Differentiable Conjugacy of Anosov Diffeomorphisms on Three Dimensional Torus

We consider two C^2 Anosov diffeomorphisms in a C^1 neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are C^1+ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.


Introduction
Consider an Anosov diffeomorphism f of a compact smooth manifold. Structural stability asserts that if a diffeomorphism g is C 1 close to f then f and g are topologically conjugate. The conjugacy h is unique in the homotopy class of identity. h It is known that h is Hölder continuous. There are simple obstructions for h being smooth. Namely, let x be a periodic point of f , f p (x) = x. Then g p (h(x)) = h(x) and if h were differentiable then Df p (x) = (Dh(x)) −1 Dg p (h(x))Dh (x) i.e. Df p (x) and Dg p (h(x)) are conjugate. We see that every periodic point carries a modulus of C 1 -differentiable conjugacy.
Suppose that for every periodic point x, f p (x) = x, differentials of return maps Df p (x) and Dg p (h(x)) are conjugate then we say that periodic data (p. d.) of f and g coincide.
Suppose that p. d. coincide, is h differentiable? A positive answer for Anosov diffeomorphisms of T 2 was given in [LMM], [L]. De la Llave [L] observed that the answer is negative for Anosov diffeomorphisms of T d , d ≥ 4. He constructed two diffeomorphisms with the same p. d. which are only Hölder conjugate.
We provide positive answer to the previous question in dimension three under an extra assumption.
The authors would like to thank A.Katok for suggesting us the problem, numerous discussions and constant encouragement.

Formulation of the main result
Let f be an Anosov diffeomorphism of T d . It is known [M] that f is topologically conjugate to a linear torus automorphism L. It is also known that Anosov diffeomorphisms of T 3 are the only Anosov diffeomorphisms on three dimensional manifolds [Fr], [N].
Let L be a hyperbolic automorphism of T 3 with real eigenvalues. It is easy to show that absolute values of these eigenvalues are distinct. For the sake of notation we also assume that the eigenvalues are positive. This is not restrictive.
We will always assume that the Anosov diffeomorphisms that we are dealing with are at least C 2 .
Theorem 1. Given L as above there exists a C 1 -neighborhood U of L such that any f and g in U having the same p. d. are C 1+ν conjugate, ν > 0.
Remark. The constant ν depends on the size of U and provided sufficient smoothness of f and g can be made as close as desired to log λ 3 / log λ 2 (see the definition in the next section) by shrinking the size of U.
Remark. We don't know how to bootstrap regularity of h to the regularity f and g like it was done in dimension two.
A result about integrability of central distribution [BI] allows to show a stronger statement.
Theorem 2. Let f and g be Anosov diffeomorphisms of T 3 and where h is a homeomorphism homotopic to identity. Suppose that p. d. coincide.
Also assume that f and g can be viewed as partially hyperbolic diffeomorphisms: Analogous conditions with possibly different set of constants hold for a g-invariant Remark. Here and further in the paper we assume that the unstable distribution has dimension two. Obviously one can formulate the counterpart of Theorem 2 in the case when stable distribution has dimension two.

