Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian manifold and a uniformly quasiconformal transversely symplectic Anosov flow we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed that the flow is smoothly conjugate to an algebraic one. To establish the latter, we prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz-continuous invariant 1-form. Several features of the reasoning are interesting: The use of exterior calculus for Lipschitz-continuous forms, that the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.


INTRODUCTION
1.1. Statement of main result. For Anosov systems (Definition 1.2), both diffeomorphisms and flows, interesting phenomena of smooth and geometric rigidity have been observed in connection with the degree of (transverse) regularity of the (weak) stable and unstable subbundles of these systems. The seminal result was the study of volume-preserving Anosov flows on 3-manifolds by Hurder and Katok [13], which showed that the weak-stable and weak-unstable foliations are C 1+Zygmund and that there is an obstruction to higher regularity whose vanishing implies smoothness of these foliations. This, in turn, happens only if the Anosov flow is smoothly conjugate to an algebraic one. The cocycle obstruction described by K atok and Hurder was first observed by Anosov and is the first nonlinear coefficient in the Moser normal form. Therefore one might call it the KAM-cocycle. This should not be confused with "KAM" as in "Kolmogorov-Arnold-Moser", and Hurder and Katok refer to this object as the Anosov-cocycle.
In [7] we showed some analogous rigidity features associated with the longitudinal direction, i.e., associated with various degrees of regularity of the sum of the strong stable and unstable subbundles: For a volume-preserving Anosov flow on a 3-manifold the strong stable and unstable foliations are Zygmund-regular [16, Section II.3, (3·1)], see Definition 1.3, and there is an obstruction to higher regularity, which admits a direct geometric interpretation and whose vanishing implies high smoothness of the joint strong subbundle and that the flow is either a suspension or a contact flow. In the papers announced here [8,9] we push this to higher-dimensional systems: To show that 3. implies 4. we study the canonical 1-form (Definition 1.2) of the time-change of a geodesic flow or of the suspension of an infranilmanifoldautomorphism, and because we only have Lipschitz-continuity at our disposal, we need to explore how smooth-rigidity results can be pushed to the lowest conceivable regularity. This requires two main results. On one hand, a Lipschitzcontinuous 1-form whose exterior derivative is invariant under the geodesic flow of a negatively curved locally symmetric space must be (a constant multiple of) the canonical 1-form for the flow. On the other hand, an essentially bounded 2-form invariant under an infranilmanifoldautomorphism is smooth. A special case of the second result is that in which the 2-form arises as the exterior derivative of a Lipschitz-continuous 1-form, in which case it vanishes. Thus, both results involve exterior calculus of Lipschitz-continuous 1-forms. It is clear that this has to be done with care, and we invoke results to the effect that, for instance, the classical Stokes Theorem holds for Lipschitz-continuous forms [4].
Given this common motivation, it is surprising that our two separate results involve rather different arguments for geodesic flows on one hand and suspensions on the other hand.
A particular point of interest is in this respect that while, like with many other results in hyperbolic dynamics, ergodic theory enters the proof in the case of geodesic flows only to the extent that we use ergodicity of the geodesic flow, it turns out that for the case of a suspension we use in an essential way that the infranilmanifoldautomorphism is indeed mixing rather than merely ergodic. This reflects the need to deal with parabolic effects due to the nilpotent part.
1.2. Background and terminology. We now introduce the notions that play a role in this result and the proof.  ([14]). An Anosov flow on a manifold M is a smooth flow ϕ t with • an invariant decomposition T M = X ⊕ E u ⊕ E s (where X =φ = 0 is the generator of the flow and E u and E s are called the unstable and stable subbundles) and • a Riemannian metric on M such that Dϕ t ↾ E s and Dϕ −t ↾ E u are contractions whenever t > 0.
The definition of Anosov diffeomorphism is analogous with t ∈ Z and X absent.
The canonical 1-form A of an Anosov flow ϕ t is defined by A(X ) = 1 and E u , E s ⊂ ker A. A canonical time-change is defined using a closed 1-form α by replacing the generator X of the flow by the vector field X /(1 + α(X )), provided α is such that the denominator is positive. (See Section 2 for more on canonical time-changes.) The subbundles are invariant and (Hölder-) continuous with smooth integral manifolds W u and W s that are coherent in that q ∈ W u (p) ⇒ W u (q) = W u (p). W u and W s define laminations (continuous foliations with smooth leaves).

