cx gives measures absolutely continuous with respect to the Lebesgue measure on W cu (y).
Observe that in this case the partition in center manifolds (orbits of the flow) is not measurable, but because the measure is invariant by the flow, the local disintegration in the center leaves in a foliated box is actually absolutely continuous with respect to Lebesgue.
If we have some periodic compact center leaf W c (p) of period k, we have a natural f k -invariant measure µ c p supported in this leaf, for the class A is just µ c p and for the class B is the absolutely continuous measure induced by the flow.
6.1. Invariance principle. Let P : M × RP 2d−1 → M be the projection to the first coordinate, P (x, v) = x, and let m be an F -invariant measure on M × RP 2d−1 such that P * m = µ. By Rokhlin [15] we can disintegrate the measure m into {m x , x ∈ M } with respect to the partition We recall the following result whose proof can be found in [4]. We say that a disintegration of m is u/c-invariant if, for every x and y in the support of µ and in the same unstable manifold, µ c x -almost every t ∈ W c (x) (Lebesgue almost every for class B), . Analogously, we call a disintegration s/c-invariant changing unstable by stable. We say that the disintegration is su/c-invariant if the disintegration is both u/c and s/c-invariant. This property is going to play a major role in our argument so we now prove that if m is su-invariant, then m admits a disintegration su/c-invariant.
We separate the argument for each class of partially hyperbolic map that we consider. 7. Disintegration for class A. In this section we deal with the disintegration of m when f is of class A, as described in section 3.1.
We have two disintegrations, one of m with respect to P, as defined in the previous section, and one of µ with respect toP = {W c (p),p ∈M }, as explained in remark 4.
We introduce a third disintegration {mx,x ∈M } with respect to the partition It is easy to see thatmx Because π * µ has local product structure and f has zero center Lyapunov exponent, by the Invariance Principle for smooth cocycles([5, Proposition 4.8]), there exists a continuous disintegration {µ c x } which is su-invariant everywhere in the support ofμ. This means that: (a') µ c z = (h s y,z ) * µ c y for everyỹ,z ∈M in the same stable set, the restriction of f to W c (x). By the f -invariance of µ we have that forμ-almost everyx ∈M . From the continuity of the disintegration it follows that, this relation is true for every x ∈ suppμ. So, for everyf k -fixed pointp ∈M Proposition 7.1. If m is su-invariant then m admits a continuous disintegration {mx,x ∈M } in supp(μ) with respect toP that is su/c-invariant.
Proof. We need to prove that for everyx ∈ supp(μ) andỹ ∈ supp(μ) in the same * -leaf, * ∈ {s, u}, Let us prove for * = u, the case * = s is analogous. Takex ∈M andỹ ∈M in the same unstable leaf and such that µ c x -almost every x ∈ W c (x) belongs to M u and µ c y -almost every y ∈ W c (ỹ) belongs to M u . Then proving that the disintegration {mx} is u-invariant. Analogously we can find a total measure set such thatmx is s-invariant. Using [5, Proposition 4.8] we conclude that m admits a disintegration continuous in supp(μ) with respect to the partitionP, which is s and u invariant. Now the continuity implies that (4) is true for everỹ x ∈ supp(μ). Thus for everyx ∈ supp(μ) andỹ ∈ supp(μ) in the same unstable leaf (analogously for the stable holonomies) as claimed.

Disintegration for class B.
This section is devoted to prove the existence of an su/c-invariant disintegration of m for the time one maps of Anosov flows, we prove a more general version of this result. We say that a measure µ has a good product structure if it has projective product structure and also for every foliated box the disintegration in center manifolds {W c Cx (y)} y∈Cx is absolutely continuous with respect to the Lebesgue measure. Every SRB measure for the flow has a good projective product structure, the absolute continuity in the center manifold is just because the measure is invariant by the flow and the projective product structure is because of the absolute continuity of the center stable foliation (see [19,Proposition 3.4] for a proof of this fact). There exist M s and M u of total measure with s-invariance and u-invariance respectively. Take M = M s ∩ M u , x ∈ supp(µ) and a foliated box C x ⊂ M ; via a local chart we can write C = Σ × D c R where Σ is a transversal section to the center foliation and D c R is a disc of radius R in a center manifold, let π : C x → Σ be the natural projection given by the center discs, observe that the center stable and center unstable manifolds gives a product structure of Σ = Σ s × Σ u and by hypothesis µ Σ ∼ µ s × µ u .
