COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS

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1 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS BORIS KALININ AND VICTORIA SADOVSKAYA Abstract. We consider Hölder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we establish a continuous version of Zimmer s Amenable Reduction Theorem. For cocycles over hyperbolic systems we also obtain polynomial growth estimates for the norm and quasiconformal distortion from the periodic data. 1. Introduction A linear cocycle over a dynamical system f : M M is an automorphism F of a vector bundle E over M that covers f. In the case of a trivial vector bundle M R d, a linear cocycle can be represented by a matrix-valued function A : M GL(d, R) via F (x, v) = (f(x), A(x)v). In smooth dynamics linear cocycles arise naturally from the differential. They play important role in study of smooth systems and group actions, especially in aspects related to rigidity. In this paper we consider Hölder continuous linear cocycles with one Lyapunov exponent over hyperbolic and partially hyperbolic diffeomorphisms. An important motivation comes from the restriction of the derivative to Hölder continuous invariant sub-bundles such as center, stable, and unstable bundles. In hyperbolic case we studied such cocycles in [KS09, KS10]. We concentrated on obtaining conformality of the cocycle from its periodic data and applying this to local and global rigidity of Anosov systems [KS09, GKS]. From a different angle, such cocycles over hyperbolic and partially hyperbolic systems were considered in [V, ASV]. In particular, it was shown that cocycles with more than one Lyapunov exponent are generic in various cases, for example in a neighborhood of a fiber bunched cocycle. These results indicated that having one exponent is an exceptional property. In this paper we show that it is true in a very strong sense by developing a structural theory for such cocycles. We expect that these results will be useful in the study of partially hyperbolic systems and in the area of rigidity of hyperbolic systems and actions. In the base we consider a partially hyperbolic diffeomorphism f which is is volumepreserving, accessible, and center bunched. This is the same setting as in the latest Date: November 14, Supported in part by NSF grant DMS Supported in part by NSF grant DMS

2 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 2 results on ergodicity of partially hyperbolic diffeomorphisms [BW]. We assume that the cocycle F over f is fiber bunched, i.e. non-conformality of F in the fiber is dominated by the expansion/contraction along the stable/unstable foliations of f in the base. This or similar conditions play a role in all results on noncommutative cocycles over hyperbolic or partially hyperbolic systems. If F is the restriction of the derivative of f to the center sub-bundle, fiber bunching for F corresponds to the strong center bunching for f. Thus our results apply to this setup. For fiber bunched cocycles with one Lyapunov exponent, we prove continuity of measurable F -invariant sub-bundles and conformal structures. We use this to establish a continuous version of Zimmer s Amenable Reduction Theorem. It gives the existence of a continuous flag of sub-bundles such that the induced cocycles on the factor bundles are conformal. This result is new even for cocycles over hyperbolic systems. In this case we apply the theorem to cocycles with one exponent at each periodic orbit and obtain polynomial growth estimates of quasiconformal distortion. We formulate the results in Section 3 and give the proofs in Section Definitions and notations In this parer M denotes a compact connected smooth manifold Partially hyperbolic diffeomorphisms. (See [BW] for more details.) A diffeomorphism f of M is said to be partially hyperbolic if there exist a nontrivial Df-invariant splitting of the tangent bundle T M = E s E c E u, and a Riemannian metric on M for which one can choose continuous positive functions ν < 1, ˆν < 1, γ, ˆγ such that for any unit vectors v s E s (x), v c E c (x), and v u E u (x) (2.1) Df(v s ) < ν(x) < γ(x) < Df(v c ) < ˆγ(x) 1 < ˆν(x) 1 < Df(v u ). The sub-bundles E s, E u, and E c are called, respectively, stable, unstable, and center. E s and E u are tangent to the stable and unstable foliations W s and W u respectively. An su-path in M is a concatenation of finitely many subpaths which lie entirely in a single leaf of W s or W u. A partially hyperbolic diffeomorphism f is called accessible if any two points in M can be connected by an su-path. We say that f is volume-preserving if it preserves a probability measure µ in the measure class of a volume induced by a Riemannian metric. It is conjectured that an (essentially) accessible f is ergodic with respect to such µ. This was established if f is C 2 and center bunched [BW]. The diffeomorphism f is called center bunched if the functions ν, ˆν, γ, ˆγ can be chosen to satisfy (2.2) ν < γˆγ and ˆν < γˆγ. This implies that Df E c (Df E c) 1, which is a measure of non-conformality of f on E c, is dominated by the contraction on E s and expansion on E u. If f is C 1+δ, the ergodicity holds under strong center bunching assumption [BW]: (2.3) ν θ < γˆγ and ˆν θ < γˆγ,

