Linear cocycles over hyperbolic systems and criteria of conformality

In this paper we study Holder continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant sub-bundles and conformal structures. We use these results to obtain criteria for cocycles to be isometric or conformal in terms of their periodic data. We show that if the return maps at the periodic points are, in a sense, conformal or isometric then so is the cocycle itself with respect to a Holder continuous Riemannian metric.


Introduction
Linear cocycles over a dynamical system f : M → M appear naturally in various areas of dynamics and applications. Examples include derivative cocycles as well as stochastic processes and random matrices. A linear cocycle over f is an automorphism F of a vector bundle E over M that projects to f . In the case of a trivial vector bundle M × R d , any linear cocycle can be identified with a matrix-valued function A : M → GL(d, R) via F (x, v) = (f (x), A(x)v).
In this paper we take f to be a transitive Anosov diffeomorphism of a compact manifold M. However, our techniques can be applied to hyperbolic sets and some symbolic dynamical systems. We consider a finite dimensional vector bundle P : E → M and a Hölder continuous linear cocycle F : E → E over f (see Section 3 for definitions). One of the primary examples of such cocycles comes from the differential Df or its restriction to a Hölder continuous invariant sub-bundle of T M. Such cocycles play a crucial role in smooth dynamics of hyperbolic systems.
We establish several properties of Hölder continuous linear cocycles under various conditions of relative pinching. These properties, which include existence and continuity of measurable invariant sub-bundles and conformal structures, are of independent interest and we formulate them in the next section. As the main applications we obtain conditions on F at the periodic points of f which guarantee that the cocycle is conformal or isometric. Our first theorem establishes a general criterium, and Theorem 1.3 below gives a stronger result specific to bundles with 2-dimensional fibers. We note that the assumptions of the theorems are independent of the choice of a continuous Riemannian metric on E. Theorem 1.1. Let F : E → E be a Hölder continuous linear cocycle over a transitive C 2 Anosov diffeomorphism f . Suppose that there exists a constant C per such that for each periodic point p, the quasiconformal distortion satisfies K F (p, n) def = F n p · (F n p ) −1 ≤ C per whenever f n p = p. Then F is conformal with respect to a Hölder continuous Riemannian metric on E.
Also, if there exists a constant C ′ per such that for each periodic point p, max{ F n p , (F n p ) −1 } ≤ C ′ per whenever f n p = p, then F is an isometry with respect to a Hölder continuous Riemannian metric on E.
For a cocycle on a trivial bundle M × R d given by A : M → GL(d, R) the theorem implies cohomology to a cocycle with values in the conformal or orthogonal subgroup. This means that there exists a Hölder continuous function C : M → GL(d, R) such that B(x) = C −1 (f x)A(x)C(x) is in the corresponding subgroup for all x ∈ M. The matrix C(x) can be obtained as the unique positive square root of the symmetric positive definite matrix that defines the Riemannian metric at x.
Continuous reduction to orthogonal or conformal cocycles is very useful, in particular, since cocycles with values in compact groups are relatively well understood. Some definitive results on cohomology of such cocycles were obtained in [15,17,18,21]. These results can be easily extended to cocycles with values in the conformal group. However, the question of existence of such a reduction is highly nontrivial. Even under much stronger assumption that F n x are uniformly bounded for all x ∈ M and n ∈ Z, the question remained open since it was formulated in [21]. Under assumptions on periodic data only, no reduction was known until recent progress in [6] even for the simplest case when F n p = Id for all periodic points. Theorem 1.1 can be compared to recent results by R. de la Llave and A. Windsor [14,Theorems 6.3,6.8] who obtained similar conclusions for the cocycle given by the restriction of the derivative of an Anosov map to a Hölder continuous invariant sub-bundle of T M. The main difference is that our theorem does not have any extra assumptions on growth or pinching of the cocycle which are present in [14] and in most other results in the theory of non-commutative cocycles.
Conformality arises naturally in connection with smooth rigidity for Anosov systems [22,9,24,11,20,7], in particular, some of our results are motivated by the study of derivative cocycles in [8]. It is well known that a C 1 small perturbation g of an Anosov diffeomorphism f is conjugate to f by a Hölder homeomorphism h. If h is C 1 then Df n p and Dg n hp are conjugate by Dh p for any periodic point p. Therefore the conjugacy of Df n p and Dg n hp gives a necessary condition for h to be C 1 . This condition is sufficient for systems with one-dimensional stable and unstable distributions, but not in higher dimensions [11]. The question of sufficiency of this condition is often referred to as local rigidity. Knowing that Df and Dg are conformal on the stable/unstable distribution, or on a smaller invariant distribution, helps bootstrap regularity of h along the corresponding foliation. Thus, given certain conformality of f one would like to obtain similar conformality of g. This motivates the question whether a cocycle is conformal given that the return maps F n p at the periodic points are conjugate to conformal maps. This question is also natural from the point of view of cohomology of cocycles. The following proposition shows, however, that the answer is negative in dimension higher than two.
