Uniformly quasiconformal partially hyperbolic systems

arXiv:1601.07485v1 [math.DS] 27 Jan 2016

UNIFORMLY QUASICONFORMAL PARTIALLY HYPERBOLIC SYSTEMS CLARK BUTLER AND DISHENG XU A BSTRACT. We study smooth volume-preserving perturbations of the time-1 map of the geodesic flow ψt of a closed Riemannian manifold of dimension at least three with constant negative curvature. We show that such a perturbation has equal extremal Lyapunov exponents with respect to volume within both the stable and unstable bundles if and only if it embeds as the time-1 map of a smooth volumepreserving flow that is smoothly orbit equivalent to ψt . Our techniques apply more generally to give an essentially complete classification of smooth, volumepreserving, dynamically coherent partially hyperbolic diffeomorphisms which satisfy a uniform quasiconformality condition on their stable and unstable bundles and have either uniformly compact center foliation or are obtained as perturbations of the time-1 map of an Anosov flow.

1. I NTRODUCTION A surprising number of rigidity problems originally posed in negatively curved geometry turn out to have solutions that are dynamical in nature. We review one such story here: Sullivan proposed, following work of Gromov[17] and Tukia[33], that closed Riemannian manifolds of constant negative curvature and dimension at least 3 should be characterized up to isometry by the property that the geodesic flow acts uniformly quasiconformally on the unstable foliation[32]. Informally, the uniform quasiconformality property states that the flow does not distort the shape of metric balls inside of a given horosphere over a long period of time. Sullivan’s conjecture was partially confirmed by the work of Kanai [23] who showed that among contact Anosov flows the geodesic flows of constant negative curvature manifolds are characterized up to C1 orbit equivalence by a uniform quasiconformailty. Later the minimal entropy rigidity theorem of Besson, Courtois, and Gallot [5] completed the proof of Sullivan’s conjecture among many other outstanding conjectures in negatively curved geometry. From a geometric perspective this completes the story, but from a dynamical perspective this raises many new questions. Already in the work of Kanai we see that the dynamical version of this rigidity result holds for a larger class of Anosov flows than just geodesic flows. Sadovskaya initiated a program to extend these results further to smooth volume-preserving Anosov flows and diffeomorphisms [29], which was completed in a series of works by Fang ([12], [13], [14]) who obtained the following remarkable result: all smooth volume-preserving Anosov flows which are uniformly quasiconformal on the stable and unstable foliation are smoothly orbit equivalent either to the suspension of a hyperbolic toral automorphism or the geodesic flow on the unit tangent bundle of a constant negative curvature closed Riemannian manifold. Thus we see that not even the contact structure of the flow is necessary to obtain dynamical rigidity for uniformly quasiconformal Anosov flows. 1

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In a different direction one can ask whether the uniform quasiconformality condition can be relaxed to a condition that is more natural from the perspective of ergodic theory. This direction was pursued by the first author, who showed that for geodesic flows of 14 -pinched negatively curved manifolds, uniform quasiconformality can be derived from the significantly weaker dynamical condition of equality of all Lyapunov exponents with respect to volume on the unstable bundle [10]. Our principal goal is to show that for all of the rigidity phenomena derived from uniform quasiconformality above, not even the structure of an Anosov flow is necessary. Let us be more precise: consider a closed Riemannian manifold X of constant negative curvature with dim X ≥ 3. Let T 1 X be the unit tangent bundle of X and let ψt : T 1 X → T 1 X denote the time-t map of the geodesic flow. This flow preserves a smooth volume m on T 1 X known as the Liouville measure. Consider any smooth diffeomorphism f which is C1 -close to the time-1 map ψ1 and which preserves the volume m. By the work of Hirsch, Pugh, and Shub [19], f is partially hyperbolic, meaning that there is a D f -invariant splitting T ( T 1 X ) = Eu ⊕ Ec ⊕ Es where Eu is exponentially expanded by D f , Es is exponentially contracted by D f , and the behavior of D f on the 1-dimensional center direction Ec (which is close to the flow direction for ψt ) is dominated by the expansion and contraction on Eu and Es respectively. We give a more precise definition in Section 2. We then choose a continuous norm k · k on Eu and define the extremal Lyapunov exponents of f on Eu by Z λu+ ( f ) = inf

n ≥1 M

λu− ( f ) = sup n ≥1

Z

M

log k D f n | Eu k dm,

log k( D f n | Eu )−1 k−1 dm.

We define λs+ ( f ) and λs− ( f ) similarly with Es replacing Eu . T HEOREM 1. There is a C2 -open neighborhood U of ψ1 in the space of C ∞ volumepreserving diffeomorphisms of T 1 X such that if f ∈ U and both of the equalities λu+ ( f ) = λu− ( f ) and λs+ ( f ) = λs− ( f ) hold then there is a C ∞ volume-preserving flow ϕt with ϕ1 = f . Furthermore ϕt is smoothly orbit equivalent to ψt . This theorem improves on the techniques used in the previous rigidity theorems in several fundamental ways. We are able to deduce uniform quasiconformality of the action of D f on Eu and Es from equality of the extremal Lyapunov exponents entirely outside of the geometric context considered in [10] by using new methods. We then use this uniform quasiconformality to completely reconstruct the smooth flow ϕt in which f embeds as the time-1 map. We emphasize that for a typical perturbation f of ψ1 the foliation W c tangent to Ec (which is our candidate for the flowlines of ϕt ) is only a continuous foliation of T 1 X with no transverse smoothness properties. This is one of the many reasons that strong rigidity results in the realm of partially hyperbolic diffeomorphisms are quite rare. Our inspiration was an impressive rigidity theorem of Avila, Viana and Wilkinson which overcame this obstacle to show that if we take X to be a negatively curved surface instead and f a C1 -small enough C ∞ volume-preserving perturbation of the time-1 map ψ1 such that the center foliation of f is absolutely continuous, then f is also the time-1 map of a smooth volume-preserving flow [3]. Our result can be viewed in an appropriate sense as the higher dimensional analogue of this theorem.

UNIFORMLY QUASICONFORMAL PARTIALLY HYPERBOLIC SYSTEMS

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We now explain the organization of the paper. The techniques used in the proof of Theorem 1 have much more general applications which can also be applied to the study of C ∞ volume-preserving partially hyperbolic diffeomorphisms which satisfy a uniform quasiconformality condition on their stable and unstable bundles and either have uniformly compact center foliation or are obtained as a perturbation of the time-1 map of an Anosov flow. These results are stated in Theorems 2 and 4 and Corollary 3 of Section 2 after we introduce some necessary terminology. In Section 3 we show that under a Lyapunov stability type assumption on the action of a partially hyperbolic diffeomorphism f on its center foliation, uniform quasiconformality implies that the holonomy maps of the center-stable and center-unstable foliations of f are quasiconformal. We use this to show that the center foliation of f is absolutely continuous. In Section 4 we prove Theorems 2 and 4 and Corollary 3 by using the quasiconformality obtained in Section 3 to improve the regularity of the center-stable and center-unstable holonomy maps to show that these maps are actually C ∞ . In Section 5 we finish the proof of Theorem 1 by deducing uniform quasiconformality from the condition of equality of extremal Lyapunov exponents. The arguments in Section 5 do not rely on the results of Sections 3 and 4 and may be read independently of the rest of the paper. Acknowledgments: We thank Amie Wilkinson for numerous useful discussions regarding the content of this paper. These discussions resulted in significant simplification of the proof of Theorem 1. The second author would like to thank his director of thesis Professor Artur Avila for his supervision and encouragement. This work was partially completed while both authors were visiting the Instituto Nacional de Matemática Pura e Aplicada, the second author being supported by réseau franco-brésilien en mathématiques. The first author was supported by the National Science Foundation Graduate Research Fellowship under Grant # DGE-1144082. 2. S TATEMENT OF R ESULTS C1

A diffeomorphism f : M → M of a closed Riemannian manifold M is partially hyperbolic if there is a D f -invariant splitting TM = Es ⊕ Ec ⊕ Eu of the tangent bundle of M such that for some k ≥ 1, any x ∈ M, and any choice of unit vectors vs ∈ Exs , vc ∈ Exc , vu ∈ Exu ,

k D f k (vs )k < 1 < k D f k (vu )k, k D f k (vs )k < k D f k (vc )k < k D f k (vu )k. By modifying the Riemannian metric on M if necessary we can always assume k = 1 in the above definition. We will always require that the bundles Es and Eu are nontrivial. We will also always require that M is connected. We define for x ∈ M, n ∈ Z, K u ( x, n) =

sup{k D f n (vu )k : vu ∈ Eu ( x ), kvu k = 1} , inf{k D f n (vu )k : vs ∈ Eu ( x ), kvu k = 1}

and define K s ( x, n) similarly with Eu replaced by Es . The quantities K u and K s measure the failure of the iterates of D f to be conformal on the bundles Eu and Es respectively. We say that f is uniformly u-quasiconformal if dim Eu ≥ 2 and K u is uniformly bounded in x and n. Similarly we say that f is uniformly s-quasiconformal if dim Es ≥ 2 and K s is uniformly bounded in x and n. If f is both uniformly

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u-quasiconformal and s-quasiconformal then we simply say that f is uniformly quasiconformal. Our definition of uniform quasiconformality for partially hyperbolic systems extends previous definitions of uniform quasiconformality which were considered for Anosov diffeomorphisms and Anosov flows. If the center bundle Ec is trivial or if f embeds as the time-1 map of an Anosov flow (so that Ec is tangent to the flow direction) then these definitions reduce to the standard notions of uniform quasiconformality for Anosov systems defined by Sadovskaya [29]. If dim Eu = 1 then K u ≡ 1 for any choice partially hyperbolic f , so the boundedness of K u does not give new information about f . This is the reason we require dim Eu ≥ 2 in the definition of uniform u-quasiconformality; the uniform quasiconformality conditions are only interesting when the bundles in question have dimension at least 2. We define a C ∞ diffeomorphism f to be volume-preserving there is an f -invariant probability measure m on M which is smoothly equivalent to the Riemannian volume. It is not hard to show using Kingman’s subadditive ergodic theorem [25] that when f is ergodic with respect to m we have log K u ( x, n) = λu+ ( f ) − λu− ( f ) for m-a.e. x ∈ M. n→∞ n lim

We refer to [22] for more details on this equality. Thus asymptotic subexponential growth of K u is equivalent to the equality λu+ ( f ) = λu− ( f ). Theorem 1 asks in part for the deduction of a uniform bound K u ( x, n) ≤ C from this asymptotic subexponential growth condition. Fang proved that all volume-preserving C ∞ uniformly quasiconformal diffeomorphisms are C ∞ conjugate to a hyperbolic toral automorphism [12]. This generalized the classification result of Sadovskaya which held under the additional assumption that f was symplectic [29]. Theorem 2 and Corollary 3 below extend this classification to cover a certain C1 -open set of C ∞ volume-preserving partially hyperbolic diffeomorphisms. Before stating these theorems we need to introduce a few more basic notions from partially hyperbolic dynamics. We refer the reader to [9] for a deeper discussion of partial hyperbolicity and the properties that follow. We will assume for the rest of the paper that f is C ∞ . Then the bundles Es and Eu are tangent to foliations W s and W u known respectively as the stable and unstable foliations. These foliations have C ∞ leaves but the distributions Es and Eu which they are tangent to are themselves typically only Holder ¨ continuous. We say that f is dynamically coherent if there are also f invariant foliations W cs and W cu with C1 leaves which are tangent to Es ⊕ Ec and Ec ⊕ Eu respectively. It follows by intersecting these two foliations that there is an f -invariant foliation W c with C1 leaves which is tangent to Ec . For r ≥ 1 we write that a map is Cr +α if it is Cr and the rth-order derivatives are uniformly Holder ¨ continuous of exponent α > 0. For a foliation W of an ndimensional smooth manifold M by k-dimensional submanifolds we define W to be a Cr +α foliation if for each x ∈ M there is an open neighborhood Vx of x and a Cr +α diffeomorphism Ψ x : Vx → D k × D n−k ⊂ R n (where D j denotes the ball of radius 1 centered at 0 in R j ) such that Ψ x maps W to the standard smooth foliation of D k × D n−k by k-disks D k × {y}, y ∈ D n−k . This is the notion of regularity of a


5

foliation considered by Pugh, Shub, and Wilkinson in their analysis of regularity properties of invariant foliations for partially hyperbolic systems [27]. We say that f is r-bunched if there is some k ≥ 1 such that for any unit vectors vs ∈ Exs , vc ∈ Exc , vu ∈ Exu ,

k D f k (vs )kk D f k (vc )kr < 1, k D f k (vu )kk D f k (vc )kr < 1, and furthermore for any pair of unit vectors v1c , v2c ∈ Ec ( x ),

k D f k (vs )k · k D f k (v1c )kr k D f k (vu )k · k D f k (v1c )kr < 1, < 1. k D f k (v2c )k k D f k (v2c )k The case r = 1 corresponds to the center-bunching condition considered by Burns and Wilkinson in their proof of the ergodicity of accessible, volume-preserving, center-bunched C2 partially hyperbolic diffeomorphisms [9]. When f is smooth and dynamically coherent, the r-bunching inequalities imply that the foliations W cs and W cu have uniformly Cr +α leaves for some α > 0 [27]. We say that f is ∞-bunched if it is r-bunched for every r ≥ 1. If f is ∞-bunched and dynamically coherent then the leaves of W cs and W cu are C ∞ . A natural situation in which the ∞-bunching condition holds is when there is a continuous Riemannian metric on Ec with respect to which D f | Ec is an isometry. More generally if f is center bunched, accessible, and volume-preserving and all of the Lyapunov exponents of f with respect to volume on Ec are zero, then by the results of Kalinin and Sadovskaya f is ∞-bunched [22]. Finally, when f is dynamically coherent we say that the center foliation W c is uniformly compact if all of the leaves of W c are compact with a uniform bound on their diameters in the induced Riemannian metric on Ec from TM. T HEOREM 2. Let f be a C ∞ dynamically coherent volume-preserving partially hyperbolic diffeomorphism. Suppose that f is uniformly quasiconformal, r-bunched for some r ≥ 1, and has uniformly compact center foliation. Then (1) There is an α > 0 such that W cs , W c , and W cu are Cr +α foliations of M and both W s and W u are C1+α foliations of M, (2) There is a closed Cr Riemannian manifold N, a Cr +α submersion π : M → N with fibers given by the W c foliation, and a Cr +α volume-preserving uniformly quasiconformal Anosov diffeomorphism g : N → N such that g ◦ π = π ◦ f . (3) If f is ∞-bunched then the statements of (1) and (2) are true with r = ∞. Furthermore W s and W u are also C ∞ foliations of M and g may be taken to be a hyperbolic automorphism of a torus N. When the center bundle of f is one-dimensional we can derive a sharper result as a corollary. We define a smooth diffeomorphism f : M → M to be an isometric extension of another smooth diffeomorphism g : N → N if there is a smooth submersion π : M → N satisfying g ◦ π = π ◦ f and such that this submersion has compact fibers and there is a smoothly varying family of Riemannian metrics {d x } x ∈ N on the fibers {π −1 ( x )} x ∈ N such that the induced maps f x : π −1 ( x ) → π −1 ( g( x )) are isometries with respect to these metrics. C OROLLARY 3. Let f be a C ∞ dynamically coherent volume-preserving partially hyperbolic diffeomorphism with dim Ec = 1. Suppose that f is uniformly quasiconformal and that f has uniformly compact center foliation. Then f is an isometric extension of a hyperbolic toral automorphism.