DEFINITION 1.3. A function f between metric spaces is said to be Hölder continuous if there is an
for all x ∈ U and sufficiently small h . To specify a value of Z we may refer to a function as being Z -Zygmund. The function is said to be "little Zygmund" (or For maps between manifolds these definitions are applied in smooth local coordinates. Zygmund regularity implies modulus of continuity O(|x log |x||) and hence H -Hölder continuity for all H < 1 [16, Section II.3, Theorem (3·4)]. It follows from Lipschitz-continuity and hence from differentiability. Being "little Zygmund" implies having modulus of continuity o(|x log |x||) and follows from differentiability but not from Lipschitz-continuity.
The regularity of the unstable subbundle E u is usually substantially lower than that of the weak-unstable subbundle E u ⊕ E ϕ . The exception are geodesic flows, where the strong unstable subbundle is obtained from the weak-unstable subbundle by intersecting with the kernel of the invariant contact form. This has the effect that the strong-unstable and weak-unstable subbundles have the same regularity. However, time changes affect the regularity of the strong-unstable subbundle, and this is what typically keeps its regularity below C 1 . In [7] we presented a longitudinal KAM-cocycle that is the obstruction to differentiability, and we derived higher regularity from its vanishing.

THEOREM 1.4 ([7, Theorem 3]).
Let M be a 3-manifold, k ≥ 2, ϕ : R×M → M a C k volume-preserving Anosov flow. Then E u ⊕E s is Zygmund-regular, and there is an obstruction to higher regularity that can be described geometrically as the curvature of the image of a transversal under a return map. This obstruction defines the cohomology class of a cocycle (the longitudinal KAM-cocycle), and the following are equivalent: 5. ϕ is a suspension or contact flow.
In 5. no stronger rigidity should be expected because E u ⊕ E s is smooth for all suspensions and contact flows. See [15,3] for applications of this to magnetic flows.
The work by Hurder and Katok in [13] inspired developments of substantial extensions to higher dimensions, see, for example, [12]. The present work extends our earlier work to higher-dimensional systems in this "longitudinal" context. This requires somewhat stringent assumptions, however.
In the 3-dimensional case we showed that smoothness of E u ⊕ E s implies that ϕ is a suspension or contact flow, but in the present situation we obtain more detailed information because of the quasiconformality-assumption. This uses a rigidity theorem by Fang: This also serves to illustrate that the assumption of uniform quasiconformality is quite restrictive. We should also point out that our result about rigidity of the situation with E u ⊕ E s ∈ C 1 overlaps with a closely related one by Fang, although the proof is independent: Indeed, by Theorem 1.6, ϕ is smoothly orbit equivalent to either the geodesic flow of a real hyperbolic manifold or a suspension of a symplectic automorphism of an n-torus. To show that the flow is, after rescaling, smoothly conjugate to one of these models, use the orbit equivalence to regard the canonical form for ϕ as an invariant form for the algebraic system and then apply Theorem 1.8 or Theorem 1.9.
After introducing some background on canonical time-changes, we outline how to establish Theorem 1.8 and Theorem 1.9.

CANONICAL TIME-CHANGES
We make a few remarks here about canonical time-changes (Definition 1.2) because these are not frequently encountered in the literature; they may serve to show why this is a natural class of time-changes to expect in rigidity results for flows.
Proof. Smoothness of f and 1 + d f (X ) > 0 ensure that Ψ is a diffeomorphism. Now we write x t = ϕ t (x) and use the chain rule to compute Proof. Writing β = α + d f with smooth f we observe that which implies that L X A = 0, i.e., A is X -invariant and hence proportional to the canonical 1-form of X . But A(X ) = 1 then implies that A is equal to the canonical 1-form of X .