For any ǫ > 0 there exists a Lipschitz continuous linear cocycle F : E → E, which is ǫ-close to the identity, such that for all periodic points p ∈ M the return maps F n p : E p → E p are conjugate to orthogonal maps, but F is not conformal with respect to any continuous Riemannian metric on E.
We note that, for a given p, having a uniform bound on F n p · (F n p ) −1 for all periods n is equivalent to each of the following three statements: F n p is diagonalizable over C with its eigenvalues equal in modulus; F n p is conjugate to a conformal linear map; there exists an inner product on E p with respect to which F n p is conformal. In fact, the periodic assumption in the first part of Theorem 1.1 is equivalent to having such inner products for all periodic points uniformly bounded. In the context of local rigidity, additional assumptions were made to ensure such boundedness, for example that all return maps F n p are scalar multiples of the identity [11,7,12]. Our next result for two-dimensional bundles does not require any extra assumptions. It can be applied, in particular, to the study of local rigidity without restrictive assumptions on the structure of Df n p .
Theorem 1.3. Let F : E → E be a Hölder continuous linear cocycle over a transitive C 2 Anosov diffeomorphism f . Suppose that the fibers of E are two-dimensional. If for each periodic point p ∈ M, the return map F n p : E p → E p is diagonalizable over C and its eigenvalues are equal in modulus, then F is conformal with respect to a Hölder continuous Riemannian metric on E.
Moreover, if for each periodic point p ∈ M, the return map F n p : E p → E p is diagonalizable over C and its eigenvalues are of modulus 1, then F is isometric with respect to a Hölder continuous Riemannian metric on E.
The proof of this result overcomes essential difficulties and substantially differs from the proof of Theorems 1.1. We use Zimmer's Amenable Reduction Theorem to recast the problem as one of continuity of measurable invariant conformal structures and of measurable invariant sub-bundles. We note that such results on continuity of measurable invariant objects are rare beyond the case of group valued functions with compact or abelian range.
In the next section we formulate our main technical results. In Section 3 we briefly introduce the main notions used in this paper. The proofs of all the results are given in Section 4.

Properties of cocycles
In this section we formulate our main technical results which are of independent interest. We consider various conditions of relative pinching and establish properties of cocycles including existence and continuity of measurable invariant sub-bundles and conformal structures. We make the following Standing assumptions. In the statements below, f is a transitive C 2 Anosov diffeomorphism of a compact manifold M, P : E → M is a finite dimensional Hölder continuous vector bundle over M, and F : E → E is a Hölder continuous linear cocycle over f with Hölder exponent β (see Section 3 for definitions).
In the first proposition we obtain uniform relative pinching of the cocycle from asymptotic data at the periodic points. We denote by λ + (F, p) and λ − (F, p) the largest and smallest Lyapunov exponents of F at p, and by λ + (F, µ) and λ − (F, µ) the largest and smallest Lyapunov exponents of an ergodic invariant measure µ given by (3.3).
|n| for all x ∈ M and n ∈ Z. We can apply this proposition to the case when at each periodic point p there is only one Lyapunov exponent, i.e. all eigenvalues of F n p : E p → E p are of the same modulus. In this case we see that for any ergodic invariant measure for f , the cocycle F has only one Lyapunov exponent and (2.1) is satisfied with γ = 0.
In the proposition below, κ is the exponent in the Anosov condition (3.1) for f , and β is a Hölder exponent for F in (3.2). We show that under sufficient pinching, the iterates of the cocycle at the points on the same local stable manifold W s loc remain close. The same holds for the inverse map and the points on the same local unstable manifold W u loc . To consider the compositions (F n x ) −1 • F n y and (F −n x ) −1 • F −n y we identify E x and E y for y close to x using local coordinates. This identification is Hölder.
Proposition 2.2. Suppose that for some 0 < ǫ < κβ/3 there exists C ǫ such that Then there exist C > 0 and δ 0 > 0 such that for any δ < δ 0 and n ∈ N (a) for any x ∈ M and y ∈ W s loc (x) with dist (x, y) ≤ δ we have Next we establish continuity of measurable invariant conformal structures and subbundles. In our statements, we consider ergodic f -invariant measures on M with full support and local product structure. Examples include the measure of maximal entropy, and more generally Gibbs (equilibrium) measures of Hölder continuous potentials. A measure µ has local product structure if it is locally equivalent to the product of its conditional measures on the local stable and unstable manifolds. Proposition 2.3. Suppose that F satisfies the conclusion of Proposition 2.2, and µ is an ergodic f -invariant measure on M with full support and local product structure. Then any F -invariant measurable conformal structure on E defined µ almost everywhere is Hölder continuous with exponent β.
