Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori

Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$. The main result establishes $C^{1+\nu}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same constant periodic eigenvalue data that are only H\"older conjugate. We show that these examples exhaust all possible ways to perturb $C^{1+\nu}$ conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of $\mathbb T^4$ $C^1$-close to the original example.

1. Introduction and statements 2 1.1. Positive answers 2 1.2. When the coincidence of periodic data is not sufficient 4 1.3. Additional moduli of C 1 conjugacy in the neighborhood of the counterexample of de la Llave 5 1.4. Organization of the paper and a remark on terminology 6 1. 5

Introduction and statements
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, i.e., The conjugacy h is unique in the neighborhood of identity. It is known that h is Hölder-continuous. There are simple obstructions to the smoothness of h. Namely, if x is a periodic point of f with period p, that is, f p (x) = x, then g p (h(x)) = h(x). If h were differentiable, then i.e., Df p (x) and Dg p (h(x)) are conjugate. We see that every periodic point carries a modulus of smooth conjugacy.
Suppose that for every periodic point x of period p, the differentials of the return maps Df p (x) and Dg p (h(x)) are conjugate. Then we say that the periodic data (p. d.) of f and g coincide.
Question 1. Suppose that the p. d. coincide. Is then h differentiable? If it is, how smooth is it?
1.1. Positive answers. We describe situations when the p. d. form a full set of moduli of C 1 conjugacy.
The only surface that supports Anosov diffeomorphisms is the two-dimensional torus. For Anosov diffeomorphisms of T 2 , the complete answer to Question 1 was given by de la Llave, Marco and Moriyón.
De la Llave [L92] also 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 describe this example in Section 2.
In dimension three, the only manifold that supports Anosov diffeomorphisms is the three-dimensional torus. Moreover, all Anosov diffeomorphisms of T 3 are topologically conjugate to linear automorphisms of T 3 . Nevertheless, the answer to Question 1 is not known.
Conjecture 1. Let f and g be topologically conjugate C r , r > 1, Anosov diffeomorphisms of T 3 with coinciding p. d. Then the conjugacy h is at least C 1 .
There are partial results that support this conjecture.
Theorem ( [GG08]). Let L be a hyperbolic automorphism of T 3 with real eigenvalues. Then 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.
Theorem ( [KS07]). Let L be a hyperbolic automorphism of T 3 that has one real and two complex eigenvalues. Then any f sufficiently C 1 close to L that has the same p. d. as L is C ∞ conjugate to L.
In higher dimensions, not much is known. In recent years, much progress has been made (see [L02,KS03,L04,F04,S05,KS07]) in the case when the stable and unstable foliations carry invariant conformal structures. To ensure existence of these conformal structures one has to at least assume that every periodic orbit has only one positive and one negative Lyapunov exponent. This is a very restrictive assumption on the p. d.
In contrast to the above, we will study the smooth-conjugacy problem in the proximity of a hyperbolic automorphism L : T d → T d with a simple spectrum. Namely, with the exception of Theorem B, we will always assume that the eigenvalues of L are real and have different absolute values. For the sake of notation we assume that the eigenvalues of L are positive. This is not restrictive.
Let l be the dimension of the stable subspace of L and k be the dimension of the unstable subspace of L, so k + l = d. Consider the L-invariant splitting along the eigendirections with corresponding eigenvalues µ l < µ l−1 < . . . < µ 1 < 1 < λ 1 < λ 2 < . . . < λ k .
Let U be a C 1 -neighborhood of L. The precise choice of U is described in Section 6.1. The theory of partially hyperbolic dynamical systems guarantees that for any f in U the invariant splitting survives (e. g. see [Pes04]); that is, . ⊕ E f k . We will see in Section 6.1 that these one-dimensional invariant distributions integrate uniquely to foliations U f l , U f l−1 ,... We will assume that f has the following property: Theorem A. Let L be a hyperbolic automorphism of T d , d ≥ 3, with a simple real spectrum. Assume that the characteristic polynomial of L is irreducible over Z.
There exists a C 1 -neighborhood U ⊂ Diff r (T d ), r ≥ 2, of L such that any f ∈ U satisfying Property A and any g ∈ U with the same p. d. are C 1+ν conjugate.