RIGIDITY RESULTS
The results that imply Theorem 1.8 and Theorem 1.9 are of independent interest and will therefore appear in a separate publication from the application [9] to quasiconformal Anosov flows.

REMARK 3.2.
Note that the Lipschitz assumption ensures that d A is defined almost everywhere and essentially bounded [10]. This is all we use. For comparison, we state an earlier result of Hamenstädt:

THEOREM 3.3 ([11, Theorem A.3]). If the Anosov splitting of the geodesic flow of a compact negatively curved manifold is C 1 and A is a C 1 1-form such that d A is invariant, then d A is proportional to the exterior derivative of the canonical 1-form of the geodesic flow.
Proof of Theorem 1.8 from Theorem 3.1. The hypotheses of Theorem 1.8 and of Theorem 3.1 imply smoothness; we need to show that the vector field X that generates the time-change agrees with a canonical time-change of a constantly scaled version of X 0 , where X 0 generates the geodesic flow. Rescale X 0 to X 0 /κ, where κ ∈ R is defined by d A = κd B and then apply the canonical time-change defined by the 1-form α := A − κB . Since the resulting vector field is a scalar multiple of X , the claim follows from where we used B (X 0 ) = 1. That the last denominator is A(X 0 ) and hence positive justifies the use of α to define a canonical time-change.

THEOREM 3.4. Let ψ be a hyperbolic automorphism of a torus or a infranilmanifold Γ\M . Then any essentially bounded invariant 2-form is almost everywhere equal to an M -invariant (hence smooth) closed 2-form.
If, in addition, the form is exact, then it vanishes almost everywhere.

REMARK 3.5.
We point out that in this proof we use that the automorphism is mixing (rather than just ergodic). The need for this is an interesting side-light on how parabolic effects enter into our considerations.
Proof of Theorem 1.9 from Theorem 3.4. Denote by A the canonical form of the time-change and by B the canonical form of the suspension. Theorem 3.4 applied to A implies that A is smooth and closed, and hence so is α:= A −B since B is also closed. Writing X 0 for the suspension vector field we find that the canonical time-change of X 0 is the given vector field since by construction A(X ) ≡ 1.

PROOF OF THEOREM 3.1
Using exterior calculus (carefully!) we show that A is a contact form, i.e., A ∧ n i =1 d A is a volume, and that X 0 is in the kernel of d A. By duality, for every ξ there is a ψ(ξ) such that d A(ξ, ·) = d B (ψ(ξ), ·); this is defined whenever d A is, and we choose ψ(X 0 ) = X 0 and ψ(ker B ) ⊂ ker B . We next show that ψ a.e.
= κ Id +N , where κ ∈ R {0} and N is a nilpotent operator. The main effort is now directed at showing that N = 0, i.e., that d A a.e. = κ d B (smoothness of A can then be obtained via some delicate exterior calculus). To that end we L 1 -approximate N by a continuous operator, take the Birkhoff average of this approximation (which does not change the L 1 -distance to N ) and show that it is defined and continuous everywhere and so intertwined with the flow that one can apply arguments from [1,2] to conclude that it vanishes identically. Thus N is arbitrarily L 1 -close to 0 and hence vanishes itself.
Taking Birkhoff averages of the continuous L 1 -approximation F of N requires substantial technical underpinnings because the Birkhoff Ergodic Theorem applies to scalar functions. Therefore we show that we can choose a measurable orthonormal frame field for E u on M that consists of vector fields ξ chosen in such a way that • ξ is continuous and D-parallel along unstable leaves that are homeomorphic to Euclidean space, i.e., nonperiodic leaves, • ξ ∈ E j for j = 1 or j = 2, • if ξ ∈ E j then the Lie bracket with the generator X of the geodesic flow is [X , ξ] = D X ξ + j ξ = j ξ (the last equality uses that ξ is D-parallel) and hence γ t (ξ) = e j t ξ. Here D is a Kanai connection constructed for this purpose (as in [1,2]), and its properties produce such a frame field. We use that the geodesic flow of a locally symmetric space admits an invariant splitting of the unstable subbundle into fast-and slow-unstable bundles corresponding to the exponents 1 and 2; we write E u = E 1 ⊕ E 2 . In the constant-curvature case we have E 2 = 0.
We next show that the Birkhoff average of F has a continuous extension F to the entire manifold that is parallel along stable and unstable manifolds. We then follow arguments in [1,2] to show that this implies F = 0 and hence N = 0.