3 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 3 where θ (0, δ) satisfies (2.4) νγ 1 < κ θ and ˆνˆγ 1 < ˆκ θ, for some functions κ and ˆκ such that for all x in M κ(x) < Df(v) if v E s (x) and Df(v) < ˆκ(x) 1 if v E u (x). It is known that the first inequality in (2.4) implies that E c E s is θ-hölder, the second one yields the same for E c E s, and thus (2.4) implies that E c is θ-hölder Hölder continuous vector bundles and linear cocycles. We consider a finite dimensional Hölder continuous vector bundle P : E M. By this we mean that there exists an open cover {U i } of M and a system of local coordinates φ i : P 1 (U i ) U i R d such that the coordinate changes φ j φ 1 i : (U i U j ) R d (U i U j ) R d (x, v) (x, L x (v)) are homeomorphisms with linear automorphisms L x depending Hölder continuously on x. That is, there exist C, β > 0 such that L x L y C dist(x, y) β for all i, j and all x, y U i U j. In various estimates, we will identify the fibers at nearby points using local coordinates. We also 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. Let f be a diffeomorphism of 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 fx is a linear isomorphism which depends Hölder continuously on x, i.e. there exist C, β > 0 such that F x F y C dist(x, y) β for all nearby x, y M Conformal structures. (See [KS10] for more details.) A conformal structure on R d, d 2, is a class of proportional inner products. The space C d of conformal structures on R d can be identified with the space of real symmetric positive definite d d matrices with determinant 1, which is isomorphic to SL(d, R)/SO(d, R). The group GL(d, R) acts transitively on C d via X[C] = (det X T X) 1/d X T C X, and C d carries a GL(d, R)-invariant Riemannian metric of non-positive curvature. For a vector bundle E M we can consider a bundle C over M whose fiber C x is the space of conformal structures on E x. Using a background Riemannian metric on E, C x can be identified with the space of symmetric positive linear operators on E x with determinant 1. We equip the fibers of C with the Riemannian metric as above. A continuous (measurable) section of C is called a continuous (measurable) conformal structure on E. An invertible linear map A : E x E y induces an isometry from C x to C y via A(C) = (det(a A)) 1/d (A 1 ) C(A 1 ), where C is a conformal structure viewed as an operator. If F : E E is a linear cocycle over f, we say that a conformal structure τ on E is F -invariant if F (τ(x)) = τ(f(x)) for all x M.

4 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 4 3. Statements of results Standing assumptions. Unless stated otherwise, in this paper M is a compact connected smooth manifold, f : M M is an accessible partially hyperbolic diffeomorphism preserving volume µ which is either C 2 and center bunched, or C 1+δ and strongly center bunched, P : E M is a finite dimensional Hölder continuous vector bundle over M, F : E E is a Hölder continuous linear cocycle over f with Hölder exponent β. First we establish continuity of measurable invariant conformal structures for fiber bunched cocycles. A cocycle F over a partially hyperbolic diffeomorphism f is fiber bunched if for some β-hölder norm on E (3.1) F (x) F (x) 1 ν(x) β < 1 and F (x) F (x) 1 ˆν(x) β < 1 for all x in M. This condition allows to establish convergence of certain iterates of the cocycle along the stable and unstable leaves. Theorem 3.1. If F is fiber bunched, then any F -invariant µ-measurable conformal structure on E is continuous. We denote the iterate F f n 1 x... F fx F x by F n x. A cocycle F is called uniformly quasiconformal if the quasiconformal distortion (3.2) K F (x, n) def = F n x (F n x ) 1 is uniformly bounded for all x M and n Z. The cocycle is said to be conformal with respect to some Riemannian metric on E if K F (x, n) = 1 for all x and n. Corollary 3.2. If F is uniformly quasiconformal then it preserves a continuous conformal structure on E, equivalently, F is conformal with respect to a continuous Riemannian metric on E. Next we address continuity of measurable invariant sub-bundles. If a cocycle has more than one Lyapunov exponent, then the corresponding Lyapunov sub-bundles are invariant and measurable, but not continuous in general. We show that for a fiber bunched cocycle with only one exponent measurable invariant sub-bundles are continuous. We denote by λ + (F, µ) and λ (F, µ) the largest and smallest Lyapunov exponents of F with respect to µ. We recall that for µ almost every x M, 1 (3.3) λ + (F, µ) = lim n n log F x n 1 and λ (F, µ) = lim n n log (F x n ) 1 1 (see [BP, Section 2.3] for more details). Theorem 3.3. Suppose that F is fiber bunched and λ + (F, µ) = λ (F, µ). Then any µ-measurable F -invariant sub-bundle of E is continuous. Using Theorems 3.1 and 3.3 together with Zimmer s Amenable Reduction Theorem we obtain the following description of fiber bunched cocycles with one exponent.