Remark.
1. We will see in Section 4.1 that irreducibility of the characteristic polynomial of L is necessary for f to satisfy A. Formally, we could have omitted the irreducibility assumption above. Theorem B below shows that the irreducibility of L is a necessary assumption for the conjugacy to be C 1 . We believe that Theorem A holds when L is irreducible without assuming that f satisfies A. 2. ν is a small positive number. It is possible to estimate ν from below in terms of the eigenvalues of L and the size of U. 3. Obviously an analogous result holds on finite factors of tori. But we do not know how to prove it on nilmanifolds. The problem is that for an algebraic Anosov automorphism of a nilmanifold, various intermediate distributions may happen to be nonintegrable.
Theorem A is a generalization of the theorem from [GG08] quoted above. Our method does not lead to higher regularity of the conjugacy (see the last section of [GG08] for an explanation). Nevertheless we conjecture that the situation is the same as in dimension two.
Conjecture 2. In the context of Theorem A one can actually conclude that f and g are C r−ε conjugate, where ε is an arbitrarily small positive number.
Simple examples of diffeomorphisms that possess Property A include f = L and any f ∈ U when max(k, l) ≤ 2 (see Section 4.1). In addition, we construct a C 1open set of Anosov diffeomorphisms of T 5 and T 6 close to L that have Property A. It seems that this construction can be extended to any dimension.
We describe this open set when l = 2 and k = 3. Given f ∈ U, denote by D wu f the derivative of f along V f 1 . Choose f ∈ U in such a way that ∀x = x 0 , D wu f (x) > D wu f (x 0 ), where x 0 is a fixed point of f . Then any diffeomorphism sufficiently C 1 close to f satisfies Property A.
1.2. When the coincidence of periodic data is not sufficient. First let us briefly describe the counterexample of de la Llave.
Let L : T 4 → T 4 be an automorphism of product type, L(x, y) = (Ax, By), (x, y) ∈ T 2 × T 2 , where A and B are Anosov automorphisms. Let λ, λ −1 be the eigenvalues of A and µ, µ −1 the eigenvalues of B. We assume that µ > λ > 1. Consider perturbations of the formL = (Ax + ϕ(y), By), where ϕ : T 2 → R 2 is a C 1 -small C r -function, r > 1. Obviously the p. d. of L andL coincide. We will see in Section 2 that the majority of perturbations (2) are only Hölder conjugate to L. The following theorem is a simple generalization of this counterexample.
Theorem B. Let L : T d → T d be a hyperbolic automorphism such that the characteristic polynomial of L factors over Q. Then there exist C ∞ -diffeomorphisms L : T d → T d andL : T d → T d arbitrarily C 1 -close to L with the same p. d. such that the conjugacy betweenL andL is not Lipschitz.
Remark. In the majority of cases, one can takeL = L. The need to takeL and L both different from L appears, for instance, when L(x, y) = (Ax, Ay). It was shown in [L02] that the p. d. form a complete set of moduli for the smoothconjugacy problem to L. This is a remarkable phenomenon due to the invariance of conformal structures on the stable and unstable foliations. Nevertheless we still have a counterexample if we move a little bit away from L.
Next we study the smooth conjugacy problem in the neighborhood of (1) assuming that µ > λ > 1. We show that the perturbations (2) exhaust all possibilities. Before formulating the result precisely let us move to a slightly more general setting. Let A and B be as in (1) with µ > λ > 1. Consider the Anosov diffeomorphism where g is an Anosov diffeomorphism sufficiently C 1 -close to B, so L can be treated as a partially hyperbolic diffeomorphism with the automorphism A acting in the central direction. Consider perturbations of the form L = (Ax + ϕ(y), g(y)).
As before, it is obvious that the p. d. of L andL coincide. In Section 8 we will see that L andL with nonlinear g also provide a counterexample to Question 1.
Theorem C. Given L as in (3) with µ > λ > 1, there exists a C 1 -neighborhood U ⊂ Diff r (T 4 ), r ≥ 2, of L such that any f ∈ U that has the same p. d. as L is C 1+ν -conjugate, ν > 0, to a diffeomorphismL of type (4).

1.3.
Additional moduli of C 1 conjugacy in the neighborhood of the counterexample of de la Llave. Let L be given by (1) with µ > λ > 1 and let U be a small C 1 -neighborhood of L. It is useful to think of diffeomorphisms from U as partially hyperbolic diffeomorphisms with two-dimensional central foliations.
According to the celebrated theorem of Hirsch, Pugh and Shub [HPS77], the conjugacy h maps the central foliation of f into the central foliation of g. Assume that the p. d. of f and g are the same. We will show that h is C 1+ν along the central foliation. As described above, it can still happen that h is not a C 1diffeomorphism. This means that the conjugacy is not differentiable in the direction transverse to the central foliation. The geometric reason for this is a mismatch between the strong stable (unstable) foliations of f and g -the conjugacy h does not map the strong stable (unstable) foliation of f into the strong stable (unstable) foliation of g.
Motivated by this observation, we now introduce additional moduli of C 1 -differentiable conjugacy. Roughly speaking, these moduli measure the tilt of the strong stable (unstable) leaves when compared to the model (1).
We define these moduli precisely. Let W ss L , W ws L , W wu L and W su L be the foliations by straight lines along the eigendirections with eigenvalues µ −1 , λ −1 , λ and µ respectively. For any f ∈ U these invariant foliations survive. We denote them by W ss f , W ws f , W wu f and W su f . We will also write W s f and W u f for two-dimensional stable and unstable foliations.
Let h f be the conjugacy to the linear model, Fix orientations of W σ L , σ = ss, ws, wu, su. Then for every x ∈ T 4 there exists a unique orientation-preserving isometry I σ (x) : W σ L (x) → R, I σ (x)(x) = 0, σ = ss, ws, wu, su. Define . The geometric meaning is transparent and illustrated on Figure 1. The image of the strong unstable manifold h f (W su f (h −1 f (x))) can be viewed as a graph of the t 00 00 11 11 x Figure 1. The geometric meaning of Φ u f . Herex = I su (x) −1 (t).
Clearly, Φ s/u f are moduli of C 1 -conjugacy. Indeed, assume that f and g are C 1 conjugate by h. Then h(W su f ) = h(W su g ) and h(W ss f ) = h(W ss g ) since strong stable and unstable foliations are characterized by the speed of convergence which is preserved by C 1 -conjugacy. Hence Φ It is possible to choose a subfamily of these moduli in an efficient way. We say that f and g from U have the same strong unstable foliation moduli if The definition of the strong stable foliation moduli is analogous.
Theorem D. Given L as in (1) with µ > λ > 1, there exists a C 1 -neighborhood U ⊂ Diff r (T 4 ), r ≥ 2, of L such that if f, g ∈ U have the same p. d. and the same strong unstable and strong stable foliation moduli, then f and g are C 1+ν conjugate.
Remark. In this case C 1+ν -differentiability is in fact the optimal regularity. 1.4. Organization of the paper and a remark on terminology. In Section 2 we describe the counterexample of de la Llave in a way that allows us to generalize it to Theorem B in Section 3. Sections 2 and 3 are independent of the rest of the paper.
In Sections 4 and 5 we discuss Property A and construct examples of diffeomorphisms that satisfy Property A. These sections are self-contained. Section 6 is devoted to the proof of our main result, Theorem A. It is selfcontained but in number of places we refer to [GG08], where the three-dimensional version of Theorem A was established.
Theorem C is proved in Section 7. It is independent of the rest of the paper with the exception of a reference to Proposition 10.
The proof of Theorem D appears in Section 8 and relies on some technical results from [GG08].
Throughout the paper we will prove that various maps are C 1+ν -differentiable. This should be understood in the usual way: the map is C 1 -differentiable and the derivative is Hölder-continuous with some positive exponent ν. The number ν is not the same in different statements.
When we say that a map is C 1+ν -differentiable along a foliation F , we mean that restrictions of the map to the leaves of F are C 1+ν -differentiable and the derivative is a Hölder-continuous function on the manifold, not only on the leaf.
1.5. Acknowledgements. The author is grateful to Anatole Katok for numerous discussions, advice, and for introducing him to this problem. Many thanks go to Misha Guysinsky and Dmitry Scheglov for useful discussions. The author also would like to thank the referees for providing helpful suggestions and pointing out errors. It was pointed out that tubular minimality of a foliation is equivalent to its transitivity. All these suggestions led to a better exposition.