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We make some comments on Theorem 2 and Corollary 3 before proceeding. If M = N × S for pair of compact smooth manifolds N and S, g0 : N → N is an Anosov diffeomorphism and f 0 : M → M is a smooth extension of g0 such that f 0 is an r-bunched volume-preserving partially hyperbolic diffeomorphism (r ≥ 1) with center leaves of the form W c ( x, s) = { x } × S for ( x, s) ∈ N × S, then the center leaves of f 0 are normally hyperbolic and uniformly compact. Thus there is a C1 open neighborhood U of f 0 in the space of C ∞ volume-preserving diffeomorphisms of M such that if f ∈ U then f is dynamically coherent and has uniformly compact center foliation. This follows from the theory of normally hyperbolic invariant manifolds developed by Hirsch, Pugh, and Shub [19]. Hence, with the exception of the uniform quasiconformality hypothesis, the hypotheses of Theorem 2 and Corollary 3 are not particularly restrictive among partially hyperbolic diffeomorphisms. The limiting factor for the smoothness of the foliations W cs and W cu in Theorem2 turns out to be the regularity of the leaves of the foliations themselves. Corollary 26 below shows that the holonomy maps of W cs and W cu between local unstable/local stable leaves respectively are C ∞ . In fact they are analytic maps in an appropriate choice of coordinates. The r-bunching inequalities in the hypotheses of Theorem 2 are only required to obtain that the leaves of the foliations W cs and W cu are Cr +α ; they are never used directly in the proof. The classification results of Sadovskaya and Fang for C ∞ volume-preserving uniformly quasiconformal Anosov diffeomorphisms rely fundamentally on the smooth conjugacy classification theorem of Benoist and Labourie for Anosov diffeomorphisms with C ∞ stable and unstable foliations [4]. However, the regularity of the uniformly quasiconformal Anosov diffeomorphism g obtained from Theorem 2 is limited by the regularity of the center foliation, which in turn is limited by the r-bunching hypothesis. The most we can obtain with our methods is that g is Cr +α . This is the reason we can only derive the stronger results of part (3) of Theorem 2 under the ∞-bunching hypothesis on f . Finally we observe that the conclusions of Theorem 2 imply in particular that the center foliation of f is absolutely continuous with respect to volume. We refer to Definition 12 below for our definition of absolute continuity of foliation. Pugh and Wilkinson showed that an isometric extension of a hyperbolic automorphism of the two-dimensional torus T2 can be perturbed to make the center Lyapunov exponent nonzero and thus cause the center foliation to fail to be absolutely continuous [30]. Corollary 3 shows that it is not possible to make such a perturbation of an isometric extension of a uniformly quasiconformal hyperbolic automorphism of a higher dimensional torus which maintains uniform quasiconformality on both the stable and unstable bundles. For our next theorem we consider partially hyperbolic diffeomorphisms which are obtained as perturbations of the time-1 maps of Anosov flows. Let ψt : M → M be a C ∞ volume-preserving Anosov flow with stable and unstable bundles of dimension at least 2. ˆ of M such that the lift of ψt to an T HEOREM 4. Suppose that there is a finite cover M ˆ Anosov flow ψt : M → M has no periodic orbits of period ≤ 2. Then there is a C1 -open neighborhood U of ψ1 in the space of volume-preserving C ∞ diffeomorphisms of M such that if f ∈ U and f is uniformly quasiconformal then the


7

invariant foliations W cs , W c , and W cu of f are C ∞ and there is a C ∞ volume-preserving uniformly quasiconformal Anosov flow ϕt : M → M with ϕ1 = f . Before making further comments on this theorem we recall the notion of orbit equivalence of Anosov flows. Two C ∞ Anosov flows ϕt , ψt : M → M are Cr orbit equivalent (r ∈ [0, ∞]) if there is a Cr map h : M → M such that for every x ∈ M and t ∈ R, h( ϕt ( x )) lies on the ψt -orbit of h( x ). From the classification of C ∞ volume-preserving uniformly quasiconformal Anosov flows obtained by Fang [14] we conclude that the flow ϕt obtained in the conclusion of Theorem 4 is C ∞ orbit equivalent either to the suspension flow of a hyperbolic toral automorphism or the geodesic flow on the unit tangent bundle of a constant negative curvature Riemannian manifold. ˆ for which the lift ψˆ t The hypothesis in Theorem 4 that there is a finite cover M has no periodic orbits of period ≤ 2 is very mild. It always holds if ψt is C0 orbit equivalent to the suspension flow of an algebraic Anosov diffeomorphism or the geodesic flow of a closed negatively curved Riemannian manifold. We expect that Theorem 4 holds without this hypothesis, however this hypothesis does simplify some constructions in the proofs, particularly in Section 4.1. We recall now the definitions of su-paths and accessiblity for a partially hyperbolic diffeomorphism which will play a crucial role in the proof of Theorem 1 in Section 5. For a partially hyperbolic diffeomorphism f : M → M an su-path in M is a piecewise C1 curve γ in M such that γ decomposes into finitely many C1 subcurves γxi xi+1 connecting xi to xi+1 and such that each curve γxi xi+1 is contained in a single W s or W u leaf. We define f to be accessible if any two points in M can be joined by an su-path. A notable aspect of Theorems 2 and 4 and Corollary 3 is that their hypotheses do not include any accessibility or ergodicity assumptions on f with respect to the volume m. This requires us to take some additional care at certain points in the proof. The accessibility hypotheses is used strongly in the rigidity theorem of Avila-Viana-Wilkinson and ergodicity with respect to volume is used in the classification results of Sadovskaya and Fang. The results of Corollary 3 and Theorem 4 suggest that it may be possible to obtain a global smooth classification of C ∞ volume-preserving, dynamically coherent, uniformly quasiconformal partially hyperbolic diffeomorphisms with onedimensional center in terms of the classification of uniformly quasiconformal Anosov diffeomorphisms and Anosov flows. We give an example which illustrates some of the difficulties in obtaining a classification beyond these theorems. Consider the 5 × 5 integer matrix   0 1 0 0 0  0 0 1 0 0    A :=   0 0 0 1 0 ,  0 0 0 0 1 −1 1 0 −3 1 and let f A : T5 → T5 be the induced linear map of A on the 5-torus T5 = R5 /Z5 . By numerical computation the five complex eigenvalues of A satisfy

| λ1 | = | λ2 | > | λ3 | > 1 > | λ4 | = | λ5 | , λ1 = λ2 ∈ / R,

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λ3 = λ4 ∈ / R. Thus f A is a hyperbolic toral automorphism which may also be viewed as a partially hyperbolic diffeomorphism with splitting TT5 = Eu ⊕ Ec ⊕ Es , where Eu is the real part of the complex eigenspaces corresponding to the pair of conjugate complex eigenvalues λ1 and λ2 , Ec is the eigenspace corresponding to λ3 , and Es is the real part of the complex eigenspaces corresponding to λ4 and λ5 . We conclude f A is a smooth, volume-preserving, dynamically coherent uniformly quasiconformal partially hyperbolic diffeomorphism with one-dimensional center. We then pose the following problem, P ROBLEM 5. Is there a C1 -open neighborhood U of f A in the space of smooth volumepreserving diffeomorphisms of T5 such that if f ∈ U is uniformly quasiconformal then the invariant foliations W cs , W c , and W cu of f are smooth? We expect the answer to Problem 5 to be “no” but the difficulty of constructing nontrivial uniformly quasiconformal perturbations of f A is a significant obstruction to confirming our suspicions. We note that each f ∈ U is an Anosov diffeomorphism if U is chosen small enough. 3. Q UASICONFORMALITY

OF THE

C ENTER H OLONOMY

We fix M to be a closed Riemannian manifold with distance d and let f : M → M be a C ∞ dynamically coherent partially hyperbolic diffeomorphism. For ∗ ∈ {s, c, u, cu, cs} we let d∗ denote the induced Riemannian metric on the leaves of the foliation W ∗ . We write W ∗ ( x ) for the leaf of W ∗ passing through x ∈ M. We write diam∗ for the diameter of a subset of W ∗ measured with respect to the d∗ metric. For r > 0 we write Wr∗ ( x ) for the open ball of radius r in W ∗ ( x ) centered at x in the d∗ metric. We can find small constants R ≥ r > 0 with the property that for any x ∈ M, y ∈ Wrcs ( x ) and z ∈ Wru ( x ) the local leaves Wrcs (z) and W Ru (y) intersect in exactly one point which we denote by hcs xy ( z ). This defines the local center-stable holonomy map between local unstable leaves of f . Similarly we require that if x ∈ M, y ∈ Wrcu ( x ) and z ∈ Wrs ( x ) then the local leaves Wrcu (z) and W Rs (y) intersect in exactly one point which we denote by h cu xy ( z ), and use this to define the local center-unstable holonomy. We introduce some useful shorthand related to these holonomy maps. The center-stable holonomy maps and center-unstable holonomy maps will sometimes be referred to as cs-holonomy and cu-holonomy respectively. When the domain and range are understood we will omit the subscripts on h cs and hcu . We will write ∗ ( x ) for any open ball of the form W ∗ ( x ) with r ≤ t ≤ R. Hence it makes sense Wloc t u ( x ) → W u ( y ) for the cs-holonomy maps. in our shorthand to write hcs : Wloc loc Our starting point is the following non-stationary smooth linearization lemma of Sadovskaya applied to the unstable foliation W u which is uniformly contracted by f −1 , P ROPOSITION 6. [29, Proposition 4.1] Suppose that f is a C ∞ uniformly u-quasiconformal partially hyperbolic diffeomorphism. Then for each x ∈ M there is a C ∞ diffeomorphism Φ x : Exu → W u ( x ) satisfying (1) Φ x ◦ D f x = f ◦ Φ f ( x ), (2) Φ x (0) = x and D0 Φ x is the identity map,


9

(3) The family of diffeomorphisms {Φ x } x ∈ M varies continuously with x in the C ∞ topology. The family {Φ x } x ∈ M satisfying (1), (2), and (3) is unique. The bundle Eu is a Holder ¨ continuous subbundle of TM with some Holder ¨ exponent β > 0 [27]. Therefore the restriction D f | Eu of the derivative of f to the unstable bundle is a Holder ¨ continuous linear cocycle over f in the sense of Kalinin-Sadovskaya [22]. For x, y ∈ M two nearby points we let Ixy : Exu → Eyu be a linear identification which is β-Holder ¨ close to the identity. The diffeomorphism f is uniformly u-quasiconformal if and only if, in the terminology of [22], the cocycle D f | Eu is uniformly quasiconformal. The following proposition thus applies to D f |Eu , u ( x ), the limit P ROPOSITION 7. [22, Proposition 4.2] For y ∈ Wloc u , lim D f fn−n y ◦ I f −n x f −n y ◦ D f x−n | Eu := Hxy

n→∞

exists uniformly in x and y and defines a linear map from Exu to Eyu with the following properties for x, y, z ∈ M, u = Id and H u ◦ H u = H u ; (1) Hxx yz xy xz u = Dfn u (2) Hxy ◦ H ◦ D f x−n for any n ≥ 0. f −n y f −n x f −n y u − I k ≤ Cd ( x, y ) β, β the exponent of H¨ (3) k Hxy older continuity for Eu . xy

Furthermore H u is the unique collection of linear identifications with these properties. s s ( x ) then the limit lim −n n Similarly if y ∈ Wloc n → ∞ D f y ◦ I f n x f n y ◦ D f x | E u : = Hxy exists u u u s and gives a linear map from Ex to Ey with analogous properties. H and H are known as the unstable and stable holonomies of D f | Eu respectively. Using property (2) of the unstable and stable holonomies of D f | Eu from Proposition 7 we may uniquely extend H u and H s to be defined for any y ∈ W u ( x ) and any y ∈ W s ( x ) respectively. The transition maps between the charts given by Proposition 6 are affine with derivatives given by the unstable holonomy H u , P ROPOSITION 8. Suppose that f is uniformly u-quasiconformal and let {Φ x } x ∈ M be the 1 u charts of Proposition 6. Then for each x ∈ M and y ∈ W u ( x ) the map Φ− y ◦ Φ x : Ex → u u Ey is an affine map with derivative Hxy. Proof. For any n ≥ 0 and any v ∈ Exu we use the defining properties of the charts {Φ x } x ∈ M to write −n −1 1 n Dv ( Φ− y ◦ Φ x ) = Dv ( D f f −n y ◦ Φ f −n y ◦ Φ f −n x ◦ D f x )

= D f fn−ny ◦ DD f −n (v) (Φ−f −1n y ◦ Φ f −n x ) ◦ D f x−n , We have a bound

−n 1 ◦ Φ ( x ), f −n (y)) β , −n x ) − I f −n x f −n y ≤ C ( v ) d ( f

DD f −n( v) (Φ− − n f f y

with the constant C (v) depending only on the distance of v from the origin in Exu , because the charts {Φ x } x ∈ M vary continuously in the C ∞ topology. From the existence of this bound and the proof of [22, Proposition 4.2] we conclude that 1 u lim D f fn−ny ◦ DD f −n ( v) (Φ− ◦ Φ f −n x ) ◦ D f x−n = Hxy f −n y

n→∞

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1 u u This implies that Dv (Φ− y ◦ Φ x ) = Hxy for every v ∈ Ex , from which it follows that 1 u u u Φ− y ◦ Φ x is an affine map from Ex to Ey with linear part Hxy .