5 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 5 Theorem 3.4 (Continuous Amenable Reduction). Suppose that F is fiber bunched and λ + (F, µ) = λ (F, µ). Then there exists a flag of continuous F -invariant sub-bundles (3.4) E 1... E k 1 E k = E, and continuous conformal structures on E 1 and factor bundles E i+1 /E i, i = 1,..., k 1, invariant under F E1 and the factor-cocycles induced by F on E i+1 /E i, respectively. In the case of E 1 = E, the cocycle is conformal on E with respect to some continuous Riemannian metric. If there are d = dim E x continuous vector fields which give bases for all E i, then the theorem implies that F is continuously cohomologous to a cocycle with values in a maximal amenable subgroup of GL(d, R), see Remark 4.8. However, triviality of E alone is insufficient for such reduction even if E = M 2 R 2, see [S]. Remark 3.5. If f is C 1+δ and satisfies the strong center bunching condition, then the above results apply to F = Df E c. Indeed, the condition (2.4) implies that E c is θ-hölder and (2.3) yields fiber bunching (3.1) for F = Df E c. We also show that the assumptions that the cocycle is fiber bunched and has only Lyapunov exponent with respect to the invariant volume µ can be replaced by the following assumption. Corollary 3.6. Suppose that λ + (F, η) = λ (F, η) for every ergodic f-invariant measure η. Then the conclusions of Theorem 3.1, Theorem 3.3, and Theorem 3.4 hold. This corollary relies on a certain estimate for subadditive sequences of continuous functions. In Proposition 4.9 we establish a definitive version of this useful result. We obtain Hölder continuity of the invariant structures under a stronger accessibility assumption. The diffeomorphism f is said to be locally α-hölder accessible if there exists n such that for all sufficiently close x, y M there is an su-path {x = x 0, x 1,..., x n = y} such that dist W i(x i 1, x i ) C dist(x, y) α for i = 1,..., n. Here the distance between x i 1 and x i is measured along the corresponding stable or unstable leaf W i. Corollary 3.7. If f is locally α-hölder accessible then the invariant conformal structures and sub-bundles in Theorems 3.1, 3.3, 3.4 and Corollaries 3.2, 3.6 are Hölder continuous. The Hölder exponent is αβ, except for invariant conformal structures on sub-bundles and factor bundles in Theorem 3.4 for which it is α 2 β. Remark 3.8. If f is an Anosov diffeomorphism, the corollary above applies with α = 1 due to the local product structure of stable and unstable manifolds. Moreover, in this case preservation of a volume µ can be replaced by transitivity, and measurability with respect to µ can be replaced with measurability with respect to any ergodic measure with local product structure. Indeed, Theorems 3.1, 3.3 and Corollary 3.2 for this case were proved in [KS10]. Theorem 3.4 and Corollary 3.6 can be deduced as in this paper.

6 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 6 The reduction in Theorem 3.4 can be used to obtain uniform polynomial growth estimates of the quasiconformal distortion K F (x, n) for a hyperbolic f. Note that λ + (F, η) = λ (F, η) for all measures η is a necessary condition for such an estimate. We do not obtain a similar result for partially hyperbolic f due to an unresolved basic problem whether a real-valued cocycle is cohomologous to 0 if it has zero integrals for all invariant measures. Theorem 3.9 (Polynomial Growth). Let f be a transitive Anosov diffeomorphism. Suppose that for every f-periodic point the invariant measure µ p on its orbit satisfies λ + (F, µ p ) = λ (F, µ p ). Then there exists m < dim E x and C such that K F (x, n) Cn 2m for all x M and n Z. Moreover, if λ + (F, µ p ) = λ (F, µ p ) = 0 for every µ p, then there exists m < dim E x and C such that F n x C n m for all x M and n Z. One can take m = k 1, which is the number of non-trivial sub-bundles in (3.4). 4. Proofs 4.1. Stable holonomies. Convergence of products of the type (F n y ) 1 F n x has been observed for various types group-valued cocycles whose growth is slower than the expansion/contraction in the base (see e.g. [NT, PW]). It is also related to existence of strong stable/unstable manifolds for the extended system on the bundle. We follow the notations and terminology form [V, ASV] for linear cocycles, where it is more convenient to use the following notion of holonomy. Definition 4.1. A stable holonomy for a linear cocycle F : E E is a continuous map H s : (x, y) H s xy, where x M, y W s (x), such that (i) H s xy is a linear map from E x to E y, (ii) H s xx = Id and H s yz H s xy = H s xz, (iii) H s xy = (F n y ) 1 H s f n x f n y F n x for all n N. Unstable holonomy are defined similarly. The following proposition establishes existence (cf. [ASV, Proposition 3.4]) and some additional properties of the holonomies. In our context, to consider compositions of maps defined on fibers at nearby points we identify such fibers using local coordinates. The holonomy does not depend on a particular choice of identifications and, in fact, is unique by (c). We denote by Wloc s (x) a sufficiently small ball around x in the leaf W s (x). Proposition 4.2. Suppose that the cocycle F is fiber bunched. C > 0 such that for any x M and y Wloc s (x), (a) (F n y ) 1 F n x Id Cdist(x, y) β for every n N; Then there exists