By the absolute continuity of the center foliation and the continuity of the Jacobians of the stable and unstable holonomies of f we have that the disintegration Σ → M(D c R ), x → µ c x is of the form µ c x = ρµ c where µ c is the Lebesgue measure in D c R and ρ is continuous. We write x ∼ s y for x, y ∈ Σ in the same center stable manifold, and x ∼ u y if they are in the same center unstable manifold. Take r > 0 smaller than R such that for every x ∼ s y, h s x,y : D c r → D c R is well defined. Fix some x s ∈ Σ s such that µ u ({x s } × Σ u ∩ π(M )) = 1 and fix some x u ⊂ Σ u such that µ c (x s ,x u ) ((x s , x u ) × D c R ∩ M ) = 1, by the absolute continuity in the center direction this implies that Lebesgue almost every t ∈ (x s , x u ) × D c R is in M ; for simplicity denote x 0 = (x s , x u ).
Take > 0 such that If m t is well defined for almost every t ∈ D c r then by the absolute continuity of the holonomies (H s x,y m c x ) t is defined for almost every t ∈ h s x,y (D c r ). Analogously we can define (H u x,y m c x ) t Now define a new disintegration of m in the box Σ × D c in the following way: Fix the restriction of the original disintegration defined for almost every (x 0 , t) ∈ {x 0 }× D c r , (in particular m c x0 is well defined) extend the disintegration to every y ∼ u x 0 and t ∈ h u x0,y (D c r ) by (H u x0,y m c x0 ) t . Lets call this disintegration m c y , and then extend it for every z ∼ s y ∼ u x 0 by (H s y,z m c y ) t for almost every t ∈ h s y,z h u x0,y (D c r ). Now by definition of this disintegration is well defined in Σ × D u , and it coincides with the original disintegration almost everywhere, lets denote this new disintegration by t → m s t . By construction this disintegration is s/c-invariant. We claim that this new disintegration, with respect to the partition {x} × D c × RP 2d−1 , defined by is continuous. Assuming this claim, we can define analogously, reducing if necessary, a disintegration x → m u x that is u/c-invariant on Σ × RP 2d−1 . We have that m cs x = m cu x for Lebesgue almost every x ∈ Σ, then as both are continuous we have the equality for every x ∈ Σ. Then by the uniqueness of the Rokhlin disintegration, for every x ∈ Σ, m s t = m u t for µ c almost every t ∈ D c . As this boxes cover supp(µ) we conclude the theorem.
We are left to prove the claim. We will prove that x → m cs x is uniformly continuous varying y ∼ s x and z ∼ u y. Fix some continuous ϕ : D c × RP 2d−1 → R and let y n → x, y n ∼ s x.
where h n (t) = h s yn,x (t), and H n,t = H s (x,hn(t)),(yn,t) so we can write the integral as changing variables we have: where J µ c h −1 n is the Jacobian of h −1 n with respect to the Lebesgue measure µ c . Hence using that h −1 n → id W c (x) , J µ c h −1 n → 1, H n,t → id RP 2d−1 uniformly and the continuity of ρ and ϕ, we conclude that ϕ(t, v)dm cs yn → ϕ(t, v)dm cs y . For z n ∼ u x, such that z n → x, we take z n ∼ s y n ∼ u x 0 and x ∼ s x ∼ u x 0 (see figure 1), by construction y n → x , y n ∼ u x ∼ u x 0 and we have u-invariance on W u (x 0 ) so using the same arguments as before, changing stable by unstable ϕ(t, v)dm cs yn → ϕ(t, v)dm cs x .

MAURICIO POLETTI
Now observe that where h s n = h s yn,zn and H s n,t = H s (yn,h s n (t)),(zn,t) are such that h s n → h s x ,x and H s n,t → H s (x ,h s (t)),(x,t) . Therefore, using the same calculations as before, it follows that ϕ(t, v)dm cs zn → ϕ(t, v)dm cs x . This proves the claim and concludes the proof. 9. Some technical results. We can characterize the cocycles accumulated by cocycles with zero exponents, and also, in the SL(2, R) case, the discontinuity points for the Lyapunov exponents using the next proposition: is accumulated by cocycles with zero Lyapunov exponents, then there exists some F A -invariant measure m, su-invariant, that projects to µ. Also, if the cocycles takes values in SL(2, R) and A is a discontinuity point for the Lyapunov exponents, then every F A -invariant measure m that projects to µ is also su-invariant Proof. Take (A k ) k∈N converging to A such that L(A k , µ) = 0. Take m k to be F A k -invariant measures projecting to µ, then by the invariance principle [5], this measures are su-invariant. Take a subsequence such that m k converges to some m, by theorem A.1 this measure is su-invariant.