We now set k := dim Eu and recall that our assumption that f is uniformly uquasiconformal requires that k ≥ 2. We recall the notion of a quasiconformal map between domains in R k where we equip R k with the Euclidean norm k · k, D EFINITION 9. Let h : U → V be a homeomorphism between two open subsets U, V of R k .The linear dilatation of h at x ∈ U is defined to be maxky− x k=r k f (y) − f ( x )k Lh ( x ) = lim sup minky− x k=r k f (y) − f ( x )k r →0 For K ≥ 1 we define h to be K-quasiconformal if Lh ( x ) ≤ K for every x ∈ U. Each of the normed vector spaces Exu (with norm induced from the Riemannian metric on TM) carries the linear structure of R k with a norm that is uniformly comparable to the Euclidean norm on R k . Hence K-quasiconformality can also be defined for homeomorphisms between open subsets of Exu and Eyu for x, y ∈ M. It is this sense of K-quasiconformality which is used in Lemma 10 below, which is the main result of this section. L EMMA 10. Let f be a C ∞ dynamically coherent partially hyperbolic diffeomorphism. Suppose that f is uniformly u-quasiconformal and that there exists C > 0 such that if c ( x ) then y ∈ Wloc dc ( f n ( x ), f n (y)) ≤ C, for every n ≥ 0. cs ( x ), the Then there is a constant K ≥ 1 such that for any two points x ∈ M, y ∈ Wloc homeomorphism 1 u −1 u Φy−1 ◦ hcs ◦ Φ x : Φ− x (Wloc ( x )) → Φy (Wloc ( y ),

is K-quasiconformal. Proof. By hypothesis there is a constant C > 0 such that for any x ∈ M and y ∈ c ( x ) we have d ( f n ( x ), f n ( y )) ≤ C for every n ≥ 0. Since the stable foliation Wloc c s W is contracted by f n for n ≥ 0, this implies that there is a (possibly larger) cs ( x ), constant C > 0 such that for every x ∈ M and y ∈ Wloc dcs ( f n ( x ), f n (y)) ≤ C, for every n ≥ 0. The compactness of M and the continuity of the cs-holonomy maps implies uniform continuty of these holonomy maps between local leaves of W u which are a bounded distance apart. More precisely, for each ζ ≥ 1 there is a constant L = L(ζ ) ≥ 1 such that for any x, y, z ∈ M with y ∈ W cs ( x ) and z ∈ W u ( x ) such that ζ −1 ≤ d u ( x, z) ≤ ζ and dcs ( x, y) ≤ C then L(ζ )−1 ≤ du (y, hcs (z)) ≤ L(ζ ). n u For each n ≥ 0 we define the center-stable holonomy map hcs n : f (Wloc ( x )) → n u cs n cs − n cs f (Wloc (y)) by hn = f ◦ h ◦ f , where h is the local center-stable holonomy u ( x ) to W u ( y ). By the previous remark we conclude that if z ∈ W u ( x ) from Wloc loc loc cs ( x ) satisfies ζ −1 ≤ d ( f n ( z ), f n ( x )) ≤ ζ for some n ≥ 0 then and y ∈ Wloc u

(3.1)

n L(ζ )−1 ≤ du ( f n (y), hcs n ( f ( z ))) ≤ L( ζ ).


11

cs ( x ). We first estimate the dilatation of the homeoNow fix x ∈ M and y ∈ Wloc − 1 cs morphism Φy ◦ h ◦ Φ x at 0. Using the equivariance properties of the charts from Proposition 6, we have for every n ≥ 0, 1 cs −1 −n n Φ− ◦ hcs y ◦ h ◦ Φ x = Φy ◦ f n ◦ f ◦ Φx n = D f y−n ◦ Φ−f n1(y) ◦ hcs n ◦ Φ f n(x) ◦ D f x 1 cs u Note that Φ− y ◦ h ◦ Φ x (0) = 0. Fix r > 0 and let v ∈ Ex be any vector with − 1 kvk = r. Set ζ := supx ∈ M max{k D f | Eu k, k D f | Eu k}. Then there is an integer n(v) ≥ 0 such that ζ −1 ≤ k D f n( v) (v)k ≤ ζ If w ∈ Exu is any other vector with kwk = r, we have by the definition of the quasiconformal distortion K u that

ζ −1 · K u ( x, n(v))−1 ≤ k D f n( v) (w)k ≤ ζ · K u ( x, n(v)). Since f is uniformly u-quasiconformal we conclude that there is a constant κ ≥ 1 such that for every x ∈ M, and every v, w ∈ Exu with kvk = kwk = r, κ −1 ≤ k D f n( v)(w)k ≤ κ. In other words, we can choose n(v) = n(kvk) to only depend on the norm of v. For definiteness we take n(kvk) to be the maximal integer satisfying the above inequality. The charts {Φ x } x ∈ M are uniformly C1 on the balls of radius κ in Exu as x ranges over M. By applying (3.1) in the coordinates given by Φ f n(kvk)( x ) and Φ f n(kvk)( y) we conclude that there is a constant K ≥ 1 independent of x, y and v (as long as kvk = r) such that

n (k v k) (v)))) ≤ K K −1 ≤ Φ−n1(kvk) (hcs n (k v k) ( Φ f n (k v k)( x ) ( D f x f

(y)

D f −n(kvk)

Since the linear map has uniformly bounded dilatation by our assumption that f is uniformly u-quasiconformal, we conclude after taking ratios that for a possibly larger constant K and any pair of vectors v, w ∈ Exu with kvk = kwk = r, 1 cs kΦ− y ( h ( Φ x ( v )))k 1 cs kΦ− y ( h ( Φ x ( w )))k

≤ K2 .

This holds for every r > 0, no matter how small r is. We thus conclude that the 1 cs 2 linear dilatation of Φ− y ◦ h ◦ Φ x at 0 is bounded above by K for any x ∈ M and cs y ∈ Wloc ( x ). 1 cs To bound the dilatation of Φ− y ◦ h ◦ Φ x at points other than 0 in a ball of u u ( x ), bounded radius centered at 0 in Ex , we write for z ∈ Wloc 1 cs −1 −1 cs −1 Φ− y ◦ h ◦ Φ x = ( Φy ◦ Φh cs ( z ) ) ◦ ( Φh cs ( z ) ◦ h ◦ Φz ) ◦ ( Φz ◦ Φ x ) 1 The dilatation of Φ− ◦ hcs ◦ Φz at 0 is bounded above by K 2 , by our above reah cs ( z ) 1 −1 soning. By Proposition 8 the maps Φ− y ◦ Φh cs ( z ) and Φz ◦ Φ x are both affine u u maps with linear parts Hyhcs ( z) and Hzx respectively. Since we are working on balls of bounded radius centered at 0 in Exu and Eyu respectively and the unstable holonomies are linear maps depending continuously on the base points which

12

C. BUTLER AND D. XU

are thus a bounded distance from the identity, we conclude that after possibly in1 cs −1 creasing the constant K the linear dilatation of Φ− y ◦ h ◦ Φ x at Φ x ( z ) is bounded above by K 3 . This gives us the required quasiconformality assertion of the lemma. We next recall some standard analytic properties of quasiconformal mappings. A homeomorphism h : U → V between open domains of R k is absolutely continuous if it preserves the collection of zero sets of k-dimensional Lebesgue measure. There is a natural Lebesgue measure class on the space of affine lines in R k given by the identification of this space with all translates of lines in R k , i.e., with RP k−1 × R k . Such a homeomorphism is absolutely continuous on lines if for each of the coordinate directions e1 , . . . , ek in R k we have that for almost every line ℓ ⊂ R k parallel to ei the restriction of h to a homeomorphism ℓ ∩ U → h(ℓ ∩ U ) takes subsets of ℓ ∩ U of 1-dimensional Lebesgue measure zero to zero measure sets of h(ℓ ∩ U ), where h(ℓ ∩ U ) is equipped with the 1-dimensional Hausdorff measure in R k . Here the “almost everywhere” quantifier on the space of lines parallel to ei (which we identify with R k−1 ) is taken with respect to the Lebesgue measure on R k−1 . By Fubini’s theorem if h is ACL then h is absolutely continuous. Let volk denote the standard Lebesgue measure on R k . For an absolutely continuous homeomorphism h : U → V we define the Jacobian of h to be the RadonNikodym derivative of h∗ (volk ) with respect to volk and denote it by Jac(h). We let k · k∞ denote the L∞ norm on measurable functions f : R k → R,

k f k∞ = inf sup | f ( x )| V x ∈V

where the infimum is taken over all measurable subsets V of R k with volk (R k \V ) = 0. A standard reference for the claims in Proposition 11 as well as a more precise discussion of the ACL property is Väisälä’s book [34]. P ROPOSITION 11. Suppose that h : U → V is a K-quasiconformal homeomorphism between open subsets of R k , k ≥ 2. Then h is ACL, differentiable volk -a.e. in U, and we also have k Dh x k∞ · k( Dh x )−1 k∞ ≤ K. We next discuss the notion of absolute continuity of a foliation. Let m be a measure on M which is equivalent to the Riemannian volume. Let W be a kdimensional foliation of an n-dimensional Riemannian manifold M which is tangent to a continuous subbundle E of M. For each y ∈ M we let Wr (y) denote the ball of radius r in the induced Riemannian metric on the leaf W(y) through y which is centered at y. Then there is a family of conditional measures {mW x }x∈ M of m on the foliation W with the following properties: for each x ∈ M we have W mW x ( M \W( x )) = 0, the function x → m x is constant on the leaves of W , and if Sx denotes a small (n − k)-dimensional disk passing through x and transverse to W and [ Vx := Wr ( y ) , y∈Sx

denotes an open neighborhood of x, then up to scaling mW y on each local leaf Wr ( y ) W the family {my |Wr ( y) }y∈Sx coincides with the classically defined notion of disintegration of a measure with respect to a measurable partition given by Rokhlin [28]. The family {mW x } x ∈ M is uniquely defined up to m-null sets of M and up to


13

scaling each of the measures on a given leaf of W by a positive constant. We refer to [3, Section 3] for the proof of the existence and uniqueness of the disintegration claimed in this paragraph. For a submanifold S of M we let νS be the induced Riemannian volume on S from M. We define a k-dimensional foliation W to be strongly absolutely continuous if for any pair of nearby smooth transversal (n − k)-dimensional submanifolds S1 and S2 for W the W -holonomy map hW : S1 → S2 is absolutely continuous with respect to the measures νS1 and νS2 , i.e., h∗ (νS1 ) is absolutely continuous with respect to ν S2 . Every C1 foliation is strongly absolutely continuous. The most important examples of strongly absolutely continuous foliation for purposes are the stable and unstable foliations W s and W u of a partially hyperbolic diffeomorphism; strong absolute continuity of these foliations is well-known and a proof may be found in [1]. What we call “strong absolute continuity” is the notion of absolute continuity used in [3], but this notion of absolute continuity is too strong for our purposes. We define a weaker notion of absolute continuity below, D EFINITION 12. A foliation W is absolutely continuous if for each x ∈ M there is an open neighborhood V of x and a strongly absolutely continuous foliation F of V transverse to W such that for any pair of points y, z ∈ V the W -holonomy map hW : F (y) → F (z) is absolutely continuous with respect to the induced Riemannian volumes on F (y) and F (z) respectively. This definition is weaker because we only require the existence of a particular foliation F transverse to W for which the W -holonomy maps between any pair of leaves are absolutely continuous. We emphasize that the transverse foliation F need not be smooth in our definition. Given a foliation W we say that m has Lebesgue disintegration along W if for ma.e. x ∈ M the conditional measure mW x on the leaf W( x ) is equivalent to the induced Riemannian volume on W( x ) from M. Our definition of absolute continuity is designed such that the following proposition is true, P ROPOSITION 13. Let W be an absolutely continuous foliation. Then m has Lebesgue disintegration along W . Proof. Fix a point x ∈ M and let V be an open neighborhood of x on which there is a strongly absolutely continuous foliation F transverse to W for which the W holonomy maps between any two F -leaves are absolutely continuous. In the case that F is a C1 foliation, the proof that the conclusion of the proposition holds is given by [7, Proposition 6.2.2]. However the only property of the transversal foliation F which is used in that proof is the strong absolute continuity, for completeness we give a detailed proof using only this strong absolute continuity property. Without loss of generality we assume that V has a local product structure, i.e. for any x ′ , x ′′ ∈ V, the local leaves Floc ( x ′ ) ∩ V and Wloc ( x ′′ ) ∩ V intersect at exactly one point in V. If we denote Floc ( x ′ ) ∩ V, Wloc ( x ′′ ) ∩ V by FV ( x ′ ) and WV ( x ′′ ) respectively for any x ′ , x ′′ ∈ V, then we have (3.2)

V = ∪y∈WV ( x ) FV (y) = ∪s∈FV ( x ) WV (s)

Since F is a strongly absolutely continuous foliation, there exists a positive measurable conditional density δy (·) for νWV ( x )-almost every y ∈ WV ( x ) such that for

14

C. BUTLER AND D. XU

any measurable subset A ⊂ V we have m( A) =

(3.3)

Z

Z

WV ( x ) FV ( y )

1 A (y, z)δy (z)dνFV ( y) (z)dνWV ( x ) (y)

where we recall from above that νS denotes the induced Riemannian volume on the submanifold S. Let py (·) denote the holonomy maps along the leaves of W from FV ( x ) to FV (y), and let qy (·) denote the Jacobian of py . We have (3.4)

Z

FV ( y )

1 A (y, z)δy (z)dν FV ( y) (z) =

Z

FV ( x )

1 A ( py (s))δy ( py (s))qy (s)dν FV ( x ) (s),

and by changing the order of integration in (3.3) we get (3.5)

m( A) =

Z

Z

FV ( x ) WV ( x )

1 A ( py (s))δy ( py (s))qy (s)dνWV ( s) (y)dν FV ( x ) (s).

Let p¯ s (·) denote the holonomy map along the leaves of FV from WV (s) to WV ( x ). Since F is a strongly absolutely continuous foliation the map p¯ s (·) is absolutely continuous and thus admits a Jacobian q¯s with respect to the induced volumes on WV (s) and WV ( x ) respectively. We transform the inner integral in (3.5) into an integral over WV (s) by making the change of variables r = py (s). Note that y = y(r ) is uniquely determined by r and is continuous as a function of r. Therefore we have Z

=

(3.6)

Z

WV ( x ) WV (s)

1 A ( py (s))δy ( py (s))dνWV ( s) (y) 1 A (r )qy(r )(s)δy(r ) (r )q¯s (r )dνWV ( s) (r ).

Combining this with (3.5) we get (3.7)

m( A) =

Z

Z

FV ( x ) WV ( s )

1 A (s, r )qy(r )(s)δy(r )(r )q¯s (r )dνWV ( s) (r )dν FV ( x ) (s),

which implies the statement of the Lemma.