The counterexample on T 4
Here we describe the example of de la Llave of two Anosov diffeomorphisms of T 4 with the same p. d. that are only Hölder conjugate. Understanding of the example is important for the proof of Theorem B.
We look for the conjugacy h of the form The conjugacy equation h •L = L • h transforms into a cohomological equation on ψ ϕ(y) + ψ(By) = λψ(y).
We get a continuous solution to (9), Hence the conjugacy is indeed given by the formula (8).
In the following proposition we denote by the subscript u the partial derivative in the direction of u.
Proposition 1. Assume that µ > λ > 1. Then function ψ is Lipschitz in the direction of u if and only if i.e., the series on the left converges in the sense of distribution convergence and the limit is equal to zero.
Proof. First assume (11). Let us consider the series (10) as a series of distributions that converge to ψ. Then, as a distribution, ψ u is obtained by differentiating (10) termwise: Applying (11), we get Since µ > λ the above series converges and the distribution is regular. Hence ψ is differentiable in the direction of u. Now assume that ψ is u-Lipschitz. By differentiating (9), we get a cohomological equation on ψ u , ϕ u (x) + µψ u (By) = λψ u (y), that is satisfied on a B-invariant set of full measure. We solve it using the recurrent formula ψ u (y) = − 1 µ ϕ u (B −1 y) + λ µ ψ u (B −1 y). Hence On the other hand we know that as a distribution ψ u is given by (12). Combining (12) and (13) we get the desired equality (11).
If µ = λ then the argument above works only in one direction. We will see that in this case L andL do not provide a counterexample since the p. d. are different.
Proposition 2. Assume that µ = λ. Then (11) is a necessary assumption for ψ to be Lipschitz in the direction of u.
Proof. As in the proof of Proposition 1, viewed as distribution, ψ u is given by Assume that ψ is u-Lipschitz. Then, analogously to (13), we get Note that in the sense of distributions, ψ(B N ) → 0 as N → ∞ since B is mixing. Hence, as a distibution, ψ u is given by Combining (14) and (16), we get (11).
By rewriting condition (11) in terms of Fourier coefficients of ϕ, one can see that it is an infinite codimension condition. Moreover, one can easily construct functions that do not satisfy (11); one only needs to make sure that some Fourier coefficients of the sum (11) are nonzero. For instance, for any ε > 0 and positive integer p, the function ϕ(y) = ϕ(y 1 , y 2 ) = ε sin(pπy 1 ) works. Thus the correspondingL is not C 1 -conjugate to L. Note thatL may be chosen arbitrarily close to L.

Remark.
1. Perturbations of the general type (2) can be treated analogously by decom- The assumption µ ≥ λ > 1 is crucial in this construction. 3. By choosing appropriate λ and µ, one can get any desired regularity of the conjugacy (see [L92] for details). For example, if µ 2 > λ > µ > 1, the conjugacy is C 1 but not C 2 .
From now on let us assume that µ = λ. As we remarked in the introduction, L andL do not provide a counterexample. Indeed, the derivative ofL in the basis {v, u,ṽ,ũ} is  Let x be a periodic point,L p (x) = x. Then the derivative of the return map at x is  We see that it is likely to have a Jordan block while L is diagonalizable. Hence L andL have different p. d. It is still easy to construct a counterexample in a neighborhood of L. Let L = (Ax + ξ(y)v, By) and let h(x, y) = (x + ψ(y)v, y) be the conjugacy betweenL andL is necessary for φ to be Lipschitz in the direction of u.
The proof of Proposition 3 is exactly the same as the one of Proposition 2. Now take ϕ that does not satisfy (11) as before and take ξ = 2ϕ. Then obviously the condition of Proposition 3 is not satisfied. Hence h is not Lipschitz. By looking at (18) it is obvious that our choice of ξ guarantees that the Jordan normal forms of the derivatives of the return maps at periodic points ofL andL are the same.
Remark. Due to the special choice of ξ it was easy to ensure that the p. d. ofL andL are the same. We could have taken a different and somewhat more general approach. It is possible to show that for many choices of ϕ, the sum that appears over the diagonal in (18) is nonzero for every periodic point x. All of the corresponding diffeomorphisms will have the same p. d. with a Jordan block at every periodic point.