It is not known in general whether any measurable invariant conformal structure is continuous. Some results were established when the conformal structure is bounded [20] or belongs to L p for sufficiently large p [13].
Combining Propositions 2.1, 2.2, and 2.3 we see that if at each periodic point there is only one Lyapunov exponent, or if the largest and the smallest exponents are sufficiently close, then any F -invariant measurable conformal structure on E is Hölder continuous.
We recall that a cocycle F is said to be uniformly quasiconformal if the quasiconformal distortion K F (x, n) is uniformly bounded for all x ∈ M and n ∈ Z, see Section 3.5 for details. In the next proposition we apply observations made by D. Sullivan [22] and P. Tukia [23] for quasiconformal group actions to our case. We state this result in greater generality than our standing assumptions. We note that the converse statement is also true. Proposition 2.4. Let f be a diffeomorphism of a compact manifold M and let F : E → E be a continuous linear cocycle over f . If F is uniformly quasiconformal then it preserves a bounded measurable conformal structure τ on E.
Under our standing assumptions, Proposition 2.3 implies that τ is Hölder continuous. We can normalize it by a Hölder continuous function on M to obtain a Riemannian metric with respect to which F is conformal, which yields the following corollary.
Corollary 2.5. If F is uniformly quasiconformal then it preserves a Hölder continuous conformal structure on E, equivalently, F is conformal with respect to a Hölder continuous Riemannian metric on E.
This corollary and Propositions 2.1 and 2.2 enable us to prove Theorem 1.1. Now we address continuity of measurable invariant sub-bundles. Note that the assumptions in the next proposition are stronger than those in Proposition 2.3. However, they are satisfied if at each periodic point there is only one Lyapunov exponent. Proposition 2.6. Suppose that for any ǫ > 0 there exists C ǫ such that Then any measurable F -invariant sub-bundle in E defined almost everywhere with respect to a measure with local product structure and full support is Hölder continuous.
Combining Propositions 2.3 and 2.6 with Zimmer's Amenable Reduction Theorem we obtain the following description of cocycles with slowly growing quasiconformal distortion. We use it in the proof of Theorem 1.3.
Proposition 2.7. Suppose that for any ǫ > 0 there exists C ǫ such that K F (x, n) ≤ C ǫ e ǫ|n| for all x ∈ M and n ∈ Z. Then either F preserves a Hölder continuous conformal structure on E or F preserves a Hölder continuous proper non-trivial subbundle E ′ of E and a Hölder continuous conformal structure on E ′ .
We note that the alternatives are not mutually exclusive. If E ′ is one-dimentional then having a conformal structure on it becomes trivial.

Preliminaries
In this section we briefly introduce the main notions used in this paper.

3.1.
Anosov diffeomorphisms. Let f be a diffeomorphism of a compact Riemannian manifold M. It is called Anosov if there exist a decomposition of the tangent bundle T M into two invariant continuous subbundles E s and E u , and constants C > 0, κ > 0 such that for all n ∈ N, The distributions E s and E u are called stable and unstable. These distributions are tangential to the foliations W s and W u respectively. Local stable and unstable leaves W s loc (x) and W u loc (x) are the connected components of x in the intersection of W s (x) and W u (x) with a small ball around x.

Hölder continuous vector bundles.
Let M be a compact smooth manifold. We consider a finite dimensional Hölder continuous vector bundle P : E → M over M. By this we mean that there exists an open cover {U i } of M and a system of local coordinates φ i : are homeomorphisms with liner automorphisms L x depending Hölder continuously on x. That is, there exist C, β > 0 such that for all i, j and all x, y ∈ U i ∩ U j .
We will sometimes identify the fibers at nearby points using the local coordinates. We equip E with a background Hölder continuous Riemannian metric, i.e. a family of inner products on the fibers E x depending Hölder continuously on x.
3.3. Linear cocycles and Lyapunov exponents. Let f be a diffeomorphism of a compact smooth manifold M and P : E → M be a finite dimensional Hölder continuous vector bundle over M. A Hölder continuous linear cocycle over f is a homeomorphism F : E → E such that P • F = f • P and F x : E x → E f x is a linear isomorphism which depends Hölder continuously on x, i.e. there exist C, β > 0 such that for all nearby x, y ∈ M, Here F x and F y are viewed as matrices using local coordinates. Note that the second term on the left is not necessary for a continuous F . Indeed, F −1 x is then automatically continuous in x and bounded on M, so we can estimate We consider the standard notion of Lyapunov exponents for such a cocycle F (see [3, Section 2.3] for more details). We emphasize that the Lyapunov exponents of F are defined for vectors in the linear spaces E x . Note that for any measure µ on M the vector bundle E is trivial on a set of full measure. By Oseledets's Multiplicative Ergodic Theorem the Lyapunov exponents of F , as well as Lyapunov decomposition of E, are defined almost everywhere for every ergodic f -invariant measure µ on M; in particular, they are defined at every periodic point. We are primarily interested in the largest and the smallest Lyapunov exponents of µ which can be defined as follows: It is known that C d becomes a Riemannian symmetric space of non-positive curvature when equipped with a certain GL(d, R)-invariant metric. The distance to the identity in this metric is given by where λ 1 , . . . , λ d are the eigenvalues of C (see [23, p.327] for more details and [16, p.27] for the formula). The distance between two structures C 1 and C 2 can be computed It is easy to check the following relation between this metric and the operator norm We also note that C −1 ≤ C d−1 . Thus a subset of C d is bounded with respect to this distance if and only if it is bounded with respect to the operator norm. We also note that on any bounded subset of C d this distance is bi-Lipschitz equivalent to the distance induced by the operator norm on matrices.