C OROLLARY 14. Let f be a C ∞ dynamically coherent partially hyperbolic diffeomorphism. (1) Suppose that f is uniformly u-quasiconformal and there exists C > 0 such that if c ( x ) then y ∈ Wloc dc ( f n ( x ), f n (y)) ≤ C, for every n ≥ 0. Then m has Lebesgue disintegration along W cs -leaves. (2) Suppose that f is uniformly quasiconformal and there exists C > 0 such that if c ( x ) then y ∈ Wloc dc ( f n ( x ), f n (y)) ≤ C, for every n ∈ Z. Then for ∗ ∈ {cs, c, cu}, m has Lebesgue disintegration along W ∗ leaves. Proof. By Lemma 10 and Proposition 11 the hypotheses of (1) imply that the local W cs -holonomy maps between local leaves of the unstable foliation W u are absolutely continuous. Since the stable and unstable foliations W s and W u are strongly absolutely continuous this implies by Proposition 13 that m has Lebesgue disintegration along W cs leaves.


15

Under the hypotheses of (2) we may also apply Lemma 10 to the local csholonomies of f −1 , i.e., to the cu-holonomies of f . This implies that the local W cu -holonomy maps between local leaves of the unstable foliation W s are absolutely continuous with bounded Jacobians as well, and thus by Proposition 13 (using W s as the transverse strongly absolutely continuous foliation) we conclude that m also has Lebesgue disintegration along W cu leaves. For each x ∈ M the cu-leaf W cu ( x ) is foliated by W u -leaves and this foliation is strongly absolutely continuous when we consider W cu ( x ) to be the ambient manifold. The holonomy of the W c foliation between local W u leaves inside of W cu ( x ) coincides with the W cs holonomy in M. Thus by Lemma 10 and Proposition 11 these local W c -holonomy maps are absolutely continuous. cu Let mcu x denote the conditional volume of m on W ( x ). Since the holonomy c u cu maps of W between local W leaves inside of W ( x ) are absolutely continuous we can apply Proposition 13 again to obtain that mcu x has Lebesgue disintegration along W c leaves inside of W cu ( x ). Since this holds for every x ∈ M and m has Lebesgue disintegration along W cu leaves it follows that m has Lebesgue disintegration along W c leaves. 4. H IGHER REGULARITY OF

THE CENTER FOLIATION

In this section we will prove higher regularity properties of the W c , W cs , and foliations under stronger assumptions on f . Unless stated otherwise, in all of the claims of this section we assume that f is a C ∞ dynamically coherent partially hyperbolic diffeomorphism which is uniformly quasiconformal. We further assume that f preserves an invariant measure m which is smoothly equivalent to the Riemannian volume on M.

W cu

4.1. A fiber bundle construction. We first formulate an additional condition on f which is related to the proof of Theorem 4. Suppose that dim Ec = 1, Ec is orientable, f (W c ( x )) = W c ( x ) for every x ∈ M, and f has no fixed points. Each center leaf W c ( x ) has as its universal cover a copy fc ( x ) of R with orientation determined by the orientation of Ec . The restriction W fc ( x ) with of f to W c ( x ) lifts to an orientation-preserving diffeomorphism fe of W e x be the closed segment joining e no fixed points. Fix a lift xe of x and let U f −1 ( xe) to fc ( x ). We then let Ux be the projection of this segment to W c ( x ). fe( xe) inside of W An easy exercise shows that the neighborhood Ux is independent of the chosen lift of x. It is also clear that for every x ∈ M we have f (Ux ) = U f ( x ). We say that f does not wrap if for each x ∈ M, the neighborhood Ux of x in W c ( x ) is simply connected, i.e., it is a line segment instead of a circle. For the remainder of Section 4 we will assume that f satisfies one of the following two assumptions, (A) W c is uniformly compact; or (B) dim Ec = 1, f (W c ( x )) = W c ( x ) for every x ∈ M, Ec is orientable with orientation preserved by f , f has no fixed points, and f does not wrap. In the case that f satisfies assumption (B) we set {Ux } x ∈ M to be the family of neighborhoods of points of M inside of the center foliation constructed above. When f satisfies assumption (A) we instead set Ux = W c ( x ). In both cases

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C. BUTLER AND D. XU

we have the properties that f (Ux ) = U f ( x ) and there is an R > 0 such that diamc (Ux ) ≤ R for every x ∈ M. We note that if ψ1 is the time-1 map of an Anosov flow ψt which has no periodic orbits of period ≤ 2 then assumption (B) always holds for C1 -small enough perturbations of ψ1 . We refer to the proof of Theorem 4 at the end of this section for the details behind this assertion. P ROPOSITION 15. Under assumptions (A) or (B), there is R ≥ r > 0 such that for any x ∈ M, y ∈ W c ( x ) with dc ( x, y) ≤ r, we have dc ( f n ( x ), f n (y)) ≤ R, ∀n ∈ Z Proof. Under assumption (A) we may take r = R = sup diamc (W c ( x )) < ∞ x∈ M

since we assume that the center foliation is uniformly compact so that there is a uniform bound on the diameters of center leaves. Under assumption (B) we let r = inf dc ( x, f ( x )), R = 2 sup dc ( x, f ( x )). x∈ M

x∈ M

Since M is compact and f has no fixed points we have 0 < r ≤ R/2 < ∞. If y ∈ W c ( x ) satisfies d c ( x, y) ≤ r then y ∈ Ux and we conclude that f n (y) ∈ U f n ( x ) for every n ∈ Z. Since diamc (Ux ) ≤ R for every x ∈ M, the statement of the Proposition follows. As a consequence of Proposition 15, all of the work from Section 3 applies to both f and f −1 for systems satisfying assumptions (A) or (B). In particular the u u center-stable holonomy maps hcs xy : Wloc ( x ) → Wloc ( y ) and center-unstable hos s cs cu lonomy maps hcu xz : Wloc ( x ) → Wloc ( z ) for x ∈ M, y ∈ Wloc ( x ), z ∈ Wloc ( x ) are all K-quasiconformal for some constant K ≥ 1. Consequently m has Lebesgue disintegration along each of the foliations W cs , W cu , and W c by Corollary 14. By combining this with the strong absolute continuity of the foliations W u and W s we conclude that m has Lebesgue disintegration along all of the invariant foliations W ∗ for f . We now consider the space

E = {( x, y) ∈ M2 : y ∈ Ux }, and we define F : E → E by F ( x, y) = ( f ( x ), f (y)). P ROPOSITION 16. E is a continuous fiber bundle over M with compact fibers. F preserves an invariant measure µ on E which locally decomposes as the product of the volume m on M and the conditional volume mcx on Ux . Proof. We first show that for r > 0 sufficently small and y ∈ Wru ( x ), under either c ( x ) to W c ( y ) assumption (A) or (B), the unstable holonomy map huxy from Wloc loc u can be extended to a homeomorphism h : Ux → Uy . Under either assumption (A) or (B) the neighborhoods Ux ⊂ W c ( x ) are uniformly compact (i.e., have dc diameter uniformly bounded in x) and depend continuously on x in the Hausdorff topology on sets. Thus we can choose r small enough that for any x ∈ M, y ∈ u ( z ) ∩ U consists of at most one point for any z ∈ U . Wru ( x ) we have that Wloc y x


17

Under assumption (B) this last assertion requires the condition that f does not wrap. u ( z ) ∩ U . This is obvious We claim that there is in fact exactly one point in Wloc y u c under assumption (A) since Wloc (z) must intersect W (y) if y is close enough to x, by the uniform compactness of the center foliation. Under assumption (B) let ε > 0 be a constant chosen small enough that for each x ∈ M the ε-neighborhood Uxε of Ux inside of W c ( x ) still satisfies the no wrapping condition, that is to say, Uxε remains an interval instead of a circle. Then for ε > 0 sufficiently small, r sufficiently small depending of ε, and any x ∈ M, y ∈ Wru ( x ) and z ∈ Ux the intersection u ( z ) ∩ U ε consists of exactly one point. Thus h u : U → U ε is an orientationWloc x y y preserving homeomorphism onto its image inside of Uyε . But if y ∈ Wru ( x ) for r u ( f ( x )) and W u ( f −1 ( x )) respecsufficiently small then f (y) and f −1 (y) lie in Wloc loc tively. Thus the endpoints of the interval which is the image of Ux under hu are f −1 (y) and f (y). This shows that h u actually gives a homeomorphism from Ux to Uy . Hence hu : Ux → Uy is a homeomorphism for y ∈ Wru ( x ). By similar reasoning (possibly taking r smaller) for any x ∈ M, y ∈ Wrs ( x ) the stable holonomy hs : Ux → Uy is also a homeomorphism. Finally, it is easy to see that for any y ∈ Wrc ( x ) there is a homeomorphism hc : Ux → Uy depending uniformly continuously on the pair ( x, y): in the case of assumption (A) this is trivial since Ux = Uy . In the case of assumption (B) each of the subsets Uy of W c ( x ) is an interval determined canonically by its endpoints f −1 (y) and f (y) according to the construction at the beginning of this section. These endpoints depend continuously on y ∈ W c ( x ) hence it follows that we can find a continuous family of orientation-preserving homeomorphisms hc : Uy → Ux identifying these intervals for y near x. Putting all of this together, for any z close enough to x we can find a composition of three homeomorphisms Uz → Uhs ( z) → Uhu ( hs ( z)) → Uhc ( hu ( hs ( z))) = Ux s ( z ) ∩ W cu ( x ), h u ( h s ( z )) = which depends continuously on z, where hs (z) = Wloc loc s c u Wloc (h (z)) ∩ Wloc ( x ). This proves that E is a continuous fiber bundle over M with compact fibers. We now prove the second assertion. Under assumption S (A) we consider the measurable partition of M into compact center fibers M = x ∈ M W c ( x ) and let {mcx } x ∈ M be the family of conditional measures of m on the center fibers W c ( x ) determined by this partition. Since f preserves m we have f ∗ mcx = mcf ( x ). for m-a.e. x ∈ M. In the case of assumption (B) we refer to [3, Section 3]. It is shown there that for the foliation W c of M there is a measurable family of conditional measures {mˆ cx } x ∈ M supported on the leaves W c ( x ) of the center foliation such that for y ∈ W c ( x ) the measures mˆ cx and mˆ cy coincide up to a constant factor. We normalize ˆ cx (Ux ) = 1. Furthermore, since f fixes all of the leaves these measures such that m c ˆ cx = m ˆ cx for m-a.e. x ∈ M. We define mcx of W , by [3, Proposition 3.3] we have f ∗ m c ˆ x to Ux . Since f (Ux ) = U f ( x ) we conclude that f ∗ mcx is a to be the restriction of m c ˆ cx constant multiple of m f ( x ), and by our choice of normalization of the measures m this implies f ∗ mcx = mcf ( x ) since these measures both assign mass 1 to U f ( x ).

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C. BUTLER AND D. XU

The measures mcx are equivalent to the Riemannian volume on Ux since m has Lebesgue disintegration along W c leaves. We define the measure µ on the fiber bundle E by setting, for any measurable set A ⊂ E , µ( A) =

Z

1 A ( x, y)dmcx (y)dm( x ),

where 1 A denotes the characteristic function of A. Since f preserves m and f ∗ mcx = mcf ( x ) for m-a.e. x ∈ M, we conclude that µ is F −invariant. 4.2. Conformal structures. We now introduce the bundle C Eu of conformal structures on Eu over M. For more details related to the discussion that follows we defer to [21]. The fiber C Exu over x is the space of all inner products on Exu modulo scaling by a nonzero real number, which can be identified with the nonpositively curved Riemannian symmetric space SL(k, R )/SO(k, R ). Each fiber thus carries a canonical Riemannian metric ρ x given by an isometric identification of C Exu with SL(k, R )/SO(k, R ). We will always explicitly identify C Exu with the space of inner products on Exu for which the determinant of a positively oriented orthonormal basis is 1 in the reference inner product on Exu induced from the given Riemannian metric on TM. Any linear isomorphism A : Exu → Eyu induces a map A∗ : C Eyu → C Exu by, for τx ∈ C Exu and any v, w ∈ Eyu , A∗ τx (v, w) =

τx ( A(v), A(w)) , det( A)2/k

where we recall that k = dim Eu and det( A) denotes the determinant of A in the metric induced from TM. The induced map A∗ is an isometry from (C Eyu , ρy ) to (C Exu , ρ x ). A (measurable) conformal structure on Eu is a measurable section τ : M → C Eu defined on the complement of an m-null set of M. We say that a measurable conformal structure is invariant if D f x∗ τ f ( x ) = τx for m-a.e. x ∈ M. A measurable conformal structure is bounded if there is a constant C > 0 such that ρ x (τx , τx0 ) ≤ C for m-a.e. x ∈ M, where τ 0 denotes the conformal structure on Eu induced from the Riemannian metric on TM. The condition that f is uniformly u-quasiconformal is equivalent to the existence of a constant C > 0 such that for every x ∈ M, ρ x ( D f xn )∗ τ 0f n ( x ), τx0 ≤ C ∀n ∈ Z. The following measure-theoretic lemma is vital for recovering holonomy invariance properties of measurable objects by guaranteeing simultaneous recurrence to a continuity set on a full measure set of points. Our first application will be to show that a measurable invariant conformal structure for f must be invariant under the stable and unstable holonomies H s and H u on a full measure subset of M. L EMMA 17. Let T be a measure-preserving transformation of a finite measure space ( X, µ) and let {Kn }n≥1 be a sequence of measurable subsets of X with ∑∞ n =1 µ ( X \ K n ) < ∞. Then there is a full measure subset Ω ⊆ X with the property that if x, y ∈ Ω then there is an n ∈ N and a sequence nk → ∞ with T nk ( x ) ∈ Kn , T nk (y) ∈ Kn for each nk .