Scheme of the proof
Here we outline the proof of Theorem 1. Let λ 1 , λ 2 and λ 3 be the eigenvalues of the linear automorphism L, 0 < λ 1 < 1 < λ 2 < λ 3 . We choose U in such a way that every f ∈ U is partially hyperbolic, satifying (1) with constants α,β, β, γ independent on the choice of f , 0 < λ 1 < α < 1 <β < λ 2 < β < γ < λ 3 and First we concentrate on a single diffeomorphism f in U. It is well known that distributions E s f , E u f = E wu f ⊕ E su f and E su f integrate uniquely to stable, unstable and strong unstable foliations W s f , W u f and W su f respectively. We denote by W σ f (x) the leaf of W σ f passing through x, σ = s, u, su and later wu. By W σ f (x, R) we denote the local leaf of size R, i. e., a ball of radius R inside of W σ f (x) centered at x, σ = s, u, wu, su. Let h f be conjugacy between f and L, h f • f = L • h f . Stable and unstable foliations can be characterized topologically, e.g.
In other words h f maps leaves of foliations for f into leaves of corresponding foliations for L.
We prove two simple lemmas.
Lemma 1 Lemma 3. The conjugacy h is C 1+ν along W s f . Which means that h is differentiable along the stable foliation and the derivative is a Hölder continuous function on T 3 with exponent ν Remark. The general strategy of the proof of Theorem 1 is similar to de la Llave's strategy for Anosov diffeomorphisms of T 2 [L]. One proves smoothness of h along one dimensional stable and unstable foliations. In particular proof of Lemma 3 can be carried out in the same way as in dimension two. The hard part is showing smoothness of h along two dimensional unstable foliation.
We would like to show the same for the foliation W wu f but we split the proof into two steps.
Lemma 4. The conjugacy h is uniformly Lipschitz along W wu f .
Lemma 5. The conjugacy h is C 1+ν along W wu f . After that we deal with the remaining foliation.
We would like to remark that Lemma 6 requires only the coincidence of p. d. in the weak unstable direction. It is not true in general that strong unstable foliations match.
Lemma 7. The conjugacy h is C 1+ν along W su f . Remark. Proofs of smoothness along the foliations W s f and W su f are similar and use the coincidence of periodic data in corresponding directions. Showing smoothness along the weak unstable foliation is more subtle. Now smoothness of h is a simple consequence of a regularity result. Regularity Lemma. [J] Let M j be a manifold and W s j , W u j be continuous transverse foliations with uniformly smooth leaves, j = 1, 2. Suppose that h : Moreover assume that the restrictions of h to the leaves of these foliations are uniformly C r+ν , r ∈ N, 0 < ν < 1, then h is C r+ν .
First we apply the lemma on every unstable leaf of W u f for the pair of foliations W wu f , W su f . After we know that h is C 1+ν along W u f we finish by applying the lemma to stable and unstable foliations.
The structure of the next chapter is the following. We prove Lemmas 1 and 2 in Section 4.1. Section 4.2 is devoted to the proof of Lemma 4. Sections 4.3 and 4.4 are the heart of our argument and contain proofs of Lemmas 5 and 6 correspondingly.

Proof of Theorems 1 and 2
First we prove Theorem 1.

Weak unstable foliation.
In the proofs of Lemmas 1 and 2 we work with lifts of maps, distributions and foliations to R 3 . We use the same notation for the lifts as for the objects themselves.
Denote by d(·, ·) the usual distance in R 3 and let d σ f (·, ·) be the distance in the leaves of W σ f which is defined only for pairs of points lying in the same leaf of W σ f , σ = s, u, su, wu.
Proof of Lemma 1. Let us reason by contradiction. If E wu f is not uniquely integrable then it must branch and we can find points a, b, c ∈ R 3 such that on the other hand For every . Hence a leaf of W su f can be considered as a graph of a Lipschitz function over E su L . The Lipschitz constant depends only on k. It follows that W su f is quasi-isometric: Inequalities (3), (4) and (5) sum up to a contradiction.
Proof of Lemma 2. Suppose that there are two points a and b, a ∈ W wu and since h f (a) and h f (b) lie in the same unstable leaf but not in the same weak unstable leaf we get Finally since where c 3 depends on d(a, b). Inequalities (6), (7) and (9) sum up to a contradiction.

4.2.
Affine structure on the weak unstable foliation. Let f be in U. For any x and y, y ∈ W wu f (x) define the function The following properties are easy to prove: is well defined and Hölder continuous.
Proof of Lemma 4. Fix an arbitrary point p. Let h p : W wu f (p) → W wu f (h(p)) be the restriction of h to W wu f (p). We would like to show that h p is Lipschitz with a constant that does not depend on p. Let m be the induced volume on W wu f (p). Consider the functiond f we integrate along the leaf with respect to the measure m. Functiond f has the following properties which are simple corollories of the properties of ρ f and the definition ofd f .
whenever d wu f (x, y) < K.
(D4) The functiond f is continuous. To state this property precisely we consider lift ofd f . We speak about lifts of points and leaves.
We will also needd g which is defined analogously on the leaves of W wu g and has analogous properties.
The lift of the conjugacy h satisfies the equation (8) which implies the following Also we know that weak unstable foliation is quasi-isometric which gives us the same for the distance in weak unstable foliations This tells us that h p is Lipschitz for points that are far enough. So we need to estimate d wu g (h(x), h(y)) for x and y close. Note that (D3) allows us to used g and d f in these estimates instead of d wu g and d wu f . Recall the following well-known result.
Livshitz Theorem. If f : M → M is a transitive Anosov diffeomorphism and ϕ 1 , ϕ 2 : M → R are Hölder continuous functions such that then there is a function P : M → R, unique up to a multiplicative constant, such that Moreover P is Hölder continuous.
Apply Livshitz Theorem for ϕ 1 = D wu f (·) and ϕ 2 = D wu g (h(·)). The condition of the Livshitz Theorem is satisfied because of the assumption on p. d. We have .
Choose points x and y close on the leaf W wu f (p). Choose the smallest N such Here we used (12) and (D3) ford f andd g . Function P is bounded away from zero and infinity so we get that h is uniformly Lipschitz along the weak unstable foliation.