7 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 7 (b) H s xy = lim n (F n y ) 1 F n x exists and satisfies (i), (ii), (iii) of Definition 4.1 and (iv) H s xy Id Cdist(x, y) β, (c) The stable holonomy satisfying (iv) is unique. Hxy s can be extended to any y W s (x) using (iii). Similarly, for y W u (x), the unstable holonomy H u is obtained as Hxy u = lim (F y n ) 1 Fx n. n Proof. (a) We fix x M and denote x i = f i (x). Then for any y Wloc s (x) we have (4.1) (Fy n ) 1 Fx n = (Fy n 1 ) 1 ( ) (F yn 1 ) 1 F xn 1 F n 1 x = = (Fy n 1 ) 1 (Id + r n 1 ) Fx n 1 = = (Fy n 1 ) 1 Fx n 1 + (Fy n 1 ) 1 r n 1 Fx n 1 = = n 1 = Id + (Fy) i 1 r i Fx, i where (F yi ) 1 F xi = Id + r i. i=0 Since F is fiber bunched, there is θ < 1 such that F (x) F (x) 1 ν(x) β < θ for every x in M. For the function ν we denote its trajectory product by ν i (x) = ν(x)ν(fx)... ν(f i 1 x) = ν(x 0 )ν(x 1 )... ν(x i 1 ), i N. Then one can estimate dist(f n x, f n y) dist(x, y)ν n (y), see e.g. [BW, Lemma 1.1]. Lemma 4.3. There is C 1 such that for every x M, y Wloc s (x), and i 0, (Fy) i 1 Fx i C 1 θ i ν i (y) β. Proof. Using Hölder continuity of F we obtain We estimate F xk F yk 1 + F x k F yk F yk 1 + C 2 (dist(x k, y k )) β. (F i y) 1 F i x (F y ) 1 (F y1 ) 1 (F yi 1 ) 1 F x F x1 F xi 1 i 1 F yk (F yk ) 1 k=0 i 1 k=0 i 1 F xk F yk < k=0 i 1 θ ν(y k ) β k=0 ( 1 + C 2 (dist(x k, y k )) β). Since the distance between x n and y n decreases exponentially, the second product is uniformly bounded and we obtain (Fy) i 1 Fx i C 1 θ i ν i (y) β. Since F is Hölder continuous with exponent β, we have (4.2) r i = (F yi ) 1 F xi Id (F yi ) 1 F xi F yi C 3 dist(x i, y i ) β C 3 (C 4 dist(x, y) ν i (y)) β

8 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 8 It follows from (4.2) and Lemma 4.3 that for every i 0, (4.3) (F i y) 1 r i F i x (F i y) 1 F i x r i C 1 θ i ν i (y) β C 3 C β 4 dist(x, y) β ν i (y) β = C 5 dist(x, y) β θ i. Using (4.1) and (4.3), we obtain Id (F n y ) 1 F n x n 1 n 1 (Fy) i 1 r i Fx i C 5 dist(x, y) β θ i C dist(x, y) β. i=0 i=0 (b) It follows from (4.1) and (4.3) that (F n+1 y ) 1 F n+1 x (F n y ) 1 F n x = (F n y ) 1 r n F n x C 5 dist(x, y) β θ n. Therefore {(Fy n ) 1 Fx n } is a Cauchy sequence, and hence it has a limit Hxy s : E x E y. Since the convergence is uniform on the set of pairs (x, y) where y Wloc s (x), the map H s is continuous. Clearly, the maps Hxy s are linear and satisfy Hxx s = Id and Hyz s Hxy s = Hxz. s It follows from (a) that Hxy s Id Cdist(x, y) β. We also have Hxy s = lim (Fy n ) 1 (F k n f n y ) 1 F k n f n x F x n = (Fy n ) 1 Hf s n x f n y Fx n. k (c) Suppose that H 1 and H 2 are two stable holonomies satisfying Hxy 1,2 Cdist(x, y) β. Then using Lemma 4.3 we obtain H 1 xy H 2 xy = (F n y ) 1 (H 1 f n x f n y H 1 f n x f n y) F n x C 1 θ n ν n (y) β Cdist(f n x, f n y) β C 6 θ n 0 as n, Id and hence H 1 = H Proof of Theorem 3.1. 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 2.3. Let τ be an F -invariant µ-measurable conformal structure on E. We first show that τ is essentially invariant under stable and unstable holonomies of F. Proposition 4.4. Suppose that H s is a stable holonomy for a linear cocycle F. If τ is a measurable F -invariant conformal structure then τ is essentially H s -invariant, i.e. there is a set G M of full measure such that τ(y) = H s xy(τ(x)) for all x, y G such that y W s loc(x). Proof. We denote by F n x (τ(x)) the push forward of τ(x) from C(x), the space of conformal structures on E x, to C(f n x). Since the conformal structure τ is F -invariant,