For the second part, lets suppose that an SL(2, R) cocycle A is a discontinuity point for the Lyapunov exponents. We have that L(A, µ) > 0, otherwise A is a continuity point, then by [9, Proposition 3.1] the conditional measures of every F -invariant measure m are of the form m x = aδ E u x + bδ E s x , with a + b = 1, so we can write m = am u + bm s where m u and m s are measures projecting in µ with disintegration m u x = δ E u x and m s x = δ E s x . The invariance of E u x by unstable holonomies gives that m u is u-invariant, analogously m s is s-invariant.
If A is a discontinuity point then there exist (A k ) k∈N converging to A such that L(A k , µ) does not converge to L(A, µ). Taking and m u k the u-invariant measure projecting in µ as before (but for A k ) if L(A k , µ) > 0 or any u-invariant measure if L(A k , µ) = 0, we have that does not converge to taking a subsequence we can suppose that m u k converges to some m = am u + bm s with b = 0, then we have that m is u-invariant, so m s = b −1 (m − am u ) is also u-invariant, hence every a m u + b m s is u-invariant.
As λ + (A) = −λ − (A) the proof of s-invariance follows analogously. 9.1. Symplectic transvections. The results of this section can be found in the thesis of Cambrainha [12], for completeness we rewrite the statements and the proofs. A linear map τ : R 2d → R 2d is called a transvection if there are a hyperplane H ⊂ R 2d and a vector v ∈ H such that the restriction τ | H is the identity on H and for any vector u ∈ R 2d , τ (u) − u is a multiple of v, say τ (u) − u = λ(u)v where λ is a linear functional of R m such that H ⊂ ker λ.
Lemma 9.2. Let (E, ω) be a 2d-dimensional symplectic vector space and let V and W be subspaces of E with complementary dimensions (i.e., dim(V ) + dim(W ) = 2d). Suppose that V ∩ W has dimension k > 0. Then, there exist k symplectic transvections σ 1 , . . . , σ k arbitrarily close to the identity such that Proof. We proceed by induction on k = dim(V ∩ W ). Let m = dim(V ), so that k ≤ min{m, 2d − m}.
Let us now perform the general step of the induction. Suppose now that the lemma is true for k − 1, and let V ∩ W = span{v 1 , . . . , v k }. In this case, we can find basis for V and W such that Observe that V 0 = V +W and V 1 = (span{v 1 , ..., v k }) ⊥ are co-dimension k subspaces of E. Fix u 0 / ∈ V 0 ∪ V 1 , ε > 0 and consider the symplectic transvection Any element v ∈ σ k (V ) ∩ W can be written in a similar way to (5). From this, we have that: Since our choice u 0 / ∈ V 0 implies that the vectors {v 1 , ..., v m , w k+1 , ..., w 2d−m , u 0 } are linearly independent, one has: The second and fourth equations imply that Therefore: By induction hypothesis, there are σ 1 , . . . , σ k−1 symplectic transvections arbitrarily close to the identity such that This completes the proof of the lemma.
The conclusion of the previous lemma is an open condition: B(V ) ∩ W = {0} for every B sufficiently close to the symplectic automorphism σ provided by this lemma. By recursively using this fact and the previous lemma, we deduce that: Corollary 9.3. Let (E, ω) be a 2d-dimensional symplectic vector space and let {(V j , W j ) : 1 ≤ j ≤ m} be a finite collection of pairs subspaces of E with complementary dimensions (i.e., dim(V j ) + dim(W j ) = 2d for all 1 ≤ j ≤ m). Then, there exists a symplectic automorphism σ arbitrarily close to the identity such that Proof of Theorem C. Observe that for class A, π −1 of thef -periodic points are f -periodic center leaves, so the hyperbolicity off implies the existence of periodic compact center leaves. For class B it is well know that every non-trivial basic set has a dense set of periodic orbits for the flow, then this are fixed compact center leaves (actually circles) for the time one map.
To simplify notation let us assume that there exists af -fixed pointp, i.e:f (p) =p (since all the arguments and results are not affected by taking an iterate).