19

Proof. By the Birkhoff ergodic theorem, for each n ∈ N the Birkhoff averages 1 k −1 j k ∑ j =0 1 K n ( T ( x )) converge pointwise (on a measurable set En with µ( X \ En ) = 0) as k → ∞ to a nonnegative T − invariant measurable function Pn with integral µ(Kn ). Define Ω ⊂ X by ∞ \ 3 . Ω= En ∩ x ∈ X : ∃ N ∈ N such that for n ≥ N, Pn ( x ) > 4 n =1 We claim that µ( X \Ω) = 0 Consider the sets 1 Bn = x ∈ En : 1 − Pn ( x ) ≥ . 4 By the Markov inequality we have µ ( Bn ) ≤ 4

Z

X

1 − Pn dµ = 4µ( X \Kn )

Since ∑∞ n =1 µ( X \Kn ) < ∞ by hypothesis, we conclude by the Borel-Cantelli lemma that for µ-a.e. x ∈ X there are only finitely many n such that x ∈ Bn . This implies that Ω is a full measure subset of X. Now we verify that Ω has the desired properties of the Lemma’s conclusion. If x, y ∈ Ω are any two given points then from the definition of Ω there is a common n such that Pn ( x ) > 34 and Pn (y) > 34 . We explain how to construct nk inductively from nk−1 , with our construction also showing how to construct the initial n1 from n0 = 0. Given nk−1 , we choose Nk > nk−1 large enough that 1 Nk

Nk

∑ j = n k −1

1 Kn ( T j ( x )) >

3 4 Nk − n k −1 Nk

> 78 . The existence of this Nk is guaranteed by the fact that Pn ( x ) > 34 . We conclude from these estimates that j ∈ [nk−1 , Nk ] : T j ( x ) ∈ Kn 4 > Nk − nk−1 7 and the same for y, and we also take Nk large enough that

and the same for y, where [nk−1 , Nk ] denotes the set of integers j satisfying nk−1 ≤ j ≤ Nk . It follows that there is a common nk ∈ [nk−1 , Nk ] such that T nk ( x ) ∈ Kn and T nk (y) ∈ Kn . Inducting on k completes the proof. Proposition 18 below summarizes the essential properties of invariant conformal structures for uniformly quasiconformal linear cocycles which we will need. It is a slight improvement of the proof of [22, Proposition 4.4] as it removes the assumption of ergodicity of f with respect to the volume m which was used in that proof. P ROPOSITION 18. Suppose that f is uniformly u-quasiconformal and volume-preserving. Then there is an invariant bounded measurable conformal structure τ : M → C Eu . Furu (x) thermore there is a full measure subset Ω of M such that if x, y ∈ Ω and y ∈ Wloc then u ∗ ( Hxy ) τy = τx ,

20

C. BUTLER AND D. XU

s ( x ) then and similarly if y ∈ Wloc s ∗ ( Hxy ) τy = τx .

Proof. By [21, Proposition 2.4] the uniform quasiconformality of the linear cocycle D f | Eu implies that there is an invariant bounded measurable conformal structure τ : M → C Eu . By Lusin’s theorem we can find an increasing sequence {Kn }n∈N of compact subsets of M such that τ is uniformly continuous on Kn and m( M \Kn ) < 2−n . Let Ω ⊂ M be the full measure set of points satisfying the conclusion of Lemma 17 for both f and f −1 . We also require that τ is defined and D f -invariant on Ω. s ( x ). Then there is an n > 0 and a sequence Let x, y ∈ Ω be given with y ∈ Wloc n n k k nk → ∞ such that f ( x ) and f (y) both lie in Kn for each nk . Then n

n

s ∗ u ∗ ρ x (τx , ( Hxy ) τy ) = ρ x (( D f x k )∗ τ f nk ( x ), ( Hxy ) ( D f y k )∗ τ f nk (y) )

= ρ x (( D f xnk )∗ τ f nk ( x ), ( D f xnk )∗ ( H sf nk x f nk y )∗ τ f nk (y) ) = ρ f nk ( x ) (τ f nk ( x ), ( H sf nk x f nk y )∗ τ f nk (y) ) ≤ ρ f nk ( x ) (τ f nk ( x ), ( I f nk x f nk y )∗ τ f nk (y) ) + ρ f nk ( x ) (( I f nk x f nk y )∗ τ f nk (y), ( H sf nk x f nk y )∗ τ f nk (y) ), where we recall that Ixy : Exu → Eyu is our chosen Holder ¨ continuous family of identifications of nearby fibers of Eu . The uniform continuity of τ on Kn implies that ρ f nk ( x ) (τ f nk ( x ) , ( I f nk x f nk y )∗ τ f nk ( y) ) → 0, as nk → ∞ since d( f nk ( x ), f nk (y)) → 0 as nk → ∞. Since k I f nk x f nk y − H sf nk x f nk y k → 0 uniformly as nk → ∞ we also conclude that ρ f nk ( x ) (( I f nk x f nk y )∗ τ f nk ( y), ( H sf nk x f nk y )∗ τ f nk ( y) ) → 0, s )∗ τ . as nk → ∞. Combining these two facts gives τx = ( Hxy y The same proof replacing W s by W u and nk by −nk shows that if x, y ∈ Ω with u ( x ) then τ = ( H u ) ∗ τ . y ∈ Wloc x y xy

4.3. Equivariance properties of the center holonomy. In the next proposition we u ( x ) → W u ( y ) for the center-stable holonomy between local unwrite hcxy : Wloc loc stable leaves inside the same center-unstable leaf, which coincides with the center holonomy between these leaves. P ROPOSITION 19. Let Q ={ x ∈ M : for mcx -a.e. y ∈ Ux , u u hcxy : Wloc ( x ) → Wloc (y) is differentiable at x }.

Then m( Q) = 1. u ( x ). Let Proof. For x ∈ M we let mux denote the conditional measure of m on Wloc cu m x be the conditional measure of m on the subset

Sx =

[

u (x) y ∈W loc

cu ( x ). Uy ⊂ Wloc


21

Since m has Lebesgue disintegration along each of the foliations W u , W c , and W cu , for m-a.e. x ∈ M the measure mcu x decomposes as conditional measures in two different ways, mcu x ≍

Z

Ux

muy dmcx (y) ≍

Z

u (x) Wloc

mcy dmux (y)

where we use the notation ≍ to indicate that the two measures are equivalent on Sx . By Lemma 10 and Proposition 11, for every y ∈ Ux the center holonomy map u ( y ) → W u ( x ) is differentiable at m u -a.e. z ∈ W u ( y ). Thus if we set hcyx : Wloc y loc loc Tx = {z ∈ Sx : hcxy is differentiable at z for y = huzx (z)}, cu cu then by the first expression for mcu x we have m x ( Tx ) = m x ( Sx ) for m-a.e. x ∈ M. cu Since Tx has full m x measure in Sx we conclude by the second expression for mcu x u ( x ). This immediately implies that that mcy ( Tx ∩ Uy ) = mcy (Uy ) for mux -a.e. y ∈ Wloc u ( x )) = m u (W u ( x )) from the definition of Q. Since the W u foliation is mux ( Q ∩ Wloc x loc absolutely continuous and this holds for m-a.e. x ∈ M we conclude that m( Q) = 1, i.e., Q has full volume in M.

We then define Q = {( x, y) ∈ E : x ∈ Q}. From the definition of Q and Proposition 19 we see that Q has full µ-measure inside of E . For ( x, y) ∈ Q we can c : E u → E u to be the derivative of h c at x. The map ( x, y ) → H c then define Hxy x y xy xy is clearly measurable and defined µ-a.e. by Proposition 19. Our next goal is to show that the maps H c are equivariant with respect to the stable and unstable holonomies H s and H u of D f | Eu . L EMMA 20. There is a full µ-measure subset Ω of Q such that if ( x, y), (z, w) ∈ Ω with u ( x ) and w ∈ W u ( y ) then the following equation holds, z ∈ Wloc loc (4.1)

c c u u ◦ Hxy , Hzw ◦ Hxz = Hyw

s ( x ) and w ∈ W s ( y ) then, and similarly if (z, w) ∈ Ω with z ∈ Wloc loc

(4.2)

c s s c Hzw ◦ Hxz = Hyw ◦ Hxy .

Proof. We let Λ ⊂ M be the full m-measure set of points on which the invariant bounded measurable conformal structure τ : M → C Eu of Proposition 18 is defined and invariant under both D f and the stable and unstable holonomies H s and H u . We let Ω0 ⊂ E be the set of ( x, y) ∈ E such that both x and y are in Λ. The absolute continuity of W c together with the construction of the measure µ implies that µ(E \Ω0 ) = 0. By Lusin’s theorem we can find an increasing sequence of compact subsets Kn ⊂ E such that µ(E \Kn ) < 2−n and such that H c restricts to a uniformly continuous function on each Kn . Since µ is F-invariant, by applying Lemma 17 to both F and F −1 there is a measurable set Ω with µ(E \Ω) = 0 and such that for any pair of points ( x, y), (z, w) ∈ Ω there is an n ∈ N and a pair of infinite sequences nk → ∞ ′ ′ and n′k → ∞ with F −nk ( x, y), F −nk (z, w) ∈ Kn for each nk and F nk ( x, y), F nk (z, w) ∈ Kn . We now prove that equation (4.1) holds. The proof for equation (4.2) will be u ( x ) and w ∈ completely analogous. Let ( x, y), (z, w) ∈ Ω be such that z ∈ Wloc u c − n − n Wloc (y). Since H is uniformly continuous on Kn and d( f x, f z), d( f −n y, f −n w) →

22

C. BUTLER AND D. XU

0 as n → ∞, we conclude that

u

(4.3)

H f −nk y f −nk w ◦ H cf −nk x f −nk y − H cf −nk z f −nk w → 0,

as k → ∞, where {nk } is the infinite sequence from the previous paragraph corresponding to the pair ( x, y), (z, w). For x ∈ Λ we let SExu denote the unit sphere in Exu in the metric τx . For any two points x, y ∈ Λ and an invertible linear map A : Exu → Eyu we then define SA(v) =

A(v) 1

det( A) k

,

where the determinant is taken with respect to the induced Riemannian metric τ 0 on Eu from TM. We remark that if A∗ τy = τx then SA maps SExu to SEyu and consequently SA is an isometry from Exu to Eyu when these are given the metrics 1

τx and τy respectively. This is because A then maps SExu to detτ ( A) m · SEyu , where detτ denotes the determinant of this linear map with respect to the family of inner products on Eu given by τ. Our convention for representing elements of C Exu is to take the inner product which has determinant 1 with respect to the background metric τx0 . Thus detτ = det for linear maps between fibers of Eu . We also note that A is clearly determined by SA and det( A). For ( x, y), (z, w) ∈ Ω given as in the statement of the lemma we will show that (4.4)

c u u c det( Hzw ) det( Hxz ) = det( Hyw ) det( Hxy ),

(4.5)

c u u c SHzw ◦ SHxz = SHyw ◦ SHxy .

The desired statement of the lemma follows from these two equations. From differentiating the equation f −k ◦ hcxy = hcf −k x f −k y ◦ f −k , expressing the equivariance of center holonomy with respect to the dynamics f we obtain the equation c D f y−k ◦ Hxy = H cf −k x f −k y ◦ D f x−k ,

(4.6)

which is valid for any ( x, y) ∈ Q. Taking determinants and rearranging, we conclude that det( D f y−k | Eu ) c det( Hxy ) = det( H cf −k x f −k y ) det( D f x−k | Eu ) c and then taking ratios at the iterApplying the same equation to (z, w) with Hzw ates nk gives c ) det( H cf −nk x f −nk y ) | Eu ) det( D f z−nk | Eu ) det( Hxy · · = −n −n c ) det( H cf −nk z f −nk w ) det( D f w k | Eu ) det( D f x k | Eu ) det( Hzw

−nk

det( D f y

As k → ∞ the right side converges to 1 by equation 4.3, the first factor in the u ), and the second factor converges to product on the left side converges to det( Hyw u ) −1 . Rearranging the resulting equation gives equation (4.4). det( Hxz For the second equation we first consider the following lemma which only uses the hypothesis that f is uniformly u-quasiconformal. We let k · kτ denote the norm on Exu induced by the inner product τx .


23

u ( x ) and v ∈ E u , v ′ ∈ E u . If L EMMA 21. Suppose x, y ∈ Λ, y ∈ Wloc x y

lim inf kSD f −n (v) − SH uf−n y f −n x (SD f −n (v′ ))kτ = 0,

then

u (v) SHxy

n→∞ = v′ .

u ( v ). Then we have Proof. Let w = SHxy

SD f −n (v) = SH uf−n y f −n x (SD f −n (w)), by the equivariance properties of the unstable holonomy H u . Therefore we have lim inf kSH uf−n y f −n x (SD f −n (w)) − SH uf−n y f −n x (SD f −n (v′ ))kτ = 0. n→∞

But the invariance of τ under the unstable holonomy H u on Λ and its invariance under D f imply that SD f −n and SH u are both isometries with respect to the family of metrics given by τ on the fibers Exu of the vector bundle Eu . This then implies that lim inf kw − v′ kτ = 0, n→∞

which means that w = v ′ as desired.

Now let ( x, y), (z, w) ∈ Ω be given as in the statement of the Lemma. From equation 4.3 and the equivariance properties of H u we conclude that −nk

lim kSH cf −nk z f −nk w (SD f z

k→∞

u (SHxz (v)))

−nk

− SH uf−nk y f −nk w (SH cf −nk x f −nk y (SD f x

(v)))k = 0.

Applying the equivariance relation (4.6), this implies that −nk

lim kSD f w

k→∞

c u (SHzw (SHxz (v)))

−nk

− SH uf−nk y f −nk w (SD f y

c (SHxy (v)))k = 0

Since the measurable conformal structure τ is bounded, the norms k · kτ are uniformly comparable to the norm k · k = k · kτ0 and thus this equation also holds with k · k replaced by k · kτ . We thus conclude by Lemma 21 that c u u c SHzw (SHxz (v)) = SHyw (SHxy (v)).

Since this holds for every v ∈ Exu we deduce equation (4.1) as desired. s ( x ) and w ∈ W s ( y ), To prove the second equation (4.2) where instead z ∈ Wloc loc u s we follow the exact same proof, replacing H by H and −nk by nk everywhere. We next recall the following elementary lemma from analysis, L EMMA 22. Suppose f : R k → R k is ACL and that there is a continuous map G : R k → GL(k, R ) such that D f = G almost everywhere. Then f is a C1 map and D f = G everywhere. Proof. Let f = ( f 1 , . . . , f k ), f i : R k → R. Since f is ACL each coordinate function f i is ACL. Thus there exists a full Lebsgue measure set Λ ⊂ R k such that for every x ∈ Λ and 1 ≤ i, j ≤ k, f i | x +R·e j is absolutely continuous and D f = G for almost every point (with respect to arc length) in { x + R · e j }, where e1 , . . . , ek denote the standard basis of R k .