9 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 9 and F n y induces an isometry, we obtain dist (τ(y), H s xy(τ(x))) = dist(f n y (τ(y)), F n y H s xy(τ(x))) = = dist(τ(y n ), H s x ny n F n x (τ(x))) = dist(τ(y n ), H s x ny n (τ(x n ))) dist(τ(y n ), τ(x n )) + dist(τ(x n ), H s x ny n (τ(x n ))). Since 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 hence bounded. 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. Suppose that both x and y are in G. Then there exists a sequence {n i } such that x ni S and y ni S. Since y Wloc s (x), dist(x n i, y ni ) 0 and hence dist(τ(x ni ), τ(y ni )) 0 by uniform continuity of τ on S. Since H is continuous, it is uniformly continuous on compact sets. Hence Hx s ni y ni Id 0 and, by the lemma below, dist(τ(x ni ), H s x ni y ni (τ(x ni ))) 0. We conclude that dist (τ(y), H s xy(τ(x))) = 0 and thus τ is essentially H s -invariant. Lemma 4.5. [KS10, Lemma 4.5] Let σ be a conformal structure on R d and A be a linear transformation of R d sufficiently close to the identity. Then dist (σ, A(σ)) k(σ) A Id, 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. Similarly, τ is essentially H u -invariant. Since the stable and unstable holonomies of F are continuous we conclude that τ is essentially uniformly continuous along W s and W u. Since the base system f is center bunched and accessible this implies continuity of τ on M by [ASV, Theorem E] or [W, Theorem 4.2] Proof of Proposition 3.2. We use the following proposition from [KS10] which relies on observations of D. Sullivan [Su] and P. Tukia [T]. We include the proof for completeness. Recall that a measurable conformal structure τ on E is called bounded if the distance between τ(x) and τ 0 (x) is essentially bounded on M for a continuous conformal structure τ 0 on E. Proposition 4.6. Let f be a homeomorphism 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.

10 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 10 Proof. Let τ 0 be a continuous conformal structure on E. We consider the set S(x) = { (F n x ) 1 (τ 0 (f n x)) : n Z } in C(x), the space of conformal structures on E x. Since F is uniformly quasiconformal, the sets S(x) have uniformly bounded diameters. Since the space C(x) has nonpositive 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) = { (Fx n ) 1 (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. Under our standing assumptions, Theorem 3.1 implies that τ is continuous. We can normalize it by a continuous function on M to obtain a Riemannian metric with respect to which F is conformal Proof of Theorem 3.3. Let E be a measurable F -invariant subbundle of E with dim E x = d. We consider a fiber bundle G over M whose fiber over x is the Grassman manifold G x of all d -dimensional subspaces in E x. Then F induces the cocycle F : G G over f with diffeomorphisms F x : G x G fx depending continuously on x in smooth topology. The stable holonomy H s for F induces a stable holonomy H s for F. Similarly, to the linear case, this is a family of diffeomorphisms H xy s : G x G y that satisfies properties (ii) and (iii) Definition 4.1 and depends continuously on x and y Wloc s (x). Similarly H u induces the unstable holonomy H s for F. The sub-bundle E gives rise to a µ-measurable F -invariant section φ : M G. We take m to be the lift of µ to the graph Φ of φ, i.e. for a set X G we define m(x) = µ(π(x Φ)), where π : G M is the projection. Equivalently, 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 is F -invariant. Lemma 4.7. [KS10, Lemma 4.6] There exists C > 0 such that for any x M, subspaces ξ, η G x and n Z we have (4.4) dist( F n x (ξ), F n x (η)) C K F (x, n) dist(ξ, η) The definitions of K(x, n), λ + (F, µ), and λ (F, µ) yield that for µ almost all x (4.5) 1 lim n n lim n 1 log K(x, n) = lim n n log( F x n (Fx n ) 1 ) = 1 n log F n x lim n 1 n log (F n x ) 1 1 = λ + (F, µ) λ (F, µ) = 0.