Let E be a Hölder continuous vector bundle over a compact manifold M. A conformal structure on E x is a class of proportional inner products on E x . Using the background Riemannian metric on E, we can identify an inner product with a symmetric linear operator with determinant 1 as before. For each x ∈ M, we denote the space of conformal structures on E x by C(x). Thus we obtain a bundle C over M whose fiber over x is C(x). We equip the fibers of C with the Riemannian metric defined above. A continuous (Hölder continuous, measurable) section of C is called a continuous (Hölder continuous, measurable) conformal structure on E. A measurable conformal structure τ on E is called bounded if the distance between τ (x) and τ 0 (x) is uniformly bounded on M for a continuous conformal structure τ 0 on E.
Now, let f be a diffeomorphism of M and F : E → E be a linear cocycle over f . Then F induces a natural pull-back action F * on conformal structures as follows. For a conformal structure τ (f x) ∈ C(f x), viewed as the linear operator on E f x , 3.5. Uniform quasiconformality. Let f be a diffeomorphism of a compact manifold M and F : E → E be a linear cocycle over f . For x ∈ M and n ∈ Z the quasiconformal distortion of F is defined by We say that F is uniformly quasiconformal if K F (x, n) is uniformly bounded for all x ∈ M and n ∈ Z . If K F (x, n) = 1 for all x and n, then F is said to be conformal.
Clearly, F is conformal with respect to a Riemannian metric on E if and only if it preserves the conformal structure associated with this metric. We note that the notion of uniform quasiconformality does not depend on the choice of a continuous metric. So if F preserves a continuous conformal structure on E then F is uniformly quasiconformal on E with respect to any continuous metric on E. Corollary 2.5 shows that the converse is also true if f is a transitive Anosov diffeomorphism.
[6, Theorem 1.4] Let f be a homeomorphism of a compact metric space X satisfying the closing property, let F be a Hölder GL(d, R) cocycle over f , and let µ be an ergodic invariant measure for f . Then the Lyapunov exponents λ 1 ≤ ... ≤ λ d (listed with multiplicities) of F with respect to µ can be approximated by the Lyapunov exponents of F at periodic points. More precisely, for any ǫ > 0 there exists a periodic point p ∈ X for which the Lyapunov exponents λ As stated in the remark after this theorem, it holds for any Hölder continuous linear cocycle F . Also, a transitive Anosov diffeomorphism satisfies the closing property. Thus we can apply the theorem in our setup and immediately obtain the desired result for µ. Now we prove the estimate for the quasiconformal distortion K F (x, n) using the following result.  a n+k (x) ≤ a n (f k (x)) + a k (x) for every x ∈ M, n, k ≥ 0 and such that there is a sequence of continuous functions b n , n ≥ 0 satisfying (4.2) a n (x) ≤ a n (f k (x)) + a k (x) + b k (f n (x)) for every x ∈ M, n, k ≥ 0.
If inf n 1 n M a n dµ < 0 for every ergodic f -invariant measure, then there is N ≥ 0 such that a N (x) < 0 for every x ∈ M.
To simplify the notations we write K(x, n) for K F (x, n). For a given ǫ > 0 we apply the proposition to a n (x) = log K(x, n) − (γ + ǫ)n and b n (x) = log K(x, n) + (γ + ǫ)n.
It is easy to see from the definition of the quasiconformal distortion that K(x, n + k) ≤ K(x, k) · K(f k x, n) and K(x, n + k) ≥ K(x, n) · (K(f n x, k)) −1 for every x ∈ M, n, k ≥ 0. It follows that a n+k (x) ≤ a n (f k (x)) + a k (x), i.e. the functions a n satisfy (4.1), and a n+k (x) ≥ a n (x) − b k (f n x). Hence and we obtain (4.2).
Let µ be an ergodic f -invariant measure. We note that since a n satisfy (4.1), the Subadditive Ergodic Theorem implies that inf n 1 n M a n dµ = lim n→∞ 1 n a n (x) for µ almost all x ∈ M.