24

C. BUTLER AND D. XU

Absolute continuity of f i | x +R·e j implies that fi (x + t · ej ) = fi (x) +

= fi (x) +

Z t ∂ fi 0

Z t 0

∂x j

( x + s · ei )ds

Gij ( x + s · ei )ds,

where G = ( Gij )1≤i,j≤k is the matrix representation of G in the standard basis of R k . Since both f and G are continuous this last equation holds for all x ∈ R k ∂f and t ∈ R. This proves that for each 1 ≤ i, j ≤ k the partial derivative ∂xi of f j

exists and coincides with Gij . In particular all partial derivatives of f exist and are continuous at every point in R k which implies that f is C1 and D f = G. L EMMA 23. For any x ∈ M and y ∈ Ux we have the equality c c 1 Φ− y ◦ h xy ◦ Φ x = Hxy

as maps from Exu to Eyu . The measurable function H c on Ω therefore admits a continuous extension to E and the center holonomy is linear in the charts {Φ x } x ∈ M . Proof. We first consider pairs ( x, y) ∈ Ω. Since D0 Φ x = Id Exu for every x ∈ M and hcxy ( x ) = y, the equation 1 c c D0 ( Φ − y ◦ h xy ◦ Φ x ) = Hxy ,

holds for any ( x, y) ∈ Ω. To compute the derivative at other points of Exu , we let v ∈ Exu , z = Φ x (v), and w = hcxy (z) = hczw (z). We suppose that (z, w) ∈ Ω and compute, c −1 −1 1 c −1 Dv ( Φ− y ◦ h xy ◦ Φ x ) = D0 ( Φy ◦ Φw ) ◦ D0 ( Φw ◦ h zw ◦ Φz ) ◦ Dv ( Φz ◦ Φ x ). 1 u −1 u By Proposition 8 we know that D0 (Φ− y ◦ Φw ) = Hwy and Dv ( Φz ◦ Φ x ) = Hxz . Hence 1 c u c u c Dv ( Φ− y ◦ h xy ◦ Φ x ) = Hwy ◦ Hzw ◦ Hxz = Hxy ,

whenever ( x, y), (z, w) ∈ Ω, by Lemma 20. Since Ω has full µ-measure we con1 c u u clude that for m-a.e. x ∈ M and mcx -a.e. y ∈ Ux the map Φ− y ◦ h xy ◦ Φ x : Ex → Ey c almost everywhere. is differentiable almost everywhere on Exu with derivative Hxy − 1 c By Lemma 10 the map Φy ◦ h xy ◦ Φ x is quasiconformal and therefore ACL. By 1 c 1 c Lemma 22 this implies that Φ− y ◦ h xy ◦ Φ x is a C map with derivative Hxy every− 1 c c where, i.e., Φy ◦ h xy ◦ Φ x coincides exactly with the linear map Hxy . Since Ω has full µ-measure in E and µ is fully supported we conclude that Ω is dense in E and thus the equation (4.7)

1 c c Φ− y ◦ h xy ◦ Φ x = Hxy

holds on Exu for a dense set of pairs ( x, y) ∈ E . But the left side of this equation 1 u depends uniformly continuously on the pair ( x, y) on as a map Φ− x (Wloc ( x )) → − 1 u u u Φy (Wloc (y)) between neighborhoods of 0 in Ex and Ey of uniform size. Furtherc is determined by its restriction to a map between these more the linear map Hxy c also depends uniformly continuneighborhoods. Hence we conclude that Hxy ously on the pairs ( x, y) ∈ Ω.


25

1 c When y is close to x the map Φ− y ◦ h xy ◦ Φ x is uniformly close to the linear u u c is uniformly identifications Ixy : Ex → Ey introduced in Section 3. Hence Hxy c belong to a close to Ixy for ( x, y) ∈ Ω. In particular for ( x, y) ∈ Ω the maps Hxy u uniformly bounded subset of the space of invertible linear maps Ex → Eyu . This shows that H c admits a continuous extension to E such that equation (4.7) still holds on a neighborhood of 0 in Exu for any ( x, y) ∈ E . Finally, because Φ x , hcxy , c all have the proper equivariance properties with respect to f −1 and D f −1 and Hxy which uniformly contract Eu it follows that equation 4.7 actually holds on all of Exu .

As a corollary of Lemma 23 we deduce that equations (4.1) and (4.2) from Lemma 20 actually hold on all of E because the uniform continuity of H c implies each side of these equations is uniformly continuous in the quadruple of points x, y, z, w and both of these equations hold on a dense subset of E . cs ( x ), we let z be the unique intersection point of W c ( x ) Given x ∈ M, y ∈ Wloc loc s with Wloc (y) and define cs s c Hxy = Hzy ◦ Hxz . By the observation in the previous paragraph, if we let w be the intersection of s ( x ) with W c ( y ) then we also have the equality Wloc loc cs c s Hxy = Hwy ◦ Hxw . cs depends in a uniformly continuous fashion x and y from the We note that Hxy uniform continuity of H c and H s . u u L EMMA 24. The center-stable holonomy hcs xy : Wloc ( x ) → Wloc ( y ) between two local 1 cs unstable leaves is C with derivative H . s ( x ) and h cs : W u ( x ) → W u ( y ) is Proof. We will first show that if y ∈ Wloc xy loc loc cs s differentiable at x then Dh xy ( x ) = Hxy . We will prove this by contradiction. s −1 cs u If Dx (hcs xy ) 6 = Hxy then L xy : = Φy ◦ h xy ◦ Φ x is differentiable at 0 ∈ Ex but s u D0 ( L xy ) 6= Hxy . Thus there exists v ∈ Ex with kvk = 1 and some constants ε 0 , η > 0 such that s k L xy (tv) − Hxy (tv)k ≥ ε 0 t, ∀|t| ≤ η.

By the uniform u-quasiconformality of f there is then a constant C ≥ 1 independent of n such that (4.8)

s k D f yn ( L xy (tv)) − D f yn ( Hxy (tv))k ≥ C −1 det( D f yn | Eu )ε 0 t, ∀|t| ≤ η,

and also with the properties that for for every x ∈ M and any unit vector ξ ∈ Exu , C −1 det( D f xn | Eu ) ≤ k D f xn (ξ )k ≤ C det( D f xn | Eu ), and lastly the distortion estimate det( D f yn | Eu ) ≤ C det( D f xn | Eu ) holds for y ∈ s ( x ) and n ≥ 0. Wloc By the uniform continuity of the charts Φ x in the x-variable, the W cs foliation s (z) and H s , given any ε > 0 there exists δ = δ(ε) such that for any z ∈ M, w ∈ Wloc u with ds (z, w) ≤ δ and any ξ ∈ Ez satisfying kξ k ≤ 1 we have (4.9)

s k Lzw (ξ ) − Hzw (ξ )k < ε

26

C. BUTLER AND D. XU

We choose ε < C −3 ε 0 and then choose n large enough that ds ( f n ( x ), f n (y)) ≤ δ(ε), and such that C −1 det( D f xn | Eu )−1 < η. We put z = f n ( x ) and w = f n (y). Applying the equivariance of D f with respect to the charts Φ x , the center stable holonomy hcs , and the linear stable holonomy H s in equation (4.8) we obtain s k Lzw ( D f xn (tv)) − Hzw ( D f xn (tv))k ≥ C −2 det( D f xn | Eu )ε 0 t, ∀|t| ≤ η,

Let t be the maximal number such that k D f xn (tv)k ≤ 1 and then put ξ = D f xn (tv). We conclude that equation (4.9) applies to the above and thus obtain ε > C −2 det( D f xn | Eu )ε 0 t. But we have η ≥ t ≥ C −1 det( D f xn | Eu )−1 by the uniform u-quasiconformality of f . Hence we conclude that ε > C −3 ε 0 , contradicting our choice of ε. cs ( x ) and h cs is differentiable at x. Let z = W cu ( x ) ∩ Now suppose that y ∈ Wloc xy loc s cs cs cs u u Wloc (y). Then h xy = hzy ◦ h xz . The map hcs xz : Wloc ( x ) → Wloc ( z ) coincides with u ( x ) to W u ( z ) and thus it follows from Lemma the center holonomy from Wloc loc cs 1 c at x. We conclude that 23 that h xz is a C diffeomorphism with derivative Hxz cs hzy is differentiable at z and thus by our work above the derivative of h cs zy at z s . Thus D ( h cs ) = H s ◦ H c = H cs . Since h cs is ACL from the is given by Hzy x xy zy xz xy xy cs is uniformly continuous in x and quasiconformality given by Lemma 10 and Hxy 1 y by the remarks preceding this lemma we conclude by Lemma 22 that hcs xy is C cs with derivative given by H . L EMMA 25. There is a continuous invariant conformal structure τ : M → Eu for D f | Eu which is invariant under H c , H u , and H s holonomies. Proof. By Proposition 18 there is a bounded measurable invariant conformal structure τˆ : M → Eu for D f | Eu defined on a full measure subset Ω of M such that τˆ is H u and H s invariant on Ω. For a point x ∈ M we define τx to be the barycenter in C Exu in the nonpositively curved metric ρ x of the set o n c ∗ O x = ( Hxy ) τˆy : y ∈ Ω ∩ W c ( x )

(for existence and uniqueness of barycenters of nonpositively curved metrics see [11]). The definition of τx assumes the set O x is nonempty; this will be true for m-a.e. x ∈ M because of the absolute continuity of the center foliation. The definition of τx also assumes that O x is a bounded subset of C Exu . This is clear under assumption (A) that the center foliation is uniformly compact. For the case of assumption (B) when the center foliation has mostly noncompact leaves which are fixed by f we must give a different argument for the boundedness of O x . We define o n c ∗ O xb = ( Hxy ) τˆy : y ∈ Ω ∩ Ux

where we recall that Ux is the neighborhood of x in W c ( x ) used to define the bundle E . There is a uniform bound on the ρ x -diameter of the sets O xb from the reference conformal structure τx0 on Exu . We write

Ox =

[

n ∈Z

( D f xn )∗ (O bf n ( x ) )


27

Since f is uniformly quasiconformal there is a uniform bound (independent of n ∈ Z) on the distance from ( D f n )∗ (O bf n ( x ) ) to τx0 . It follows that the set O x is bounded in C Exu . Since • H c is equivariant with respect to D f according to equation (4.6), • τˆ is D f -invariant, • f (W c ( x )) = W c ( f ( x )) for every x ∈ M, and lastly • D f ∗ is a fiberwise isometry on the conformal structure bundle C Eu , we have D f ∗ (O f ( x )) = O x and consequently D f ∗ τ f ( x ) = τx . Thus τ is also an invariant conformal structure. Clearly τ is invariant under H c holonomy because of c ◦ H c = H c for x, y, z in the same center leaf. By the the composition property Hyz xy xz c equivariance of H with respect to H u and H s given by equations (4.1) and (4.2), τ is also invariant under H s and H u holonomy on Ω. In particular τ is invariant under both H cs and H u so τ is invariant under uniformly continuous holonomies along two transverse foliations of M. It follows that τ is uniformly continuous on Ω and thus has a unique continuous extension to M which is invariant under H c , H u , and H s holonomies. By combining Lemmas 24 and 25 we derive the main result of this subsection, C OROLLARY 26. The center-stable holonomy h cs between local unstable leaves is analytic u u ∞ in the charts {Φ x } x ∈ M . Hence hcs xy : Wloc ( x ) → Wloc ( y ) is a C diffeomorphism Proof. Let τ be the continuous invariant conformal structure on Eu from Lemma 25 cs ( x ). We consider τ and τ which is invariant under H c , H u and H s . Let y ∈ Wloc x y as conformal structures ω x and ωy on the Euclidean spaces Exu and Eyu respectively by using the canonical identification for each v ∈ Exu of Tv Exu with Exu and assigning ω x to be the image of τx in Tv Exu under this identification. 1 cs ∗ u We claim that (Φ− y ◦ h xy ◦ Φ x ) ω y = ω x . To show this, let v ∈ Ex be given 1 cs u ′ and let v′ = Φ− y ( h xy ( Φ x ( v ))) ∈ Ey be its image. Let z = Φ x ( v ) and w = Φy ( v ). Similarly to Lemma 23 we write 1 cs −1 Dv ( Φ− y ◦ h xy ◦ Φ x ) = D0 ( Φy ◦ Φw ) 1 cs −1 ◦ D0 ( Φ − w ◦ h zw ◦ Φz ) ◦ Dv ( Φz ◦ Φ x ) u 1 cs u = Hwy ◦ D0 ( Φ − w ◦ h zw ◦ Φz ) ◦ Hxz u cs u = Hwy ◦ Hzw ◦ Hxz cs is the derivative of h cs at z from where in the third line we used the fact that Hzw zw Lemma 24 and that both D0 Φz = Id Ezu and D0 Φw = Id Ewu . By the invariance of τ 1 cs ∗ u under H cs and H u we conclude that Dv (Φ− y ◦ h xy ◦ Φ x ) ω y = ω x for every v ∈ Ex . Identifying Exu and Eyu with the Euclidean space R k , the inner products ω x and ωy are smoothly equivalent to the Euclidean norm on R k . Thus conformal mappings with respect to these inner products are the same as conformal mappings 1 cs with respect to the standard Euclidean metric. Since Φ− y ◦ h xy ◦ Φ x is conformal k as a map between two open subsets of R we conclude that it is analytic: for k = 2 this is a classical result in one-variable complex analysis and for k ≥ 3 this follows from Gehring’s theorem that all 1-quasiconformal mappings between subdomains of R k are the restrictions of Mobius ¨ transformations to these domains [15].