4.3.
Transitive point argument and construction of a measure absolutely continuous with respect to weak unstable foliation. We divide the proof of Lemma 5 into several steps. The conjugacy h is Lipschitz along W wu f and hence wudifferentiable at almost every point with respect to Lebesgue measure on the leaves of W wu f . It is obvious that wu-differentiability of h at x implies wu-differentiability of h at any point from the orbit {f i (x), i ∈ Z}. Moreover: Step 1. Suppose that h is wu-differentiable at x and {f i (x), i ≥ 0} = T 3 then h is C 1+ν along W wu f and (10).
The problem now is to show existence of such a transitive point x. We know that almost every point is transitive with respect to a given ergodic measure with full support. On the other hand h is wu-differentiable at almost every point with respect to Lebesgue measure on the leaves. Unfortunately it can happen that for natural ergodic "physical measures" these two "full measure" sets do not intersect. In other words weak unstable foliation is not absolutely continuous with respect to a "physical measure".
Let us explain this phenomenon in more detail. Consider a volume preserving C 1 small perturbationL of L, H • L =L • H. The Lyapunov exponents ofL are defined on a full volume set of regular points R and are given by the formula The perturbationL can be chosen in such a way that χ wu > log λ 2 (see [BB], Proposition 0.3). It is easy to show that the weak unstable foliation ofL is not absolutely continuous. Namely, let ∆ be a segment of a weak unstable leaf of L. Then by Lemma 2 H(∆) is a piece of a weak unstable leaf ofL. We show that Lebesgue measure of R ∩ H(∆) is equal to zero. For any n ≥ 0 H(L n (∆)) =L n (H(∆)) and (2) guarantees thatL n (H(∆)) can be viewed as a graph of a Lipschitz function over a leaf of the weak unstable foliation of L. Hence length(L n (H(∆)) ≤ c 1 · length(L n (∆)) = λ n 2 · length(∆), n ≥ 0. Suppose that Leb(R ∩ H(∆)) > 0 then length(L n (H(∆)) ≥ c 2 e n(χ wu −ε) , ε = 1 2 (χ wu − log λ 2 ) which contradicts the previous inequality. This observation answers a question of Hirayama and Pesin [HP] about existence of non-absolutely continuous foliations with non-compact leaves.
To overcome this problem we do Step 2. Construction of a measure µ absolutely continuous with respect to W wu f . This construction follows the lines of Pesin-Sinai [PS] construction of u-Gibbs measures. In our setup the construction is simpler so for the sake of completeness we present it here. Measure µ has full support. Thus ergodicity of µ would imply that almost every point is transitive and hence by Step 1 h would be wu-differentiable. We do not know how to show ergodicity of µ. Instead we do Step 3. Set of transitive points is a full measure µ set.
Steps 2 and 3 guarantee existence of a transitive point needed in Step 1.