Using the definitions of K(x, n), λ + (F, µ), and λ − (F, µ) we obtain that for µ almost all x and hence lim n→∞ 1 n a n (x) ≤ −ǫ < 0 for µ almost all x ∈ M. Thus all assumptions of the proposition above are satisfied and hence for any ǫ > 0 there exists N ǫ such that a Nǫ (x) < 0, i.e. K(x, N ǫ ) ≤ e (γ+ǫ)Nǫ for all x ∈ M. For any n > 0, we write n = mN ǫ + r, 0 ≤ r < N ǫ , and estimate where C ǫ = max K(x, r) with the maximum taken over all x ∈ M and 1 ≤ r < N ǫ . Since K(x, n) = K(f n x, −n) we obtain K(x, n) ≤ C ǫ e (γ+ǫ)|n| for all x in M and n in Z.

4.2.
Proof of Proposition 2.2. First we consider the case when y ∈ W s loc (x). Since at least one of the points x and y is non-periodic, we assume that x is. We denote x i = f i (x) and y i = f i (y) for i = 0, 1, ..., n. We have We estimate Since F is Hölder continuous with exponent β, we have Since y ∈ W s loc (x), for κ is as in (3.1) we obtain (4.4) r Lemma 4.2 below shows that Combining (4.3), (4.4), and (4.5) we obtain This completes the proof for the case of y ∈ W s (x).
To prove (b) we observe that F −1 satisfies the assumptions of the proposition.
Thus we can apply (a) to F −1 , which yields (b).
It remains to prove estimate (4.5). To do this, we construct special metrics on E f k x along the orbit of a non-periodic point x ∈ M. We denote x k = f k (x), k ∈ Z.
Lemma 4.1. Let f be a diffeomorphism of a compact manifold M, E be a continuous vector bundle over M, and F be a continuous linear cocycle over f . Suppose that for some ǫ > 0 there exists C ǫ such that K F (x, n) ≤ C ǫ e ǫ|n| for all x ∈ M and n ∈ Z. Then for any non-periodic point x ∈ M there exist metrics · x k on E x k , k ∈ Z, such that Moreover, there exists a constant M ǫ such that v ≤ v x k ≤ M ǫ v for all k ∈ Z and v ∈ E x k , where · is a given continuous metric on E.
Proof. We choose a unit vector u ∈ E x and set u k = F k By the assumption on K F , F m x k ( v v ) · F m x k (u k ) −1 ≤ C ǫ e |m|ǫ and hence the terms of this series are bounded by C 2 ǫ e −|m|ǫ v 2 . This implies that the series converges and v 2 x k is at least the term with m = 0, and thus v x k ≥ v .
We note that it suffices to prove the estimate for n = 1, then it automatically follows for all n. We observe that u k+1 is a unit vector parallel to F x k (u k ), and hence u k+1 = F x k (u k )/ F x k (u k ) . For any vector v ∈ E x k we estimate Here we used the estimate |j| − 1 ≤ |j − 1|. Similarly, using |j − 1| ≤ |j| + 1 we obtain

It follows that for any two vectors
Proof. We consider the metrics · x k on E x k , k = 0, . . . , i, given by Lemma 4.1. We denote by F x k k the norm of the operator F from (E x k , · x k ) to (E x k+1 , · x k+1 ) and we denote by (F x k ) −1 k the norm of the corresponding inverse operator. Since Using Hölder continuity of F in the metric · , we obtain For n = 1, the inequality (4.6) gives F x k k · (F x k ) −1 k ≤ e 3ǫ and we estimate

It follows that (F
This completes the proof of Proposition 2.2.

4.3.
Proof of Proposition 2.3. We identify the spaces of conformal structures at nearby points by identifying the fibers of E with R d using local coordinates. We use the distance between conformal structures described in Section 3.4. Let τ be an invariant µ-measurable conformal structure on E.
First we estimate the distance between the values of τ at x and at a nearby point y ∈ W s loc (x). Let x n = f n (x), y n = f n (y), and let D x := (F n x ) * be the isometry from C(f n x) to C(x) induced by F n x (see (3.6)). Since the conformal structure τ is invariant, τ (x) = D x (τ (x n )) and τ (y) = D y (τ (y n )). Using this and the fact that D y is an isometry, we obtain To estimate (τ (x n ), ((D x ) −1 • D y )(τ (x n ))) we use the following lemma.
where k(σ) is bounded on compact sets in C d . More precisely, if σ is given by a matrix C, then k(σ) ≤ 3d C −1 · C for any A with A − Id ≤ (6 C −1 · C ) −1 .