PROOF OF THEOREM 3.4
Consider a hyperbolic automorphism ψ of either a torus or a nilmanifold Γ\N . Suppose ω is an essentially bounded 2-form such that ψ * ω = ω.
One can write a 2-form locally as 1≤i < j ≤n a i j d x i ∧ d x j , and we will look for a way of doing so globally and with constant coefficients. To that end we pass to the complexification of the tangent bundle and work with a basis of N -invariant sections X i .
At one point we choose a basis in such a way that ψ is in Jordan canonical form with respect to the dual basis consisting of the forms X * i , i.e., ψ = i a i j X * i with A = (a i j ) in Jordan form (strictly triangular since we passed to the complexification). Now translate the basis and the dual basis by N to get invariant sections. With these choices, a Jordan block for eigenvalue λ l for the nth iterate is of the form A n l = λ n l M ℓ,λ l (n) with M ℓ,λ l (n) a fixed polynomial in n. Now denote by Ω(x) the matrix that represents ω x with respect to the invariant frame field. Then the matrix of ψ * ω is given by t A Ω(x)A and hence the iterated relation ω = ψ n * ω becomes Ω(ψ n (x)) = t A n Ω(x)A n , which is bounded (in n) for almost every x (since ω is essentially bounded). Fix such an x and decompose Ω into (not necessarily square or diagonal) blocks Ω i j according to the Jordan form of A, i.e., in such a way that where ℓ i and ℓ j are the sizes of the blocks A i and A j , respectively. For any i , j and x, P i j x (n) is a matrix-valued polynomial in n, and indeed it is constant: • If |λ i λ j | = 1 then P i j x (n) = 0-otherwise |(λ i λ j ) n P i j x (n)| grows exponentially and is, in particular, unbounded. • If |λ i λ j | = 1 then P i j x (n) is constant (in n)-otherwise |(λ i λ j ) n P i j x (n)| = |P i j x (n)| is unbounded.
We therefore have t M ℓ i ,λ i (n)Ω i j (x)M ℓ j ,λ j (n) = P i j x (n) = P i j x (0) = Ω i j (x), so Ω i j (ψ n (x)) = (λ i λ j ) n Ω i j (x) for all i , j . This shows that every entry of the matrix Ω is almost everywhere equal to an eigenfunction of ψ. Since ψ is mixing, all eigenfunctions are constant, and hence Ω is almost everywhere equal to an M -invariant (hence smooth) 2-form.
To prove the last assertion of Theorem 1.9, i.e., that Ω vanishes if it is exact, we introduce a notion of averaging. Let vol C :=X * 1 ∧ · · · ∧ X * n be the N -invariant complex-valued volume form defined by the dual basis we used before. By compactness, this gives a finite volume. Then for any p-form α = i 1 ,...,i p α i 1 ,...,i p X * i 1 ∧ · · · ∧ X * i p we define the average α := i 1 ,...,i p ( Γ\N α i 1 ,...,i p d vol C )X * i 1 ∧ · · · ∧ X * i p and prove that d α = d α for any 1-form α. We can write and since the coefficients here are constant, we obtain d ψ * α = ψ * d α.