Fix this compact center leaf K = W c (p), so f : K → K has an invariant measure µ c K . Now we are going to define h : K → K that is a composition of stable and unstable holonomies.  Fix the periodic pointp ∈M andz ∈M a homoclinic point forp, i.e:z ∈ W s (p) ∩ W u (p), let us call K = W c (p). Define h : K → K (see figure 2)  Observe that the restriction of f to one of this compact leaves, W c (p), is a rotation in R/T Z where T is the period of p by the flow. So after a C 1 change of coordinates we can suppose that f p = f | W c (p) = r 1 T where The restriction of the cocycle to W c (p) is a cocycle over a rotation with Lebesgue µ c p as invariant measure. Fix a flow periodic point p ∈ supp(µ), lets take z ∈ W cs (p) ∩ W cu (p), by invariance of the center stable and center unstable manifolds φ t (z) ∈ W cs (p) ∩ W cu (p) for every t ∈ R. Observe also that the orbit of z is non recurrent. Suppose that actually z ∈ W u (p) and define h(p) = W c (p) ∩ W s (z), now for t ∈ [1, T ] define h(φ t (p)) := φ t (h(p)) and observe that by the invariance of the stable and unstable manifolds φ t (h(p)) ∈ W c (φ t (p)) ∩ W s (φ t (z)) and φ t (z) ∈ W u (φ t (p)).
So h is a composition of stable and unstable holonomies. Also in the circle coordinates, identifying p with 0, where ω is such that φ ω (p) = h(p). In particular h also preserves the Lebesgue measure µ c p in the circle coordinates. Call h u p,z : W c (p) − {p} → W c (z) the map given by t → φ t (z), by the same reasoning as before this map is given by an unstable holonomy, if there is no risk of ambiguity we just write h u = h u p,z . This map is not well defined in 0 because φ T (z) = z.
Observe that as the center stable and center unstable are dense in supp(µ) we can find points z 1 , . . . , z d as before such that any pair z i , z j , with i = j, are in different orbits of the flow. Hence, we can define maps h 1 , . . . , h d with the properties above.
10.3. Weakly pinching. Now we return to the proof of Theorem C. Fix once for all some periodic compact center leaf K = W c (p), take A | K : K → Sp(2d, R), this defines a linear cocycle over fp with invariant measure µ K , observe that µ K is also h-invariant. Let λ 1 µ K (t) ≥ · · · ≥ λ k µ K (t) be the Lyapunov exponents of this cocycle. We say that A ∈ H α (M ) is Weakly pinching if L(A | K , µ K ) > 0.  Figure 4. perturbation of H some su-invariant measure, we can do this with less than d-point z i to get that supp(m t ) = ∅, for t in a positive measure set, a contradiction. Then we can find A arbitrarily close to A that does not admit any su-invariant measure. Now we can prove Theorem C.
Proof of Theorem C. By Lemma 10.1 there exists A , arbitrary close to A, weakly pinching that does not admit any su-invariant measure, by the invariance principle (theorem 6.2) A has positive exponent and also by Proposition 9.1 A is not accumulated by cocycles with zero exponents, then it is stably non-uniformly hyperbolic. 11. Proof of Theorem A and B. We need the following Proposition: Proof. To prove Proposition 11.1 let us recall some results.
We say that an f -invariant measure µ is non-periodic if for every k ∈ Z f k | supp µ is not the identity map. By Xu [21] in a very general topology (including the H α (M ) topology) there exists a dense set of cocycles with L(A, µ) > 0. So if there exist some compact periodic center leaf W c (p) such that such that µ c p is non-periodic we are done.
If we are not in the previous case we can do the following: first we need periodic points of arbitrarily large period, • for class A: We are assuming that for everyf -periodic pointp ∈M , there exist k(p) such that f k(p) p | supp µ c p = id. Asf is an hyperbolic homeomorphism there exist periodic points of arbitrary large period, then as k(p) is at least the period ofp we have arbitrary large k(p), • for class B: If for every periodic point p ∈ supp(µ), the invariant measure is periodic this means that 1 Tp is rational. T p can be taken arbitrarily large, so we have rational rotations with arbitrarily large period. Appendix A. Closedness of s and u-states. In the study of Lyapunov exponents of linear cocycles over hyperbolic or partially hyperbolic maps one of the principal tools to prove positivity, simplicity or continuity is to analyze the invariant measures of the cocycle that projects to some fixed invariant measure in the base.
With some conditions that allow the existence of linear stable and unstable holonomies, having zero exponents (in some cases also discontinuity) can be caracterized by some rigidity condition in the invariant measures of the cocycles, this is known as the Invariance Principle (see [5]). This rigidity condition says that the measures must be s and u-states, this means that the disintegration is invariant by the holonomies (see section A.1 for the precise definition).
In many works closedness of s or u-states has been proved and used for specific cases ( [8], [5], [4]). The purpose of this appendix is to give a more general proof of this fact for general partially hyperbolic maps without any extra conditions on invariant measure of the base map.