Proof of Theorem B
Here we consider L : T d → T d with a reducible characteristic polynomial. We show how to constructL andL with the same p. d. which are not Lipschitz conjugate.
Assume that all real eigenvalues of L are positive. Otherwise we may consider L 2 . Let M : R d → R d be the lift of L and let {e 1 , e 2 , . . . e d } be the canonical basis, so It is well known that the characteristic polynomial of M factors over Z into the product of polynomials irreducible over Q: Let λ be the eigenvalue of M with the smallest absolute value which is greater than one. Without loss of generality we assume that P 1 (λ) = 0.
Let V i be the invariant subspace that corresponds to the roots of P i . Then dim V i = deg P i and it is easy to show that Matrices of P i (M ) have integer entries. Hence there is a basis {ẽ 1 ,ẽ 2 , . . .ẽ d },ẽ i ∈ span Z {e 1 , e 2 , . . . e d }, i = 1, . . . d, such that the matrix of M in this basis has integer entries and is of block diagonal form with blocks corresponding to the invariant subspaces V i , i = 1, . . . r.
We consider projection of M toT d = R d /span Z {ẽ 1 ,ẽ 2 , . . .ẽ d }. Denote by N the induced map onT d . We have the following commutative diagram, where π is a finite-to-one projection.
With certain care, the construction of Section 2 can be applied to N . We have to distinguish the following cases: 1. λ and µ are real. 2. λ is real and µ is complex. 3. λ is complex and µ is real. 4. λ and µ are complex.
In the first case the construction of Section 2 applies straightforwardly. We use a function of the type (17) to produceÑ . Now we only need to make sure thatÑ can be projected to a mapL : T d → T d . Since π is a finite-to-one covering map this can be achieved by choosing suitable p in (17).
Other cases require heavier calculations but follow the same scheme as Proposition 1. We outline the construction in case 4, which can appear, for instance, if A and B are hyperbolic automorphisms of four-dimensional tori without real eigenvalues.
Let V A = span{v 1 , v 2 } be the two-dimensional A-invariant subspace corresponding to λ and V B = span{u 1 , u 2 } be the two-dimensional B-invariant subspace corresponding to µ. Then A acts on V A by multiplication by |λ|R A and B acts on V B by multiplication by |µ|R B , where R A and R B are rotation matrices expressed in the bases {v 1 , v 2 } and {u 1 , u 2 }, respectively.
We are following the construction from the previous section. Let Then we look for a conjugacy of the form

The conjugacy equation
Solving for ψ gives which we would like to differentiate along the directions u 1 and u 2 . We use the formula ϕ(By) u = ϕ 1 (By) u1 ϕ 1 (By) u2 ϕ 2 (By) u1 ϕ 2 (By) u2 = |µ| to get that, as a distribution, Now we assume that ψ is Lipschitz and we differentiate (19) along the directions u 1 and u 2 : Hence, by the recurrent formula, Combining the expressions for ψ u , we get Using Fourier decompositions, one can find functions ϕ that do not satisfy the condition above. One also needs to make sure that the choice of ϕ allows one to projectÑ down toL. We omit this analysis since it is routine. This is a contradiction and therefore ψ (and hence h) is not Lipschitz. If |λ| = |µ| but λ = µ, then the scheme above still works. Obviously, extra Jordan blocks do not appear in the normal forms at periodic points ofL.
Finally, the case λ = µ must be treated separately. We use the same trick as in Section 2 to findL andL with the same p. d. that are only Hölder conjugate. This trick also works well in the case of complex eigenvalues; we omit the details.
4. On the Property A 4.1. Transitivity versus minimality. Here we discuss Property A. Let F be a foliation of a compact manifold M . As usually F (x) stands for the leaf of F that contains x and F (x, R) stands for the ball of radius R centered at x inside of F (x).
.. V f k−1 to be tubularly minimal. We introduce the following related property: Remark. We define Property A in terms of tubular minimality rather than transitivity since we need denseness of the tubes for the proof of Theorem A.
A priori, transitivity is weaker than minimality. Hence, a priori, Property A is weaker than Property A ′ .
If, in Theorem A, we had required f to satisfy Property A ′ instead of Property A, then the induction procedure that we use (the first induction step) is much simpler. The proof of this step, assuming only Property A, requires a much more lengthy and delicate argument. It is not clear to us what the relationship is between Properties A and A ′ ; they may be equivalent. Thus, we will first provide a proof of Theorem A assuming that f has Property A ′ , then we will present a separate proof of this first induction step (namely Lemma 6.6) that uses only Property A.
Minimality of a foliation can be characterized similarly to tubular transitivity. 1 We would like to thank the referee for pointing out this fact.

Proposition 5. The foliation F is minimal if and only if for every x and every open ball
The proof is simple, so we omit it. As a corollary the foliation F is minimal if and only if for every x and every open ball B ∋ x, there exists a number R such that This is the property which we will actually use in the proof of the induction step 1.