11 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 11 Hence Lemma 4.7 implies that Lyapunov exponent of F along the fiber is zero m a.e. This together with existence of the stable and unstable holonomies for F allows us to apply [ASV, Theorem C] to the measure m and conclude that there exists a system of conditional measures m x on G x for m which are holonomy invariant and depend continuously on x M in the weak topology. Since the conditional measures m x and m x coincide for all x in a set X M of full µ measure, we see that m x = m x is the atomic measure at φ(x) for all x X. Since X is dense we obtain that m x is atomic for all x M. Indeed, for any x M we can take a sequence X x i x and assume by compactness of G that φ(x i ) converge to some ξ G x. This implies that m xi = m xi converge to the atomic measure at ξ, which therefore coincides with m x by continuity of the family { m x }. Denoting φ(x) = supp m x for x M, we obtain a continuous section φ which coincides with φ on X. This shows that E coincides µ-almost everywhere with a continuous subbundle which is invariant under the stable and unstable holonomies Proof of Theorem 3.4. We use the following particular case of Zimmer s Amenable Reduction Theorem: [HKt, Corollary 1.8], [BP, Theorem 3.5.9] Let f be an ergodic transformation of a measure space (X, µ) and let F : X GL(d, R) be a measurable function. Then there exists a measurable function C : X GL(d, R) such that the function G(x) = C 1 (fx)f (x)c(x) takes values in a maximal amenable subgroup of GL(d, R). It is known that any maximal amenable subgroup of GL(d, R) is conjugate to a group of block-triangular matrices of the form A A A = A k where each diagonal block A i is a scalar multiple of a d i d i orthogonal matrix and d d k = d. Since E can be trivialized on a set of full µ-measure [BP, Proposition 2.1.2], we can measurably identify E with M R d and view F as a function M GL(d, R). Thus can we apply the Amenable Reduction Theorem to F and obtain a measurable coordinate change function C : M GL(d, R) such that A(x) = C 1 (fx)f (x) C(x) is of the above form for µ-almost all x M. Note that the subbundle V i spanned by the first d 1 + +d i coordinate vectors in R d is A-invariant for i = 1,..., k. Denoting E i x = C(x)V i we obtain the corresponding flag of measurable F -invariant subbundles E 1 E 2 E k = E with dim E i = d d i. By Theorem 3.3 the subbundles E i are continuous. Since A 1 (x) is a scalar multiple of a d d orthogonal matrix for µ-a.e. x, we conclude that the restriction of F to E 1 is

12 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 12 conformal with respect to the push forward by C of the standard conformal structure on V 1. This gives a measurable F -invariant conformal structure τ 1 on E 1. Since F preserves E 1, so does the stable holonomy H s. Hence H s induces a stable holonomy H s,1 for the restriction of F to E 1. By Proposition 4.4 the conformal structure τ 1 is essentially invariant under H s,1. Similarly we obtain essential invariance of τ 1 under the unstable holonomy H u,1. This yields continuity of τ 1 on M as in the end of the proof of Theorem 3.1. Similarly, we can consider continuous factor-bundle E i /E i 1 over M with the natural induced cocycle F (i). Since the matrix of the cocycle induced by A on V i /V i 1 is A i, it preserves the standard conformal structure. Hence F (i) preserves the corresponding measurable conformal structure τ i on E i /E i 1. Again the holonomies H s and H u induce continuous holonomies for F (i) on E i /E i 1. As above, we conclude that τ i is essentially invariant under these holonomies and hence continuous. Remark 4.8. Suppose that E is a trivial bundle so that we can view the cocycle F as a function F : M GL(d, R). Suppose in addition that there exist d continuous vector fields X i such that E i is spanned by the first d d i of them. Then the theorem implies that F is continuously cohomologous to a cocycle with values in a maximal amenable subgroup of GL(d, R). Indeed, the vector fields give a continuous coordinate change function C : M GL(d, R) such that Ã(x) = C 1 (fx)f (x) C(x) has a block triangular form. Moreover, each diagonal block Ãi corresponds to the factor cocycle on E i /E i 1 and hence, by the theorem, it preserves a continuous conformal structure σ i on R d i, i.e. Ã i (x)(σ i (x)) = σ i (fx). Then the coordinates in R d i can be changed by the unique positive square root of the symmetric positive definite matrix that defines σ i (x). This coordinate change makes the diagonal blocks conformal. We note that this result does not hold without assuming existence of the vector fields X i. Even E = M R 2 can have a non-orientable invariant sub-bundle E 1, which makes continuous conjugacy to a triangular cocycle impossible [S] Subadditive sequences of functions. Proposition 4.9 plays a key role in the proof of Corollary 3.6. It removes extra assumptions from [R, Proposition 3.5], which was similar to a result in [Sch], but proved to be more useful for many applications. Let f be a homeomorphism of a compact metric space X. A sequence of continuous functions a n : X R is called subadditive if (4.6) a n+k (x) a k (x) + a n (f k x) for all x X and n, k N. For any Borel probability measure µ on X we denote a n (µ) = a X ndµ. If µ if f- invariant, (4.6) implies that a n+k (µ) a n (µ)+a k (µ), i.e. the sequence of real numbers {a n (µ)} is subadditive. It is well known that for such a sequence the following limit exists: χ(µ) := lim n a n (µ) n a n (µ) = inf n N n.