The precise statement of the main result is given in theorem A.
x,y (ξ) is continuous where (x, y) varies in the set of points x ∼ s y, • there exist C > 0 and γ > 0 such that H s x,y is (C, γ) Hölder for every x ∼ s y. Analogously we say that F admits unstable holonomies if for every x ∼ u y there exist H u x,y with the same properties changing stable by unstable. From now on fix f and vary the cocycles F projecting to f in a topology such that x, y, F → H s,F x,y varies continuously. Fix some f -invariant probability measure µ, as E is compact there always exists some F -invariant probability measure m that projects to µ. By Rokhlin disintegration theorem, we can disintegrate m with respect to the partition given by the fibers {x} × E, so we have x → m x defined almost everywhere.
We say that an F -invariant measure that projects to µ is an s-state if there exists a total measure subset M ⊂ M such that for every x, y ∈ M , x ∼ s y, H s x,y * m x = m y . Analogously, we say that a measure is an u-state is the same is true changing stable by unstable manifolds. We call m an su-state if it is booth s and u-state.
We want to prove that Theorem A.1. If m k are s-states for F k , that projects to µ such that F k → F and m k → m in the weak * topology then m is an s-state.
By [4, theorem 4.1] if a cocycle F has all his Lyapunov exponents equal to zero, then the F -invariant measure m is an su-state. As a corollary we have Corollary A.2. If F does not admit any su-state, then there exists a neighborhood of F with non-zero exponents.
For each z ∈ Σ, let r(z) be the largest integer such that f j (S(z)) does not intersect any S(w) for all w ∈ Σ, 0 < j ≤ r(z). Now let B 0 be the σ-algebra of sets E ⊂ M such that for every z and j as before, either E contains f j (S(z)) or is disjoint from it. A B 0 -mensurable function, is a function that is constant on the sets f j (S(z)), 0 ≤ j ≤ r(z).
For every k ∈ N, let H k : M × E → M × E be defined by (x, ξ) → (x, H kx (ξ)) where if x ∈ f j (S(z)) for some z ∈ Σ id otherwise (8) where H s,k x,z is the stable holonomy of F k . Now as in [4] we can change our cocycle byF k = H k F k (H k ) −1 , this is called a deformation cocycle of F k , such that x →F kx is B 0 measurable.
Let m k be an F k -invariant measure, definem k = H k * m k , this measure isF kinvariant. Observe that m k being an s-state implies that x → m k x is B 0 measurable. Moreover, m k is an s-state if and only if this is true for every z ∈ M and Σ z transversal to the stable foliation (this is explained in more detail in [4,Section 4.4]).
Proof. Fix ε > 0 and take a compact set K ⊂ M such that µ(K) > 1 − ε φ and φ is continuous in K × E, take φ : M × E → R be a continuous function such that φ(x, v) = φ (x, v) for every x ∈ K, v ∈ E and φ ≤ φ . Now take k sufficiently large such that φ dm k − φ dm < , then So for k sufficiently large this is less than 3ε, concluding the proof.
Observe that then by (9) and (10) we have that for k sufficiently large ϕ • H k dm k − ϕdm < 2ε.
So we are left to prove thatm k →m implies that x →m x is also B 0 measurable. Let B 0 be a σ-algebra, let µ be a measure in M . Assume that we have some measuresm k in M ×E converging in the weak * topology tom and let P : M ×E → M be the natural projection, also assume that P * m k = µ. The next lemma is a corollary of lemma A.3.
Suppose that x →m k x is B 0 measurable, this is true if and only if for every continuous function ϕ : E → R, x → ϕdm k x is B 0 measurable. First we need the next lemma Lemma A.6. Let φ k be a sequence of function in L 2 (µ) that is B 0 measurable such that φ k converges weakly to φ, then φ is B 0 measurable.
Proof. First observe that the space of B 0 measurable functions is closed and convex in L 2 (µ), lets call this space by H ⊂ L 2 (µ). Suppose that φ / ∈ H then by Hahn-Banach there exist some ρ ∈ L 2 (µ) such that ρξdµ = 0 for every ξ ∈ H and ρφdµ > 0. A contradiction because ρφ k dµ → ρφdµ Now to conclude the proof of theorem A.1 we prove: Proof. Take φ k (x) = ϕdm k x , this function bounded then it is in L 2 (µ). For any function ρ ∈ L 2 (µ) we have that So by lemma A.5 we have that where φ(x) = ϕdm x . By hypothesis φ k is B 0 measurable, then by lemma A.6 φ is B 0 measurable.