28

C. BUTLER AND D. XU

4.4. Regularity of the foliations. We now prove higher regularity of the W cu , W cs , and W c foliations under additional bunching hypotheses on f . We begin with a folklore lemma which enables us to deduce regularity properties of a foliation from regularity properties of its holonomy maps between a specific family of transversals. When this family of transversals is smooth Lemma 27 follows directly from the claims in [27]; the proof of Lemma 27 is essentially identical to the proof of [29, Lemma 3.2] which handled the specific case of weak foliations of Anosov flows which were transverse to the strong unstable foliation. L EMMA 27. Let an integer r ≥ 1 and α > 0 be given. Suppose that W and F are two transverse foliations of M such that both W and F have uniformly Cr +α leaves. We further suppose that the local holonomy maps along W between any two F -leaves are locally uniformly Cr +α . Then W is a Cr +α foliation of M. Proof. Let n = dim M and k = dim W . As in [29, Lemma 3.2], we fix a point x ∈ M together with a neighborhood V of x and choose a C ∞ coordinate chart g : V → R k × R n−k such that g(V ∩ W( x )) ⊂ R k × {0} and g(V ∩ F ( x )) ⊂ {0} × R n−k . We then define for p = (y, z) ∈ g(V ), Ψ( p) = (y, g(W)( p) ∩ g(F )(0)) = (y, h p,0 (y)), where g(W), g(F ) denote the images of our foliations under g and h p,0 (y) is the unique intersection point of g(W)( p) with g(F )(0) inside of g(V ). This map straightens the W -foliation into a foliation of R k × R n−k by k-disks D k × {z}. Since the leaves of W are uniformly Cr +α the map Ψ is Cr +α when restricted to the leaves of W , and since the holonomy maps of the W foliation between F -transversals are uniformly Cr +α the chart Ψ is also Cr +α along the leaves of F . By Journé’s lemma [20] this implies that Ψ is Cr +α . L EMMA 28. Suppose f is r −bunched for some r ≥ 1, then there is an α > 0 such that W c , W cs , and W cu are Cr +α foliations of M. If f is ∞-bunched then these foliations are all C ∞ . Proof. Since f is C ∞ and r −bunched, there is an α > 0 such that the leaves of ∗ , ∗ ∈ {cs, cu, c } are uniformly Cr +α . By Corollary 26 the cs-holonomy maps Wloc between local unstable leaves are analytic diffeomorphisms. Hence using W u as our transverse foliation F for Lemma 27 we conclude that W cs is a Cr +α foliation of M. By applying all of the results of this section to the cs-holonomy maps of f −1 instead (i.e., the cu-holonomy maps of f ) we conclude that the cu-holonomy maps are analytic between local stable leaves. Hence we also obtain that W cu is a Cr +α foliation of M. For each x ∈ M and m = dim Eu , k = dim Ec , n = dim M we can thus find a neighborhood V of x and a Cr +α foliation chart Ψ : V → D m+k × D n−m−k = D m × D n−m ⊂ R n , such that W cu is mapped to the foliation by (m + k)-cubes D m+k × {z}, z ∈ D n−m−k and W cs is mapped to the foliation by (n − m)-cubes {y} × D n−m , y ∈ D m (here D j again denotes the open unit cube in R j ). The intersection of these two foliations is the image of W c which is a foliation by k-disks {y′ } × D k × {z′ }, y′ ∈ D m ,


29

z′ ∈ D n−m−k . Thus Ψ is also a Cr +α foliation chart for Ec and therefore W c is also a Cr +α foliation of M. Our final lemma applies under both assumptions (A) and (B) in the case that the center is 1-dimensional. and is a straightforward consequence of the C1 regularity of the center foliation together with the fact that in dimension 1, length and volume are the same. L EMMA 29. If dim Ec = 1 then f is ∞-bunched and therefore W c , W cs , and W cu are C ∞ foliations of M. Furthermore there is a C ∞ norm | · | on Ec with respect to which D f | Ec acts by isometries. Proof. When dim Ec = 1, f is always 1-bunched. Hence by Lemma 28 the center foliation W c is C1+α for some α > 0. Let νxc denote the Riemannian volume on Ux ⊂ W c ( x ). Since W c is a C1+α foliation the conditional measures {mcx } x ∈ M of the volume m on the sets Ux are absolutely continuous with continuous densities with respect to νxc . Thus there are positive continuous functions ζ x : Ux → R such that dmcx = ζ x dνxc which also depend continuously on x ∈ M. Since f ∗ mcx = mcf ( x ) and f ∗ νxc (y) = k D f y | Eyc k−1 νcf ( x )(y) we thus derive the relationship ζ x (y) = k D f y | Eyc k, ζ f ( x )( f (y)) which is valid for y ∈ Ux . We set ξ ( x ) := ζ x ( x ). Thus ξ : M → (0, ∞) is a measurable function satisfying the equation, ξ (x) = k D f x | Exc k, ξ ( f ( x )) for every x ∈ M. It is then clear that D f | Ec acts by isometries with respect to the norm | · | = ξ · k · k on Ec . We will first show that ξ is continuous on M. By [27] the center bundle Ec is β-Holder ¨ continuous for some β > 0. Thus the 1-dimensional linear cocycle D f | Ec admits continuous stable and unstable holonomies Ps and Pu according to Proposition 7. Let {Kn }n≥1 be an increasing sequence of subsets of M such that ξ is uniformly continuous on Kn and m( M \Kn ) < 2−n . By Lemma 17 there is a full measure subset Ω of M such that if x, y ∈ Ω then there is an n > 0 and there are infinite sequences nk → ∞ and n′k → ∞ such that f nk ( x ), f nk (y) ∈ Kn and ′ ′ f − n k ( x ), f − n k ( y ) ∈ Kn . s ( x ), Then for x, y ∈ Ω with y ∈ Wloc n

ξ (x) ξ ( f nk ( x )) k D f x k | Exc k . = · ξ (y) ξ ( f nk (y)) k D f ynk | Eyc k Letting nk → ∞ we have

ξ ( f n k ( x )) ξ ( f n k ( y ))

n

→ 1 and

k D f x k | Ec k n

x

k D f y k | Ec k

that s ξ ( x ) = Pxy ξ ( y ).

y

s . Hence we conclude → Pxy

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C. BUTLER AND D. XU

u ( x ) we conclude instead that ξ ( x ) = P u ξ ( y ). The holonomies Similarly if y ∈ Wloc xy s u P and P are uniformly continuous on local stable and unstable leaves respectively so this shows that ξ restricted to Ω is uniformly continuous on local stable and unstable leaves. c ( x ) we recall from the construction of the measures m c in PropoFor y ∈ Wloc x c such that P c · m c and m c sition 16 that there is a unique positive constant Pxy xy x y coincide on the overlap Ux ∩ Uy of the two neighborhoods on which these measures are each supported. Because the neighborhoods {Uy }y∈W c ( x ) vary continuously over points in the same center leaf and these are the neighborhoods used to c varies uniformly normalize the measures mcy in Proposition 16 we obtain that Pxy c continuously in x and y for y ∈ Wloc ( x ). The measures νxc and νyc clearly also coincide on the overlap Ux ∩ Uy so it follows c ζ = ζ on U ∩ U . Hence for y ∈ W c ( x ) we have ξ ( y ) = P c ζ ( y ). that Pxy x y x y xy x loc Thus ξ is also uniformly continuous when restricted to the center foliation. Since ξ is uniformly continuous when restricted to each of the three foliations W s , W u , and W c we conclude ξ is continuous on M. Hence there is a continuous norm on | · | on Ec with respect to which D f | Ec acts by isometries. This implies that f is r-bunched for every r ≥ 1, i.e., f is ∞-bunched. Thus by Lemma 28 W cu , W cs , and W c are C ∞ foliations of M. This implies that the ˆ cx } x ∈ M of m from Proposition 16 on the W c are both C ∞ in conditional measures {m the basepoint x ∈ M and are C ∞ equivalent to the smooth Riemannian arclength νxc on W c ( x ); for this assertion recall that we assume that m is smoothly equivalent to the Riemannian volume on M. In the case of assumption (A) this implies without further argument that the family of conditional measures {mcx } x ∈ M used in this proof are also C ∞ in x ∈ M and are C ∞ equivalent to νxc , since there is a canonical smooth normalization of ˆ cx } x ∈ M such that mcx (W c ( x )) = 1 for each x ∈ M. In the case of the family {m assumption (B) we only need to note that the arcs Ux ⊂ W c ( x ) are determined by their endpoints in a canonical smooth fashion according to the discussion at the beginning of Section 4 and these endpoints are given by f −1 ( x ) and f ( x ), which clearly smoothly depend on x. Hence there is a smooth normalization of the famˆ cx } x ∈ M of conditional measures such that mcx (Ux ) = 1 for every x ∈ M and ily {m we obtain the same conclusion as we did in the case of assumption (A). As a consequence the family of C ∞ functions ζ x : Ux → (0, ∞) is also C ∞ in the basepoint x, so we conclude that ξ ( x ) = ζ x ( x ) is C ∞ and consequently the norm | · | on Ec is C∞ .

4.5. Proofs of Theorems 2-4. Proof of Theorem 2. Since f is dynamically coherent, r-bunched, volume-preserving, and uniformly quasiconformal with uniformly compact center foliation, we conclude from the results of Section 4 that the foliations W cs , W cu and W c are Cr +α for some α > 0. We define N to be the quotient of M by the equivalence relation x ≡ y if y ∈ W c ( x ). Since the center foliation is uniformly compact, N is a topological manifold, and since W c is a Cr foliation of M we actually conclude that N is a Cr manifold and f descends to a Cr +α Anosov diffeomorphism g : N → N. The invariance of the conformal structure τ from Lemma 25 under center holonomy implies that τ descends to a conformal structure τ¯ on the unstable bundle


31

of g acting on N. This shows that g is uniformly u-quasiconformal. An analogous argument using the invariant conformal structure on Es shows that g is also uniformly s-quasiconformal. Hence g is a Cr +α uniformly quasiconformal Anosov diffeomorphism of N. Thus g satisfies Hasselblatt’s pinching conditions for C1+α regularity of the Anosov splitting [18] and we conclude that the stable and unstable foliations W s,g and W u,g of g are C1+α . Our remaining task is to show that the stable and unstable foliations W s and W u of f are C1+α . By Lemma 27 it suffices to show that the local stable holonomy maps of W s between leaves of the center-unstable foliation W cu are C1+α . The proof of this same fact for the local unstable holonomy maps will be analogous. Since f is r-bunched for some r ≥ 1, by [27] the stable holonomy maps between W c leaves are C1+α diffeomorphisms for some (possibly smaller) α > 0. Since W s and W u project under the smooth submersion π : M → N to C1+α foliations W s,g and cu ( x ) → W cu ( y ) W u,g of N we conclude that the local stable holonomy maps Wloc loc 1 + α u cu are also C when restricted to the subfoliation W of W . Hence the local stable holonomy maps between center-unstable leaves are C1+α when restricted to the two subfoliations W c and W u of W cu so it follows by Journé’s lemma that these local stable holonomy maps themselves are C1+α and thus W s is a C1+α foliation of M. This completes the proof of claims (1) and (2) of Theorem 2. We now assume that f is ∞-bunched. Applying the results of the previous two paragraphs with r = ∞ we conclude that W cs , W cu and W c are C ∞ foliations of M, N is a C ∞ manifold, and g : N → N is a C ∞ volume-preserving uniformly quasiconformal Anosov diffeomorphism. By the classification theorem of Fang [12] g is smoothly conjugate to a hyperbolic toral automorphism and the stable and unstable foliations W s,g and W u,g of g are C ∞ . Lastly, to show that the stable and unstable foliations W s and W u of f are C ∞ we repeat the argument of the previous paragraph with the C1+α -regularity replaced with C ∞ -regularity for the stable and unstable foliations of g. Proof of Corollary 3. Since f is uniformly quasiconformal with uniformly compact center foliation and dim Ec = 1, by Lemma 29 f is ∞-bunched and there is a smooth norm | · | on Ec such that D f | Ec is an isometry with respect to this norm. Hence the conclusions of part (3) of Theorem 2 apply to f so that the foliations W c , W u , and W s of M are C ∞ , the quotient π : M → N of M by the center foliation is a torus and there is a hyperbolic toral automorphism g : N → N such that π ◦ f = g ◦ π. Since there is a smooth norm on Ec with respect to which D f | Ec acts by isometries we conclude that f is an isometric extension of g. Proof of Theorem 4. We will first prove that the foliations W cs , W cu , and W c are C ∞ for any C1 -small enough volume-preserving uniformly quasiconformal perturbation f of ψ1 under the assumption that ψt itself has no periodic orbits of period ≤ 2. We claim that there is a C1 -open neighborhood U of ψ1 in the space of smooth volume-preserving diffeomorphisms of M such that if f ∈ U then f satisfies assumption (B) of Section 4. Since ψ1 is partially hyperbolic, the center foliation W c,ψ1 for ψ1 is normally hyperbolic and every center leaf is fixed by ψ1 , by the work of Hirsch, Pugh, Shub [19] we deduce that any f which is C1 close to ψ1 is partially hyperbolic, dynamically coherent, and has the property that f (W c ( x )) = W c ( x ) for every x ∈ M. Furthermore the center bundle Ec for f is

32

C. BUTLER AND D. XU

orientable with orientation preserved by f because it is C0 close to the orientable center bundle for ψ1 . Since ψt has no periodic orbits of period ≤ 1, ψ1 has no fixed points and thus if the neighborhood U is chosen small enough then f will have ψ no fixed points as well. Finally, consider for each x ∈ M the flow line Ux of x S ψ in the center foliation W c,ψ1 of ψ1 given by Ux = t∈[−1,1] ψt ( x ). Since ψt has no ψ

periodic orbits of period ≤ 2, Ux is a subarc of W c,ψ1 ( x ) which is not a circle. The subarc of Ux of the center leaf W c ( x ) for f through x constructed at the beginning ψ of Section 4 is uniformly close to Ux and thus if f is C1 close enough to ψ1 then Ux will be a subarc of W c ( x ) instead of a circle. Thus f does not wrap and so f satisfies assumption (B) of Section 4. By Proposition 29 there is thus a C ∞ norm | · | on Ec with respect to which D f | Ec is an isometry and we also conclude that the invariant foliations W cs , W cu , and W c for f are C ∞ . Since Ec is a smooth orientable subbundle of TM we can find a smooth nonvanishing section Z : M → Ec . ˆ → M of M such that the Now suppose only that there is a finite cover p : M ˆ ˆ lift ψt of ψt to M has no periodic orbits of period ≤ 2. Let Uˆ be the C1 -open neighborhood of ψˆ 1 given by Theorem 4 applied to ψˆ t . We let U be the C1 -open neighborhood of ψ1 consisting of all smooth volume-preserving diffeomorphisms ˆ cs , W ˆ cu , and W ˆ c of the invariant ˆ →M ˆ lies in Uˆ . Then the lifts W f whose lift fˆ : M ∞ ˆ and thus since the projection p is a local foliations for f are C foliations of m smooth diffeomorphism we conclude that the invariant foliations W cs , W cu , and W c for f are C ∞ . We take Z to be a nonvanishing C ∞ section of Ec . Since f preserves the orientation of Ec and fixes every center leaf we can, by replacing Z with − Z if necessary, arrange that for every x ∈ M, Z ( x ) is oriented in the direction of f ( x ) on W c ( x ). Let ϕtZ be the C ∞ flow generated by Z. Its flowlines are the center foliation W c . There is then a positive smooth function σ : M → (0, ∞) such that ϕσZ( x ) = f ( x ) for every x ∈ M. We then set ϕt ( x ) = ϕtZ·σ( x ). Then ϕt is a C ∞ flow on M with ϕ1 = f . The fact that ϕt is a uniformly quasiconformal Anosov flow follows from the partial hyperbolicity and uniform quasiconformality estimates for its time-1 map f . Finally we show that ϕt preserves a measure smoothly equivalent to volume. For each t ∈ R, ϕt : M → M is a smooth diffeomorphism and thus the measures ( ϕˆ t )∗ m and m are smoothly equivalent with C ∞ Radon-Nikodym derivative d ( ϕt ) ∗ m := Jt . Since ϕ1 = f we have J1 ≡ 1. For every x ∈ M we clearly have dm Jt+s ( x ) = Jt ( ϕs ( x )) · Js ( x ) from the property that ϕt+s = ϕt ◦ ϕs . We claim that ϕt is topologically transitive. Since ϕ1 = f is volume-preserving the nonwandering set of ϕt is all of M. By the spectral decomposition theorem for flows [31] we can decompose M into connected components invariant under ϕt on which ϕt is topologically transitive; since M is connected we conclude that ϕt is actually topologically transitive on M. We can thus apply the following criterion for a topologically transitive Anosov flow ϕt to preserve a measure smoothly equivalent to volume: ϕt preserves a measure smoothly equivalent to volume if and only if for every periodic point p of ϕt of period ℓ( p) we have Jℓ( p) ( p) = 1 [26].