Proof of Lemma 5.
Step 1. Let us pick a point y ∈ T 3 and show that h is wu-differentiable at y and moreover where P is the same as in the proof of Lemma 4. Choose y ′ ∈ W wu f (y). Property (D1) ofd f ,d g ensures that it is enough to show thatd Fix an ε > 0 small compared tod f (y, y ′ ). Choose a small open ball B centered at y and define and (z, z ′ ) has the same orientation as (y, y ′ )}.
The condition about orientation ensures that B ′ has only one connected component. The set B ′ is a small neighborhood of y ′ because of the continuity ofd f (D4). The size of B must be chosen in such a way that (1) |P (z) − P (y)| < ε if z ∈ B, (2) |d g (h(z), h(z ′ )) −d g (h(y), h(y ′ ))| < ε where z and z ′ are the same as in definition of B ′ .
Since x is transitive there is an arbitrarily large N such that f N (x) ∈ B. Choose z on W wu f (x) such thatd f (f N (x), f N (z)) =d f (y, y ′ ) so that f N (z) ∈ B ′ by the definition. We choose N big enough so that Figure 1. Differentiability of h at the point y.
Step 2. Let x 0 be a fixed point for f and let V 0 be an open bounded neighborhood of x 0 in W wu f (x 0 ). Consider a probability measure η 0 supported on V 0 with density proportional to ρ f (x 0 , ·). For n > 0 define V n = f n (V 0 ), η n = (f n ) * η 0 so that η n is supported on V n and has density proportional to ρ f (x 0 , ·) by (P 3).
Let µ n = 1 n n−1 i=0 η i . By the Krylov-Bogoljubov theorem {µ n ; n ≥ 0} is weakly compact and any of its limits is f -invariant. Let µ be a one of those limits along a subsequence {n k ; k ≥ 1}. We would like to prove that µ has absolutely continuous conditional measures on the pieces of weak unstable foliation.
Let us be more precise. Consider a small open set X ⊂ T 3 which can be decomposed in the following way Here Y is a two dimensional transversal. To simplify the notation let W (y) = W wu f (y, R y ). Denote by µ T the transverse measure on Y : for Y ′ ⊂ Y µ T (Y ′ ) = µ(∪ y∈Y ′ W (y)). Similary define η n T and µ n T . Obviously µ n k T → µ T weakly as k → ∞. We show that for µ T almost every y, y ∈ Y the conditional measure µ y on the local leaf W (y) is absolutely continuous with respect to Lebesgue measure m y on W (y).
The conditional measures are characterized by the following property First we look at conditional measures of η n . We fix X and Y as above and we assume that the end points of V n lie outside of X. Let {a 1 , a 2 , . . . a m } = Y ∩ V n . Then the formulas for the transverse measure and conditional measures are obvious: Notice that η n y actually do not depend on n. The goal now is to show that dµ y = W (y) ρ f (y, z)dm y (z) −1 ρ f (y, ·) for almost every y. It could happen that the end points of V n lie inside of X. Support S n of η n T consists of finitely many points. Some of these points correspond to the end points of V n . Denote the set of these points by B n , |B n | ∈ {0, 1, 2}. Let A n = S n \B n then there is a natural decomposition of the transverse measure η n ρ(x 0 , z)dm a (z) δ(a) The conditional measures η n y for y / ∈ B n are given by formula (17). Since W wu f is uniformly expanding it is clear that η n T (B n ) → 0 as n → ∞. Hence 1 n and Consider a continuous function F on X.
The function F is bounded so it follows from (18) that the last limit is zero. So we get Now notice that the function that we integrate with respect to η n k (T,A) is continuous and does not depend on n k . Hence using (19) we get and by (16) we see that up to normalization the density of the conditional measure on W (y) is equal to ρ f (y, ·) for µ T a. e. y.
) is a dense irrational line in T 3 . Hence the support µ is the whole torus.
Step 3. To prove that µ a. e. point is transitive we fix a ball in T 3 and show that a. e. point visits the ball infinitely many times. Then to conclude transitivity we only need to cover T 3 by a countable collection of balls such that every point is contained in an arbitrarily small ball.
So let us fix a ball B ′ and a slightly smaller ball B, B ⊂ B ′ . Let ψ be a nonnegative continuous function supported on B ′ and equal to 1 on B. By Birkhoff ergodic theorem where I is σ-algebra of f -invariant sets. Let A = {x : E(ψ|I)(x) = 0}. Then µ(A∩B) = 0 since A ψdµ = A E(ψ|I)dµ = 0. Hence E(ψ|I)(x) > 0 for µ a.e. x ∈ B.
The next statement is a direct corollary of (D1) and (D2). There exists a δ > 0 which depends on the initial configuration {a, b, c, d, e} such that Combining (21) and (23) we get ∃δ ′ > 0 : ∀n > n 0 d wu f (f −n (b), f −n (e)) d wu f (f −n (a), f −n (c)) On the other hand we know that h is continuously wu-differentiable, hence ∀ε > 0 ∃n 0 : ∀n > n 0 and ∀i ≥ 0 well. The strong unstable foliation is orientable and the pairs (x 0 , y 0 ), (x i , z i ), i ≥ 1 Thus Lemma 1 is automatic. Proof of Lemma 2 go through with minor differences since we know that W su f is quasi-isometric. The bootstrap of regularity of h to the regularity of f and g cannot be done straightforwardly. The reason is the lack of smoothness of weak unstable foliation. Let N = [log λ 3 / log λ 2 ]. It is known [LW] that given f sufficiently C 1 -close to L the individual leaves of weak unstable foliation are C N immersed curves. In general the the leaves of weak untable foliation cannot be more than C N smooth. An example was constructed in [JPL]. Hence our method cannot lead to smoothness higher than C N .