13 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 13 Also, by the Subaddititive Ergodic Theorem, if µ is ergodic then a n (x) (4.7) lim = χ(µ) for µ-almost all x X. n n Proposition 4.9. Let f be a homeomorphism of a compact metric space X and a n : X R be subadditive sequence of continuous functions. If χ(µ) < 0 for every ergodic invariant Borel probability measure µ for f, then there exists N such that a N (x) < 0 for all x X. Proof. We denote by M the set of f-invariant Borel probability measures on X. First we note that, by the Ergodic Decomposition, if χ(µ) < 0 for every ergodic µ M, then the same holds for every µ M. First we show that there exists K such that a K (µ) < c < 0 for all µ M. Since χ(µ) < 0 there exists n µ and c µ such that a nµ (µ) < 2c µ < 0. Since a n are continuous, for every µ M there is a neighborhood V µ in the weak topology such that a nµ (ν) < c µ for every ν V µ. We choose a finite cover {V µi, i I} of M and set R = max I n i and c = max I c i. Take any µ M and let i be such that µ V µi. For any K we can write K = kn i + r, where r < n i R, and by the subadditivity we get a K (µ) ka ni (µ) + a r (µ) < kc + a r (µ). Since a r are uniformly bounded for r < R, we conclude that a K (µ) < c < 0 provided that K, and hence k, are sufficiently large. Lemma Suppose that for a continuous function φ : X R, φ(µ) < c for all µ M. Then there exists n 0 such that 1 n 1 n i=0 φ(f i x) < c for all x X and n n 0. Proof. Suppose on the contrary that there exit sequences x j X and n j such that S j = 1 nj 1 n j i=0 φ(f i x) c. Note that S j = ψ(µ j ), where µ j = 1 nj 1 n j i=0 δ(f i x j ) is a probability measure. Using compactness of the set of probability measures on X we may assume, by passing to a subsequence if necessary, that µ j weak converges to a probability measure µ. Since the total variation norm f µ j µ j 2 n j it follows that the limit µ is f-invariant. On the other hand ψ(µ) = lim ψ(µ j ) c, which contradicts the assumption. Applying the lemma to a function a K satisfying a K (µ) < c < 0 we conclude that there is n 0 such that n 1 i=0 a K(f i x) < cn for all n n 0 and x X. Let n = Km n 0. Using subadditivity repeatedly, for i = 0,..., K 1 we obtain a n (f i x) a K (f i x) + a n K (f K+i x)... Adding these K inequalities, we get a K (f i x) + a K (f K+i x) + + a K ( f (m 1)K+i x ). a n (x) + a n (fx) + + a n (f K 1 x) n 1 a K (f i x) cn. i=0

14 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 14 Let N = n + K. For i = 0,..., K 1, we obtain a N (x) a i (x) + a n+k i (f i x) a i (x) + a n (f i x) + a K i (f n+i x) =: a n (f i x) + i (x), where we set a 0 (x) = 0. Let M = max{ i (x) : 0 i K 1, x M}. Adding the inequalities, we get K a N (x) a n (x) + a n (fx) + + a n (f K 1 x) + KM cn + KM and hence a N (x) cn/k + M. Since c < 0, taking m = n/k sufficiently large, we can ensure that a N (x) < 0 for all x Proof of Corollary 3.6. Applying Proposition 4.11 below with ξ = 0 we obtain that for every ɛ > 0 there exists C ɛ such that (4.8) F n x (F n x ) 1 C ɛ e ɛ n for all x M and n Z. We consider ɛ such that e ɛ < min ν(x) β. Then (4.8) implies that the cocycle F n over f n is fiber bunched for sufficiently large n. Hence Proposition 4.2 shows existence of the stable and unstable holonomies for F n. Any measurable F -invariant conformal structure is also F n -invariant, and hence by Theorem 3.1 it is continuous. Since F has only one Lyapunov exponent, so does F n, and it follows that any measurable F -invariant sub-bundle is continuous. In the proof of Theorem 3.4 we use the measurable amenable reduction for F and then use F n to show continuity of bundles and structures. Proposition Suppose that there exists ξ 0 such that λ + (F, µ) λ (F, µ) ξ for every ergodic f-invariant measure µ. Then for any ɛ > 0 there exists C ɛ such that (4.9) K F (x, n) = F n x (F n x ) 1 C ɛ e (ξ+ɛ) n for all x M and n Z. Proof. To simplify the notations we write K(x, n) for K F (x, n). For a given ɛ > 0 we apply Proposition 4.9 to the functions a n (x) = log K(x, n) (ξ + ɛ)n, n 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) for every x M and n, k 0. It follows that the sequence of functions {a n } is subadditive. Let µ be an ergodic f-invariant measure. As in (4.5) we obtain that 1 lim n n log K(x, n) = λ +(F, µ) λ (F, µ) ξ for µ-almost all x. It follows that a n (x) lim ɛ < 0 for µ-almost all x, n n and (4.7) implies that χ(µ) < 0 for the sequence {a n }. Therefore by Proposition 4.9 there exists N ɛ such that a Nɛ (x) < 0, i.e. K(x, N ɛ ) e (ξ+ɛ)nɛ for all x M.