33

Suppose that this does not hold, then without loss of generality we can assume that there is a periodic point p for which Jℓ( p) ( p). For this point and any integer n > 0 we then have Jnℓ( p) ( p) = ( Jℓ( p) ( p))n → ∞, as n → ∞. On the other hand, let ⌊nℓ( p)⌋ denote the greatest integer smaller than nℓ( p) and let K := sup0≤t≤1 sup x ∈ M Jt ( x ). Then since J1 ≡ 1 we have Jnℓ( p) ( p) = Jnℓ( p)−⌊nℓ( p)⌋( p) ≤ K < ∞, for each integer n > 0. Thus we obtain a contradiction so that Jℓ( p) ( p) = 1 for every periodic point p and thus ϕt preserves a measure smoothly equivalent to volume on M. 5. P ROOF

OF

T HEOREM 1

Let X be a closed Riemannian manifold of constant negative curvature with dim X ≥ 3 and let T 1 X be the unit tangent bundle of X. We let π : T 1 X → X denote the standard projection of a unit tangent vector to its basepoint in X. We let ψt denote the geodesic flow on T 1 X and consider a smooth, volume-preserving perturbation f of the time-1 map ψ1 . We will establish in this section that the equalities λu+ = λu− and λs+ = λs− imply that D f | Eu and D f | Es respectively are uniformly quasiconformal for small enough volume-preserving perturbations of ψ1 . We will prove this implication for the unstable bundle Eu ; the proof for Es will be analogous. By Theorem 4 and the smooth orbit equivalence classification result of Fang [14] this suffices to complete the proof of Theorem 1 from the Introduction. We first need to recall some properties of the frame flow associated to closed Riemannian manifolds of constant negative curvature. Let X (2) be the 2-frame bundle over X which has fiber over each p ∈ X given by (2)

X p = {(v, w) ∈ Tp1 X : v is orthogonal to w}. (2)

We let ψt be the 2-frame flow on X (2) obtained by applying the geodesic flow ψt to the first vector v ∈ Tp1 X and then taking the image of w under parallel transport along the geodesic γ(s) = π (ψs (v)), s ∈ [0, t], on X. We let Eu,ψ be the unstable bundle of the geodesic flow ψt on T 1 X and we let SEu,ψ be the unit sphere inside of Eu,ψ , where we equip Eu,ψ with the Riemannian norm k · k coming from its realization as the tangent spaces of unstable horospheres in the universal cover of X. We have a smooth identification SEu,ψ → X (2) coming from this realization by identifying a unit vector v ∈ Tp1 X (2)

u,ψ

together with a unit vector w ∈ SEv to the orthonormal 2-frame (v, w) ∈ X p obtained from identifying w with its image in the tangent space of the unstable horosphere through p which is orthogonal to v. Since X has constant negative curvature the geodesic flow is conformal on unstable horospheres and therefore (2)

under this identification the 2-frame flow ψt Dψ ( w )

corresponds to the renormalized

derivative action w → k Dψt ( w)k on SEu,ψ . For a more detailed version of this dist cussion as well as the discussion in the paragraphs below we refer to [6], [8].

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C. BUTLER AND D. XU

We consider the stable and unstable holonomies H s,ψ and H u,ψ of ψt on Eu,ψ H s,ψ (·) , k H s,ψ (·)k u,ψ ′ SEv′ for v ∈

and their renormalized versions SH s,ψ (·) = u,ψ

H u,ψ (·) which k H u,ψ (·)k ′ and v ∈ W u,ψ (v)

SH u,ψ (·) =

give isometric identifications SEv → W s,ψ (v) respectively, where W s,ψ and W u,ψ denote the stable and unstable foliations of ψ respectively. An su-loop based at v ∈ T 1 X is an su-path for ψt which starts and ends at v. Based on the discussion of the previous paragraphs, given an su-loop γ for ψt u,ψ u,ψ based at a point v ∈ T 1 X we can associate an isometry Tψ (γ) : SEv → SEv obtained by composing the renormalized stable and unstable holonomy maps s,ψ u,ψ u,ψ s,ψ u,ψ u,ψ SHvi vi+1 : SEvi → SEvi+1 and SHv j v j+1 : SEv j → SEv j+1 along this loop, where u,ψ

γvi vi+1 ⊂ W s,ψ (vi ) and γv j v j+1 ⊂ W u,ψ (vi ). Thus, identifying SEv with the unit sphere Sn−1 in R n for n := dim X − 1, Tψ (γ) gives us an element of the special orthogonal group SO(n). The key observation due to Brin and Karcher [6] is that for a closed constant negative curvature manifold X and any v ∈ T 1 X there are finitely many su-loops γ1 , . . . , γk such that Tψ (γ1 ), . . . , Tψ (γk ) generate SO(n) as a Lie group when we identify Evu . Moreover the number k of loops used and the total lengths of these loops may both be taken to be bounded independently of the point v. As a consequence we have the following proposition, P ROPOSITION 30. For any δ > 0 there is a constant L > 0 and an integer ℓ > 0 such that given any v ∈ T 1 X there is a finite collection γ1 , . . . , γℓ of su-loops based at v of total u,ψ length at most L for which the collection of points { Tψ (γi )(w)}ℓi=1 is δ-dense in SEv for u,ψ

any w ∈ SEv . u,ψ

Proof. Fix a 2δ -dense collection {w j }kj=1 of points in SEv . Since there are finitely many su-loops based at v whose associated isometries generate SO(n) as a Lie group and SO(n) acts transitively on SEvu , there is a finite collection γ1 , . . . , γℓ of su-loops based at v for which each of the sets { Tψ (γi )(w j )}ℓi=1 for 1 ≤ j ≤ k is δ 2 -dense

u,ψ

in SEv . u,ψ Now let w be any point in SEv . Then there is some w j such that kw − w j k < 2δ . Since each Tψ (γi ) is an isometry we then also have k Tψ (γi )(w j ) − Tψ (γi )(w j )k < δ ℓ 2 for each 1 ≤ i ≤ ℓ. This implies that { Tψ ( γi )( w )}i =1 is a δ-dense subset of u,ψ

SEv .

C1 -small

Let f be a perturbation of the time-1 map ψ1 . If this perturbation is small enough then the linear cocycle D f | Eu is fiber bunched and consequently the conclusions of Proposition 7 apply to D f | Eu , see [22, Proposition 4.2]. Thus the linear cocycle D f | Eu admits linear stable and unstable holonomies H s and H u . For v ∈ T 1 X we define PEvu to be the projective space of Evu and we define PH s and PH u to be the induced maps of H s and H u on the projective spaces PEvu → PEvu′ for v′ ∈ W s (v) and v′ ∈ W u (v) respectively, where now W s and W u denote the stable and unstable foliations of f . We let PD f v : PEvu → PEuf( v) be the induced map from D f v . We obtain below a version of Proposition 30 which also applies to the perturbation f provided that this perturbation is small enough. We endow PEvu with the


35

Riemannian metric induced from the Riemannian metric on Eu which is in turn induced from the metric on T 1 X. Given an su-loop γ for f based at v ∈ T 1 X we associate the map T (γ) : PEvu → PEvu obtained by composing the projectivized stable and unstable holonomies PH s and PH u along the segments of this loop which lie in the stable and unstable leaves of f respectively. Unlike the case of Tψ above, T (γ) is not necessarily an isometry of PEvu . We will need the following proposition that follows from results of Katok and Kononenko, P ROPOSITION 31 ([24]). Let ψt : M → M be a contact Anosov flow on a closed Riemannian manifold. Then there is a C2 -open neighborhood V of ψ1 in the space of C2 diffeomorphisms of M and an integer J > 0 such that for every ε > 0 and every f ∈ V there exists an η > 0 such that for every p, q ∈ M with d( p, q) < η, there exists a J-legged su-path from p to q of length less than ε. Recall that for each pair of nearby points x, y ∈ T 1 X we let Ixy : Exu → Eyu be a linear identification which is Holder ¨ close to the identity. This induces an identification PIxy : PExu → PEyu that is Holder ¨ close to the identity in x and y. L EMMA 32. Given any δ > 0 there is a C2 -open neighborhood U of ψ1 such that if f ∈ U then for any v ∈ T 1 X there is a finite collection γ1 , . . . , γℓ of su-loops for f based at v such that the collection of points { T (γi )(w)}ℓi=1 is δ-dense in PEvu for any w ∈ PEvu . Proof. Let δ > 0 be given. We first apply Proposition 30 to ψt to obtain a constant L > 0 and integer ℓ > 0 such that for any v ∈ T 1 X there is a collection of su-loops σ1 , . . . , σℓ based at v of total length at most L such that { Tψ (σi )(w)}ℓi=1 is 3δ -dense u,ψ

u,ψ

in SEv for any w ∈ SEv . We apply Proposition 31 for a small ε > 0 to be determined. Given the η > 0 obtained from Proposition 31 for this ε we claim that we can find a C2 -open neighborhood U ′ of ψ1 such that for each v ∈ T 1 X there are points v1 , . . . , v ℓ satisfying d(v, vi ) < η and for each 1 ≤ i ≤ ℓ there is an su-path β i for f from v to vi such u u that the collection {PIvi v ◦ T ( β i )(w)}ℓi=1 is 2δ 3 -dense in PEv for any w ∈ PEv . This s u follows from the facts that the stable and unstable foliations W and W depend continuously on f in the C2 topology and the stable and unstable holonomies H s and H u of D f | Eu also depend continuously on f in the C2 topology[2]. Hence we obtain this statement by considering su-paths β 1 , . . . , β ℓ for f which are close enough to the su-loops σ1 , . . . , σℓ for ψt ; we can make these paths as close as desired to the loops for ψt by making the neighborhood U small enough. For each 1 ≤ i ≤ ℓ we let γi be the su-loop based at v for f obtained by concatenating β i with the J-legged su-path of length less than ε connecting v i to v given by u and H s converge Proposition 31. Since the number of legs J is fixed and both Hxy xy uniformly to the identity as y converges to x for x, y ∈ T 1 X we conclude that if ε is small enough (independent of the choice of v ∈ T 1 X) then the collection of points { T (γi )(w)}ℓi=1 is δ-dense in PEvu for any w ∈ PEvu . The use of Lemma 32 is the reason that we lose C1 -openness of the neighborhood U in Theorem 1. It is easy to see that there is a δ0 > 0 with the property that if V1 and V2 are any two proper linear subspaces of R n then the union PV1 ∪ PV2 of their projectivizations in RP n−1 is not δ0 -dense. Thus it follows that there is a δ > 0 and a C1 -open

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neighborhood U ′ of ψ1 such that for any f ∈ U ′ , any v ∈ T 1 X, and any pair of proper linear subspaces V1 and V2 in Evu , the union PV1 ∪ PV2 is not δ-dense in PEvu . We apply Lemma 32 with this δ and let f ∈ U ⊂ U ′ be a smooth volumepreserving diffeomorphism in the resulting open neighborhood with the property that λu+ = λu− . We will now show that D f | Eu is uniformly quasiconformal to complete the proof of Theorem 1. Since ψ1 is a stably accessible partially hyperbolic diffeomorphism (this well-known fact can be derived as a consequence of Proposition 31) we may assume that f is also an accessible partially hyperbolic diffeomorphism. Since the neighborhood U is chosen small enough that D f | Eu satisfies the fiber bunching condition that guarantees the existence of the stable and unstable holonomies H s and H u we conclude by the work of Avila, Santamaria and Viana [2] that the equality λu+ = λu− implies that there is a PD f -invariant probability measure µ on PEu projecting down to the invariant volume m for f on T 1 X and which has a disintegration {µv }v∈ T1 X into probability measures µv on the projective fibers PEvu which depend continuously on the basepoint v. Furthermore this disintegration is invariant under the projectived stable and unstable holonomy, that is to say, if s ) µ = µ ′ and a similar equation holds for PH u . v′ ∈ W s (v) then (PHvv ′ ∗ v v Suppose that D f | Eu is not uniformly quasiconformal. Then there is a point v ∈ T 1 X, unit vectors w1 , w2 ∈ Evu , and a sequence nk → ∞ such that

k D f n k ( w1 )k k D f n k ( w2 )k

→∞

as nk → ∞. By passing to a further subsequence and using the compactness of T 1 X we can assume that there is some z ∈ T 1 X such that f nk (v) → z as nk → ∞. Since f is accessible we can find an su-path σ connecting z to v. For nk large enough we let γnk be a J-legged su-path connecting f nk (v) to z of length at most 1, where J is given by Proposition 31. We then let T (γnk ) : PEufnk ( v) → PEzu be the map obtained by composing the s- and u-holonomies along γnk from f nk (v) to z. Let n

Ank = T (σ ) ◦ T (γnk ) ◦ PD f v k : PEvu → PEvu . The holonomy invariance of the disintegration of the PD f -invariant measure µ implies that ( Ank )∗ µv = µv for each nk . Choose a linear identification of PEvu with the real projective space RP n−1 . Then Ank gives an element of the projective linear group PSL(n, R ) for each nk . Since the transformations T (γnk ) have uniformly bounded norm together with their inverses, and since there exist unit vectors w1 , w2 ∈ Evu such that

k D f n k ( w1 )k k D f n k ( w2 )k

→ ∞, we conclude that the sequence of transformations { Ank } is not contained in any compact subset of PSL(n, R ). Hence, after passing to a further subsequence if necessary,there is a quasi-projective transformation Q of RP n−1 such that Ank converges to Q on the complement of a proper linear subspace V of RP n−1 (see [16]). Furthermore the image of Q is a proper linear subspace L of RP n−1 . Thus there is a proper linear subspace V of PEvu such that on the complement of V, Ank converges pointwise to a continuous map which has image contained inside of a proper subspace L of PEvu . Since ( Ank )∗ µv = µv for every nk , this shows that µv is supported on the union V ∪ L of two proper subspaces of PEvu . Consider any point w ∈ supp(µv ). By Lemma 32 there is a collection of su-loops γ1 , . . . , γℓ based at v such that the collection of points { T (γi )(w)}ℓi=1 is δ-dense in PEvu . But by the holonomy invariance of the disintegration of µ, if γ is an su-loop


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based at v then T (γ)(w) ∈ supp(µv ) ⊂ V ∪ L. This proves that the union V ∪ L of two proper subspaces of PEvu is δ-dense in PEvu , which contradicts our choice of δ. Thus D f | Eu is uniformly quasiconformal.

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