Proof. We write A = Id+R. Recall that the matrix C corresponding to σ is symmetric and positive definite with determinant 1. Thus there exists an orthogonal matrix Q such that Q T CQ is a diagonal matrix whose diagonal entries are the eigenvalues λ i > 0 of C. Let X be the product of Q and the diagonal matrix with entries 1/ √ λ i . Then X has determinant 1 and X[C] = X T CX = Id. Now we estimate Since R ≤ 1, we observe that B ≤ 3 X 2 · C · R . Also X 2 ≤ C −1 , as follows from the construction of X.
Since the conformal structure τ is µ-measurable, by Lusin's theorem there exists a compact set S ⊂ M with µ(S) > 1/2 on which τ is uniformly continuous and bounded.
We now show that for x n in S the term dist (τ (x n ), ((D x ) −1 • D y )(τ (x n ))) is Hölder in dist(x, y). For this we observe that the map (D y) β by the assumption. We apply Lemma 4.3 to σ = τ (x n ) and A = (F n x ) −1 • F n y . Since the conformal structure τ is bounded on S, so are C −1 and C . We obtain that where the constant k 1 depends on the set S. We conclude that if x n is in S then dist(τ (x), τ (y)) ≤ dist(τ (x n ), τ (y n ))) + k 1 · dist(x, y) β .
Let G be the set of points in M for which the frequency of visiting S equals µ(S) > 1/2. By Birkhoff Ergodic Theorem µ(G) = 1. If both x and y are in G, then there exists a sequence {n i } such that x n i ∈ S and y n i ∈ S. Since y ∈ W s loc (x), dist(x n i , y n i ) → 0 and hence dist(τ (x n i ), τ (y n i )) → 0 by continuity of τ on S. Thus, we obtain dist(τ (x), τ (y)) ≤ k s · dist(x, y) β .
Consider a small open set in M with a product structure. For µ almost all local stable leaves, the set of points of G on the leaf has full conditional measure. Consider points x, y ∈ G lying on two such local stable leaves. We denote by H x,y be the unstable holonomy map between W s loc (x) and W s loc (y). Since µ has local product structure, the holonomy maps are absolutely continuous with respect to the conditional measures. Hence there exists a point z ∈ W s loc (x) ∩ G close to x such that H x,y (z) is also in G. By the above argument, Since the points x, y, and z are close, it is clear from the local product structure that Hence, we obtain dist(τ (x), τ (y)) ≤ k 6 · dist(x, y) β for all x and y in a set of full measureG ⊂ G. We can assume thatG is invariant by considering ∞ n=−∞ f n (G). Since µ has full support, the setG is dense in M. Hence we can extend τ fromG and obtain an invariant Hölder continuous conformal structure τ on M.

Proof of Proposition 2.4.
Let τ 0 be a continuous conformal structure on E. We denote by τ 0 (x) the conformal structure on E x , x ∈ M. We consider the set , the space of conformal structures on E x . Here F * is the pull-back action given by (3.6). Since F is uniformly quasiconformal, the sets S(x) have uniformly bounded diameters. Since the space C(x) has non-positive curvature, for every x there exists a uniquely determined ball of the smallest radius containing S(x). We denote its center by τ (x).
It follows from the construction that the conformal structure τ is F -invariant and its distance from τ 0 is bounded. We also note that for any k ≥ 0 the set S k (x) = { (F n x ) * (τ 0 (f n x)) : |n| ≤ k} depends continuously on x in Hausdorff distance, and so does the center τ k (x) of the smallest ball containing S k (x). Since S k (x) → S(x) as k → ∞ for any x, the conformal structure τ is the pointwise limit of continuous conformal structures τ k (x). Hence τ is Borel measurable.

4.5.
Proof of Proposition 2.6. We consider a fiber bundle G over M whose fiber over x is the Grassman manifold G x of all k-dimensional subspaces in E x . The map Since the linear cocycle F and the bundle E are Hölder continuous, bothF andF −1 are Hölder continuous and dist C 1 (F x ,F y ) ≤ k · dist(x, y) β for all x, y ∈ M. SuchF is said to be in C β (f, G).
Lemma 4.4. There exists C > 0 such that for any x ∈ M, subspaces ξ, η ∈ G x , n ∈ Z, and ǫ > 0 we have dist(F n x (ξ),F n x (η)) ≤ C · K F (x, n) · dist(ξ, η) ≤ C · C ǫ e ǫ|n| · dist(ξ, η). Proof. Let w and v be two unit vectors in E x . We denote D = F n x . Using the formula 2 < Dw, Dv >= Dw 2 + Dv 2 − Dw − Dv 2 for the inner product, we obtain Suppose that the angle between the unit vectors w and v is sufficiently small so that it remains small when multiplied by K F (x, n). Then we obtain that the angle ∠(Dw, Dv) is also small and If the right-hand side is large the estimate is trivial, thus for any subspaces ξ, η ∈ G x we have dist(F n x (ξ),F n x (η)) ≤ C · K F (x, n) · dist(ξ, η), where the distance between two subspaces is the maximal angle. We note that the maximal angle distance is Lipschitz equivalent to any smooth Riemannian metric on the Grassman manifold, and thus we have the estimate in any smooth metric on G.