4.2.
Examples of diffeomorphisms that satisfy Property A.
Proof. Denote by F one of the foliations under consideration. Since F is a foliation by straight lines, the closure of a leaf F (x) is a subtorus of T d . This subtorus lifts to a rational invariant subspace of R d . The invariant subspace corresponds to a rational factor of the characteristic polynomial of L, but we assumed that it is irreducible over Q. Hence the invariant subspace is the whole of R d and the subtorus is the whole T d .
So we can see that the conclusion of Theorem A holds at least for f = L. We will see in Section 6.1 that for any f ∈ U, the foliations U f 1 and V f 1 are minimal. Hence the conclusion of Theorem A holds for any f ∈ U if max(k, l) ≤ 2.
It is easy to construct f = L that satisfies A when k = 3 and l = 2 since we only have to worry about the foliation V f 2 . We let f = s • L where s is any small shift .. V f k−1 arise naturally. Robust minimality of strong stable and strong unstable foliations of partially hyperbolic systems has received some attention in the literature due to its intimate connection with robust transitivity; see [Ma78] and the more recent papers [BDU02,PS06], where robust minimality of the full expanding foliation is established under some assumptions. We do not have this luxury in our setting: the expanding foliations that we are intrested in subfoliate the full unstable foliation. A representative problem here is the following.
Question 2. Let L : T 3 → T 3 be a hyperbolic linear automorphism with real spectrum λ 1 < 1 < λ 2 < λ 3 . Consider the one-dimensional strong unstable foliation. Is it true that this foliation is robustly minimal? In other words, is it true that for any f sufficiently C 1 -close to L the strong unstable foliation of f is minimal?
In addition to the simple examples above, in the next section we construct a C 1 -open set of diffeomorphisms that possess Property A. The following statement can be obtained by applying the construction and the arguments of the next section in the context of Question 2.
Proposition 7. Let L be as in Question 2. Then there exists a C 1 -open set U C 1close to L such that for every f ∈ U the strong unstable foliation of f is transitive.

An example of an open set of diffeomorphisms with Property A
Let L : T 5 → T 5 be a hyperbolic automorphism as in Theorem A, l = 2, k = 3, and let U be a C 1 -neighborhood of L chosen as in Section 6.1.
Recall that D wu Proposition 8. There exists a C 1 -neighborhoodŨ of f such that any diffeomorphism g ∈Ũ has Property A.

Remark.
A similar example can be constructed on T 6 with l = 3, k = 3. We only need to do the trick described below for both the stable and unstable manifolds of the fixed point x 0 .
Before proving the proposition let us briefly explain the idea behind the proof. We know that U g 1 and V g 1 are minimal. Hence we only need to show that the foliation V g 2 is tubularly minimal, i.e., for every x ∈ T 5 and every open ball To illustrate the idea we take g = f and x = x 0 . We work on the universal cover R 5 with lifted foliations. Let which is an open tube. We show that T contains arbitrarily long connected pieces of the leaves of V f 1 as shown on Figure 2. It will then follow that T is dense in T 5 . Indeed, the foliation V f 1 is not just minimal but uniformly minimal: for any ε > 0 there exists R > 0 such that ∀z ∈ T 5 V f 1 (z, R) is ε-dense in T 5 . This property follows from the fact that V f 1 is conjugate to the linear foliation V L 1 .
is an unbounded function of x. We make use of the affine structure on V f 1 . We refer to [GG08] for the definition of affine distance-like functiond 1 . Recall the following crucial properties ofd 1 : Using Property (D3), we can see that it is enough to show thatd 1 (x, y) is unbounded. Given x as above, pick N large enough such that the ratiõ y Figure 2. The tube T contains arbitrarily long pieces of leaves of V f 1 .
is close to 1 as shown in Figure 3. This is possible since V f 2 contracts exponentially faster than V f 1 under the action of f −1 . It is not hard to see that, given a large number n, we can pick x (and N correspondingly) far enough from x 0 such that at least n points from the orbit {x, where δ > 0 depends only on the size of B. Using (D2), we get which is an arbitrarily large number. Henced 1 (x, y) is arbitrarily large and we are done.
Remark. Although Proposition 8 deals with a pretty special situation we believe that the picture on Figure 2 is generic. To be more precise, we think that for any g ∈ U the following alternative holds. Either V g 2 is conjugate to the linear foliation V L 2 or there exists a dense set Λ such that for any x ∈ Λ and any B ∋ x the tube contains arbitrarily long connected pieces of the leaves of V g 1 .
Proof of Proposition 8. The argument is more delicate than the one presented above since we do not know that the minimum of the derivative is achieved at x y Figure 3. Illustration to the argument. Quadrilateral in the box is much smaller then the one outside.
Let B 0 be a small ball around x 0 and B 1 ⊃ B 0 a bigger ball. Condition (21) guarantees that we can choose them in such a way that with m 0 , m 1 and M satisfying where q is an integer that depends only on the size of U and the size of B 1 . After that we chooseŨ ⊂ U so the fixed point of g (that corresponds to x 0 ) is inside of B 0 and the property above persists. Namely, Note that provided that f is sufficiently C 1 -close to L and the ball B 1 is small enough, any piece of a leaf of V g 2 outside of B 1 that starts and ends on the boundary of B 1 cannot be homotoped to a point keeping the endpoints on the boundary. This is a minor technical detail that makes sure that the picture shown on Figure 4a does not occur. Thus there is a lower bound R on the lengths of pieces of leaves of V g 2 outside of B 1 with endpoints on the boundary of B 1 . Obviously, there is also an upper bound r on the lengths of pieces of leaves of V g 2 inside B 1 .
It is enough to check (22) for a dense set Λ of points x ∈ T 5 . We take Λ to be a subset of the set of periodic points of g where n(p) stands for the period of p. The set Λ consists of periodic points that spend a large but fixed percentage of time inside of B 0 . It is fairly easy to show that Λ is dense in T 5 . The proof is a trivial corollary of the specification property (e. g. see [KH95]). So we fixx 0 ∈ Λ, a small ball B centered atx 0 and y 0 ∈ B ∩ V g 1 (x 0 ) close tox 0 . Our goal now is to find x ∈ V g 2 (x 0 ) far in the tube T defined by (23) for which we can carry out estimates similar to (24).
We will be working with pieces of leaves of V g 2 . Given a piece I with endpoints z 1 and z 2 let |I| = d g 2 (z 1 , z 2 ). Let q be a number such that for any piece I, |I| = R, we have Notice that q can be chosen to be independent of g and depends only onβ 2 , R and r. Pick We fix N large and take x ∈ I N q ⊂ V g 2 (x 0 ). Let y = V g 1 (x) ∩ V g 2 (y 0 ) as before. The construction of the sequence {I i , i ≥ 1} ensures that the points f −qi (x), i = 0, . . . N − 1, are outside B 1 . This fact together with (26) and (27) allows to carry out the following estimate: The affine-like distance ratio on the right is bounded away from 0 independently of N since f −N q (x) ∈ I 1 , while the coefficient in front of it is arbitrarily large according to (25). Henced g 1 (x, y) is arbitrarily large and the projection of the tube T is dense in T 5 .