15 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 15 For any n > 0, we write n = mn ɛ + r, 0 r < N ɛ, and estimate K(x, n) K(x, r) K(f r x, N ɛ ) K(f r+nɛ x, N ɛ ) K(f r+(m 1)Nɛ x, N ɛ ) K(x, r) e (ξ+ɛ)mnɛ C ɛ e (ξ+ɛ)n, where C ɛ = max {K(x, r) : x M, 1 r < N ɛ }. Since K(x, n) = K(f n x, n), we conclude that K(x, n) C ɛ e (ξ+ɛ) n for all x in M and n in Z Proof of Corollary 3.7. By Theorem 3.1, the conformal structure τ is continuous on M. Since the stable and unstable foliations are absolutely continuous with respect to the volume, it follows from Proposition 4.4 that τ(y) = Hxy(τ(x)) s for all x M and y Wloc s (x). By Proposition 4.2, Hs xy Id C dist(x, y) β, and hence by Lemma 4.5 dist(τ(x), τ(y)) C dist(x, y) β. The same holds for all y Wloc u (x). Now local α-hölder accessibility implies that τ is αβ-hölder on M. Hölder continuity of invariant sub-bundles, as well as the subsequent results, can be obtained similarly Proof of Theorem 3.9. For Hölder continuous cocycles over hyperbolic systems, the Lyapunov exponents of any ergodic invariant measure can be approximated by Lyapunov exponents of periodic measures [K, Theorem 1.4]. Hence λ + (F, µ) = λ (F, µ) for every ergodic f-invariant measure µ and Corollary 3.6 applies. Moreover, by Corollary 3.7 and Remark 3.8 the subbundles, factor bundles, and invariant conformal structures are Hölder continuous with exponent β. We normalize the invariant conformal structures on E 1 and on the factor bundles E i /E i 1 to get Hölder continuous norms. i, i = 1,..., k, with respect to which the corresponding cocycles F (i) are conformal. Clearly, the existence of the estimates in the theorem does not depend on the choice of a continuous norm, though the constant C does. We denote by a i (x) the scaling coefficients with respect to these norms: F (i) (v) i = a i (x) v i for all v E i /E i 1 (x). These functions are positive and Hölder continuous. Also, for each periodic point p = f n p the product a i (f n 1 p) a i (fp) a i (p) is the same for all i, as otherwise Fp n would have more than one Lyapunov exponent. By the Livšic theorem [L],[KtH, ] this implies that the functions a i are cohomologous, i.e. there exist positive Hölder continuous functions φ i such that a k (x) = a 1 (x)φ i (fx)φ i (x) 1, i = 2,..., k. Choosing new norms. i = φ 1 i. i, i = 2,..., k, we make a 1 the scaling coefficient for all F (i). Hence F (x) = a 1 (x) F (x) where the cocycle F (x) induces isometries on E 1 and on each factor bundle E i /E i 1. Now inductive application of the next proposition shows that F n (x) Dn k 1. Since the inverse cocycle F 1 has the same structure we also obtain F n (x) Dn k 1, and hence K F (x, n) Cn 2(k 1). Since K F (x, n) = K F (x, n), the first part of the theorem follows.

16 COCYCLES WITH ONE EXPONENT OVER PARTIALLY HYPERBOLIC SYSTEMS 16 If λ + (F, µ) = λ (F, µ) = 0 then a 1 (x) is cohomologous to constant 1 and, by rescaling the norm we obtain that F induces isometries on E 1 and on each factor bundle E i /E i 1. Hence the second part also follows from the next proposition. Proposition Let F : E E be a continuous linear cocycle over f and let V be an F -invariant continuous subbundle. Suppose that the factor cocycle F : E/V E/V is an isometry and that for some C and k the restriction F V = F V satisfies F n V (x) Cnj for all x and n N. Then there exists a constant D such that F n (x) Dn j+1 for all x and n N. Proof. We denote by P : E E/V the natural projection and by π : E V the orthogonal projection with respect to some Riemannian metric on E. Then for any x the map v (P (v), π(v)) identifies E(x) with E/V (x) V (x), and max{ (P (v), π(v) } gives a convenient continuous norm on E. Since the linear map (x) = (π F F V π)(x) : E(x) V (fx) is identically zero on V (x) we can write it as (x) = (x) P, where the linear map (x) : E/V (x) V (fx) depends continuously on x. Thus we have π F = F V π + P, P F n = F n P, and hence π F n = (F V π + P ) F n 1 = F V (π F n 1 ) + F n 1 P = = F V ((F V π + P ) F n 2 ) + F n 1 P = = F 2 V (π F n 2 ) + F V F n 2 P + F n 1 P = = n 1 = FV n π + F n i 1 V F i P. i=0 Let K be such that (x) K for all x. Since by the assumptions F i V Cni and F i = 1, we can estimate n 1 π F n (x) Cn j + C(n i 1) j K Cn j + ncn j K Dn j+1 i=0 for some constant D independent of n and x. Since P F n = F n P 1, we conclude that F n (x) = max { P F n (x), π F n (x) } Dn j+1. References [ASV] A. Avila, J. Santamaria, M. Viana. Cocycles over partially hyperbolic maps. Preprint. [BP] L. Barreira, Ya. Pesin. Nonuniform Hyperbolicity: Dynamics of systems with nonzero Lyapunov exponents. Encyclopedia of Mathematics and Its Applications, 115 Cambridge University Press.

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