The case of n < 0 can be considered similarly.
The lemma implies that the expansion/contraction in the fiber is arbitrarily slow and, in particular, slower than the expansion/contraction of the hyperbolic system in the base. Hence the cocycleF is dominated in the sense of [1,Definition 4.1]. This notion is similar to the notion of domination or bunching for partially hyperbolic systems, the difference is that in our contextF is not a diffeomorphism, only the mapsF x are.
Dominated cocycles have Hölder continuous strong (un)stable foliations. They sub-foliate the weak (un)stable leaves which are preimages of (un)stable leaves in the base. The strong stable foliation gives rise to an s-holonomy forF , an invariant family of maps between the fibers over the same stable leaf in the base. These facts are conveniently summarized in the following proposition, whose Lipschitz version appeared in [2, Proposition 4.1]. We will refer only to part (3), which in our setting can be easily obtained from Proposition 2.2 and its proof. Indeed, the desired holonomy H s x,y is induced on the Grassmannians by lim n→∞ (F n x ) −1 • F n y . [1, Proposition 4.2] If the cocycleF ∈ C β (f, G) is dominated then there exists a unique partition W s = {W s (x, ξ) : (x, ξ) ∈ G} of the fiber bundle G such that (1) every W s (x, ξ) is a β-Hölder graph over W s (x), with Hölder constant uniform on x; (2)F (W s (x, ξ)) ⊂ W s (F (x, ξ)) for all (x, ξ) ∈ G; (3) the family of maps H s x,y : G x → G y defined for y ∈ W s (x) by (y, H s x,y (ξ)) ∈ W s (x, ξ) is an s-holonomy forF . Moreover, there is a dual statement for strong unstable leaves.
Let µ be a measure on M with local product structure and full support. A µmeasurable F -invariant sub-bundle in E gives rise to a measurableF -invariant section φ : M → G. We denote by m the lift of µ to the graph Φ of φ, i.e. for a set X ⊂ G, m(X) = µ(π(X ∩ Φ)), where π : G → M is the projection. Alternatively, m can be defined by specifying that for µ-almost every x in M the conditional measure m x in the fiber G x is the atomic measure at φ(x). Since µ is f -invariant and Φ is F -invariant, the measure m isF -invariant.
Lemma 4.4 implies that Lyapunov exponent ofF along the fiber is zero at every ξ ∈ G, in particular the exponent of m along the fiber is zero. This together with the existence of s-and u-holonomies forF allows us to apply [2, Theorem C] to the measure m and conclude that there exists a system of conditional measuresm x on G x for m which are holonomy invariant and vary continuously on x in supp µ = M. Since the systems of conditional measures {m x } and {m x } coincide on a set X of full µ measure, we see thatm x = m x is the atomic measure at φ(x) for all x ∈ X. Since X is dense we obtain thatm x is atomic for all x ∈ M. Indeed, by compactness of G, for any x ∈ M we can take a sequence X ∋ x i → x so that φ(x i ) converge to some ξ ∈ G x . This implies thatm x i = m x i converge to the atomic measure at ξ, which therefore coincides withm x by continuity of the family {m x }. Denoting φ(x) = suppm x for x ∈ M, we obtain a continuous sectionφ which coincides with φ on X. Sinceφ is invariant under s-and u-holonomies we conclude that it is β-Hölder. This yields that the invariant measurable sub-bundle in E coincides µ-almost everywhere with a Hölder continuous one.
where K(A) and K(B) are quasiconformal distortions of A and B respectively.
Multiplying by B −1 , we have A −1 = (Id + R)B −1 and hence The case of AB −1 − Id ≤ r is similar. Now we will show that F is uniformly quasiconformal using a dense orbit argument. Since f is transitive, there exists a point z ∈ M with dense orbit ø = {f k z; k ∈ Z}. We will show that the quasiconformal distortion K F (z, k) is uniformly bounded in k ∈ Z. Since ø is dense and K F (x, k) is continuous on M for each k, this implies that K F (x, k) is uniformly bounded in x ∈ X and k ∈ Z.
We consider any two points of ø with dist(f k 1 z, f k 2 z) < δ 0 , where δ 0 is sufficiently small to apply the Anosov Closing Lemma [10,Theorem 6.4.15]. We assume that k 1 < k 2 and denote x = f k 1 z and n = k 2 − k 1 , so that δ = dist(x, f n x) < δ 0 . By the Anosov Closing Lemma there exists p ∈ X with f n p = p such that dist(f i x, f i p) ≤ cδ for i = 0, . . . , n.