Proof of Theorem A
For reasons explained in Section 4 we first prove Theorem A assuming that f has Property A ′ . The only place where we use Property A ′ is in the proof of Lemma 6.6. In Section 6.6 we give another proof of Lemma 6.6 that uses Property A only.
6.1. Scheme of the proof of Theorem A. Recall the notation from 1.1 for the L-invariant splitting along the eigendirections with corresponding eigenvalues We choose a neighborhood U in such a way that, for any f in U, the invariant splitting survives: and f is partially hyperbolic in the strongest sense; that is, there exist C > 0 and constants Equivalently, the Mather spectrum of f does not contain 1 and has d connected components.
Such a choice is possible -see Theorem 1 in [JPL]. This theorem also guarantees that C 1 -size of U is rather large.
We show that the choice of U guarantees unique integrability of intermediate distributions. From now on, for the sake of concreteness, we work with unstable distributions and foliations. For The conjugacy h maps the unstable (stable) foliation of f into the unstable (stable) foliation of g. Moreover, h preserves the whole flag of weak unstable (stable) foliations.
The proof of this lemma does not use the assumption on the p. d. We only need f and g to be in U.
Lemmas 6.1, 6.2 and 6.3 can be proved under a milder assumption. Instead of requiring f and g to be in U we can require the following.
Alternative Assumption: f and g are partially hyperbolic in the sense of (30) with the rate constants satisfying We think that (⋆) is actually automatic from (30).
Remark. To carry out proofs of the Lemmas above under the Alternative Assumption one needs to transfer the picture to the linear model by the conjugacy and use the inequalities (⋆) for growth arguments. This way one uses quasi-isometric foliations by straight lines of the linear model instead of foliations of f which are a priori not known to be quasi-isometric.
Conjecture 3. Suppose that f is homotopic to L and partially hyperbolic in the strongest sense (30). Then the rate constants satisfy (⋆).
Remark. The proofs of Lemmas 6.1, 6.2 and 6.3 are the only places where we really need f and g to be in U. So, in Theorem A, the assumption that f, g ∈ U can be substituted by the alternative assumption.
Proof. By Lemma 6.3 the conjugacy between L and f takes the foliation W L 1 into the foliation W f 1 . According to Proposition 6, leaves of W L 1 are dense. Hence leaves of W f 1 are dense.
Next we describe the inductive procedure which leads to the smoothness of h along the unstable foliation.
Induction base. We know that h takes W f 1 into W g 1 .
Lemma 6.5. The conjugacy h is C 1+ν -differentiable along W f 1 , i.e., the restrictions of h to the leaves of W f 1 are differentiable and the derivative is a C ν -function on T d .
Provided that we have Lemma 6.4, the proof of Lemma 6.5 is the same as the proof of Lemma 5 from [GG08].
Induction step. The induction procedure is based on the following lemmas.
We also use a regularity result due to Journé.

Regularity Lemma ([J88]
). 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 : M 1 → M 2 is a homeomorphism that maps W s 1 into W s 2 and W u 1 into W u 2 . 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 .
Remark. There are two more methods of proving analytical results of this flavor besides Journé's. One is due to de la Llave, Marco, Moriyón and the other one is due to Hurder and Katok (see [KN08] for a detailed discussion and proofs). We remark that we really need Journé's result since the alternative approaches require foliations to be absolutely continuous while we apply the Regularity Lemma to various foliations that do not have to be absolutely continuous.
By induction h is C 1+ν -differentiable along the unstable foliation and analogously along the stable foliation. We finish the proof of Theorem A by applying the Regularity Lemma to stable and unstable foliations.
6.2. Proof of the integrability lemmas. In the proofs of Lemmas 6.1 and 6.2, we work with lifts of maps, distributions and foliations to R d . We use the same notation for lifts as for the objects themselves.
Proof of Lemma 6.1. Fix i < k. We assume that the distribution E f (1, i) is not integrable or it is integrable but not uniquely. In any case it follows that we can find distinct points a 0 , a 1 , . . . a m such that Assume thatm = 1 and let ω = ω 1 , q = q(1). The general case can be established in the same way by working with ω = ωj wherej is chosen so that q(j) > q(j) for j =j.
Proof of Lemma 6.2. The theory of partial hyperbolicity guarantees that the distributions E f (i, k), i = 1, . . . k, integrate uniquely to foliations W f (i, k). Let us fix i and j, i < j, and define is an integral foliation for E f (i, j). Unique integrability of E f (i, j) is a direct consequence of the unique integrability of E f (1, j) and E f (i, k).
6.3. Weak unstable flag is preserved: proof of Lemma 6.3.
Proof. We continue working on the universal cover. Pick two points a and b, a ∈ we have that d(h(x), h(y)) ≤ c 1 d(x, y) for any x and y such that d(x, y) ≥ 1. Hence, for any n > 0, where c 2 and c 3 depend on d(a, b). This inequality guarantees h(a) ∈ W g i (h(b)). Since the choice of a and b was arbitrary we conclude that h(W f i ) = W g i .
6.4. Induction step 1: the conjugacy preserves the foliation V m . We now prove Lemma 6.6, which is the key ingredient in the proof of Theorem A. The proof is based on our idea from [GG08] but we take a rather different approach in order to deal with the high dimension of W f . We provide a complete proof almost without referring to [GG08]. Nevertheless we strongly encourage the reader to read Section 4.4 of [GG08] first. The goal is to prove that h(V f m ) = V g m . So we will consider the foliation U = h −1 (V g m ). As in the case for usual foliation, U (x) stands for the leaf of U passing through x and U (x, R) stands for the local leaf of size R. A priori, the leaves of U are just Hölder-continuous curves. Hence the local leaf needs to be defined with a certain care. One way is to consider the lift of U and define the lift of the local leaf U (x, R) as a connected component of x of the intersection U (x) ∩ B(x, R).
We prove Lemma 6.6 by induction.
Induction base. We will be working on m-dimensional leaves of W f m . By Lemma 6.3, U subfoliates W f m . In other words, for any  Remark. Since by the induction hypothesis, h(W f (i, m − 1)) = W g (i, m − 1), we see that the leaf U (a) intersects each leaf W f (i, m − 1)(x), x ∈ W f (i, m)(a) exactly once.
Obviously (e,ẽ) has the same orientation as (d,d) and also has the advantage of lying on the leaf V f i (b). Therefore, we forget about (d,d) and work with (e,ẽ). We use an affine structure on the expanding foliation V f i . Namely we work with the affine distance-like functiond i . We refer to [GG08] for the definition. There we define the affine distance-like function on the weak unstable foliation. The definition for the foliation V f i is the same with obvious modifications. Recall the crucial properties ofd i :  Assume that (e,ẽ) has orientation opposite to (b,b) or e =ẽ. For the sake of concreteness we assume that these points lie on V f i (b) in the order b,b,ẽ, e. All other cases can be treated similarly. Theñ Remark. Notice thatd i (b,ẽ) −d i (b,b) =d i (b,ẽ) sinced i is neither symmetric nor additive. The distanced i is given by an integral of a certain density with normalization defined by the first argument. As long as the first argument (point b in the above inequality) is the same, all natural inequalities hold.
Applying (D2), we get that where c 1 does not depend on n. By property (D1) we can switch to the usual distance: where c 2 does not depend on n.
Under the action of f −1 , strong unstable leaves of W f (i + 1, m) contract exponentially faster then weak unstable leaves of V f i . Thus We have that h(e) ∈ W g (i + 1, m)(h(c)). Indeed, notice that , where the last equality is justified by the fact that h(d) ∈ V g m (h(c)). We know also that h(b) ∈ W g (i + 1, m)(h(a)). Hence, analogously to (36), we have On the other hand, we also know that h is continuously differentiable along V f i . Hence and Therefore from (37) and (38) we have which we combine with (36) to get We have reached a contradiction with (35) Remark. By the same argument one can prove that if b =b then d =d.
Lemma 6.9. Consider a weak unstable leaf W f m−1 (a) and b ∈ V f m (a), b = a. For Proof. We will be working on the universal cover R d . We abuse notation slightly by using the same notation for the lifted objects. Note that the leaves on R d are Then, obviously, Thus the supremum above is finite. Remark.
1. Given two points a, b ∈ R d let ). It is clear from the proof that constants c 1 and c 2 can be chosen in such a way that they depend only ond(a, b). 2. In the proof above we do not use the fact that both W f m−1 and V f m are expanding. We only need them to be transverse. Thus, if we substitute for the weak unstable foliation W f m−1 some weak stable foliation F , the statement still holds. 3. As mentioned earlier the assumption (29) is crucial only for Lemmas 6.1, 6.2 and 6.3. We used this assumption in the proof above only for convenience. A slightly more delicate argument goes through without using assumption (29).