Let y be a point in W s loc (p) ∩ W u loc (x). Then by Proposition 2.2, Hence by Lemma 4.6 We take m > 0 such that the set {f j (z); |j| ≤ m} is δ 0 -dense in M. Let K m = max{K F (z, j) : |j| ≤ m}. Then for any k > m there exists j, |j| ≤ m, such that dist(f k (z), f j (z)) ≤ δ 0 and hence The case of k < −m is considered similarly. Thus K F (z, k) is uniformly bounded and hence so is K F (x, k) for all x ∈ M and k ∈ Z.
Thus, F is uniformly quasiconformal on E. It follows from Corollary 2.5 that there exists a Hölder continuous metric on E with respect to which f is conformal. Now we prove the second part of the theorem. We observe that K F (p, n) = F n p · (F n p ) −1 ≤ (C ′ per ) 2 and hence F is conformal with respect to a Hölder continuous Riemannian metric g on E. This means that there exists a positive Hölder continuous function a(x) such that for each x ∈ M and each v ∈ E x It remains to renormalize the metric g. For a positive function ϕ(x), we consider a new metricg(x) = g(x)/ϕ(x). Then we have Therefore we need to find a Hölder continuous function ϕ(x) such that ϕ(f x) = a(x)ϕ(x), i.e.
Let p be a periodic point of a period n. Then a(p)a(f p) . . . a(f n−1 p) = F n p g(p) . If F n p g(p) > 1 then F mn g(p) = F n p m g(p) → ∞ as m → ∞. If F n p g(p) < 1 then Suppose that there is a continuous invariant one-dimensional sub-bundle. Hence for any point p and any n with f n p = p we obtain an invariant line for F n p : E p → E p . This implies that the eigenvalues of F n p are real. Hence, by the assumption of the theorem, they are either λ, λ or λ, −λ. If F is orientation preserving, the former is always the case. It follows that F n p = λ · Id since it is diagonalizable, and hence K F (p, n) = 1. For such F we can apply Theorem 1.1 and obtain a Hölder continuous metric on E with respect to which F is conformal.
If E is not orientable we can pass to a double cover. If F is orientation reversing, we can consider cocycle F 2 over f 2 . Thus we can always obtain an orientation preserving cocycle F ′ which by the above is conformal. This implies uniform quasiconformality of the original cocycle F . Now Corollary 2.5 yields conformality of F .
We conclude that F is conformal with respect to a Hölder continuous metric on E. The second part can be establishes in the same way as in the proof Theorem 1.1. Indeed, the assumption implies that for any periodic point p the map F n p is conjugate to an orthogonal matrix and hence there exists a constant C(p) such that max{ F n p , (F n p ) −1 } ≤ C(p) for any period n of p.

4.9.
Proof of Proposition 1.2. The idea of the example was suggested to us by M. Guysinsky and is similar to his example in [5]. We recall that since the tangent bundle is trivial, a cocycle F : E → E is defined by a function A : M → GL(d, R) via F (x, v) = (f x, A(x)v). First we construct an example for d = 3. Let S be a closed, and hence compact, f -invariant set in M that does not contain any periodic points (such sets always exist for Anosov systems and can be constructed using symbolic dynamics). Let α : M → R be given by α(x) = dist(x, S) · ǫ/(2 diamM).
Let p be a periodic point and let n be its minimal period. Since the eigenvalues of A(p, n) are of modulus 1, it is conjugate to an orthogonal matrix if and only if it is diagonalizable over C. For A(p, n) to be diagonalizable, it suffices to have three different eigenvalues, which is equivalent toα(p, n) = πk. The function α does not necessarily satisfy this condition at every periodic point, so we modify it inductively to obtain a functionα that does.
Since there are countably many periodic orbits for f , we can order them It follows from the construction thatα(x) is Lipschitz,α(x) = 0 on S, andα(p, n) = πk for any periodic point p of a minimal period n. Thus the matrix A(p, n) has three different eigenvalues: e iα(p,n) , e −iα(p,n) , 1, and hence is diagonalizable. We also note that since max |α(x) −α(x)| ≤ ǫ/2, we have max |α(x)| ≤ ǫ and thus A(x) is ǫ-close to the identity. Now we show that the cocycle F is not uniformly quasiconformal. Let x be a point in S. Then x is non-periodic andα(f n x) = 0 for every n. Hence The quasiconformal distortion of F is not uniformly bounded along the orbit of x, as for the first coordinate vector F n x v 1 = A(x, n)v 1 → ∞, while for the second coordinate vector F n x v 2 = A(x, n)v 2 = 1. It follows that F cannot be conformal with respect to any continuous Riemannian metric on E.
This example can be extended to any dimension d ≥ 4 by considering where Id d−3 is the (d − 3) × (d − 3) identity matrix.