Proof of the induction step. We will be working inside of the leaves of W f (i, m). Assume that U does not subfoliate W f (i + 1, m). Then there exists a point x 0 and x 1 ∈ U (x 0 ) close to x 0 such that x 1 / ∈ W f (i + 1, m)(x 0 ). We fix an orientation O of U and V f i that is defined on pairs of points (x, y), y ∈ U (x) and (x, y), y ∈ V f i (x). Although we denote these orientations by the same symbol it will not cause any confusion since U and V f i are topologically transverse.  Proof. Let a 0 =d(x 0 , x 1 ) (for definition ofd see the remark after the proof of Lemma 6.9). The number a 0 is positive since U (x) is transverse to W f m−1 . For any y ∈ T d , there is a unique point sh(y) ∈ U (y) such thatd(y, sh(y)) = a 0 and O(y, sh(y)) = O(x 0 , x 1 ).
The leaves of all the foliations that we consider depend continuously on the point. Therefore we can find a small ball B centered at x 0 such that ∀y ∈ B For any y ∈ T d there is a unique point sh(y) ∈ U (y) such thatd(y, sh(y)) = d(x 0 ,x 1 ) and O(y, sh(y)) = O(x 0 ,x 1 ). Then by the same argument we show an analog of (39): sh(z) U(z) Figure 8. Illustration to the argument with shifts along U (z). Foliation W f (i + 1, m) is one-dimensional here, N 1 = 3, N 2 = 2. The black segments of V f i carry known information about the orientations of ([·, sh(·)], sh(·)) and ([·, sh(·)], sh(·)). This picture is clearly impossible if sh N1 = sh N2 .
We get that To get a contradiction it remains to notice that sh N1 = sh N2 . Hence the lemma is proved.
From now on it is more convenient to work on the universal cover, although formally we do not have to do it since we are working inside of the leaves of W f (i, m), which are isometric to their lifts. Let ], x n+1 ) > δ. Lemma 6.10 also tells us that the leaves of U are monotone with respect to foliation W f (i + 1, m). Namely, for any Denote by x n , x n+1 the piece of U (x 0 ) that lies between x n and x n+1 . We know that for any n ≥ 0, x n , x n+1 is confined between the leaves W f (i, m − 1)(x n ) and W f (i, m − 1)(x n+1 ). Lemma 6.10 guarantees that x n , x n+1 is also confined between W f (i + 1, m)(x n ) and W f (i + 1, m)(x n+1 ), as shown on Figure 9. Thus, it makes sense to measure two different "dimensions" of x n , x n+1 . Namely, let a n =d(x n , x n+1 ) and b n = d f i ([x n , x n+1 ], x n+1 ). As we have remarked earlier b n > δ > 0 and a n = a 0 by the definition ofd and sh.
These "dimensions" behave nicely under the dynamics. Namely, The idea now is to show that the leaf U (f −N (x 0 )) is lying "too close" to W f (i, m − 1)(x 0 ) for N large, which would lead to a contradiction.
Take N large and let M = ⌊λ N m ⌋. Then The first equality holds since the holonomy along W f (i, m − 1) is isometric with respect tod. To ) in a similar way we need to have control over holonomies along W f (i + 1, m).
This condition ensures that the distance between x and y along W f (i, m)(x) is uniformly bounded from above. To see this we only need to bound the distance between h(x) and h(y) along W g (i, m)(h(x)). This, in turn, is a direct consequence of Lemma 6.9 applied to g since h(y) ∈ V g m (h(x)). Consider the holonomy map along W f (i + 1, m) H : T (x) → T (y). This holonomy can be viewed as the holonomy along W f (i + 1, k). Recall that W f (i + 1, k) is the fast unstable foliation. Since f is at least C 2 -differentiable, W f (i + 1, k) is Lipschitz inside of W f (i, k). Moreover, since the distance between x and y is bounded from above, the Lipschitz constant C Hol of H is uniform in x and y. For a proof, see [LY85], Section 4.2. They prove that the unstable foliation is Lipschitz within center-unstable leaves but the proof goes through for W f (i + 1, k) within the leaves of W f (i, k). b n a n x n x n+1 Figure 9. Piece x n x n+1 is "monotone" with respect to foliation W f (i, m − 1). By Lemma 6.10 x n x n+1 is also "monotone" with respect to W f (i+1, m): the intersections of x n x n+1 with local leaves of W f (i + 1, m) are points or connected components of x n x n+1 . On this picture foliations W f (i, m − 1) and W f (i + 1, m) are twodimensional. Together with the Lipschitz property of the foliation W f (i + 1, m), this provides an estimate from below on the horizontal size The holonomy constant C Hol is uniform sincê 6.5. Induction step 2: proof of Lemma 6.7 by transitive point argument. The proof of Lemma 6.7 is carried out in a way similar to the proofs of Lemmas 4 and 5 from [GG08]. Here we overview the scheme and deal with the complications that arise due to higher dimension.
First, using the assumption on the p. d., we argue that h is uniformly Lipschitz along V f m , i.e., for any point is a Lipschitz map with a Lipschitz constant that does not depend on x. At this step, the assumption on the p. d. along V f m is used. The Lipschitz property implies differentiability at almost every point with respect to the Lebesgue measure on the leaves of V f m . The next step is to show that the differentiability of h along V f m at a transitive point x implies that h is C 1+νdifferentiable along V f m . This is done by a direct approximation argument (see Step 1 in Section 4.3 in [GG08]). The transitive point x "spreads differentiability" all over the torus.
Last but not least, we need to find such a transitive point x. Ideally, for that we would find an ergodic measure µ with full support such that the foliation V f m is absolutely continuous with respect to µ. Then, by the Birkhoff Ergodic Theorem, almost every point would be transitive. Since V f m is absolutely continuous, we would then have that almost every point, with respect to the Lebesgue measure on the leaves, is transitive. Hence we would have a full measure set of the points that we are looking for.
Unfortunately, we cannot carry out the scheme described above. The problem is that the foliation V f m is not absolutely continuous with respect to natural ergodic measures (see [GG08] for detailed discussion and [SX08] for in-depth analysis of this phenomenon). Instead, we construct a measure µ such that almost every point is transitive and V f m is absolutely continuous with respect to µ. This is clearly sufficient.
The construction follows the lines of Pesin-Sinai's [PS83] construction of u-Gibbs measures. Given a partially hyperbolic diffeomorphism, they construct a measure such that the unstable foliation is absolutely continuous with respect to the measure. In fact, this construction works well for any expanding foliation. We apply this construction to m-dimensional foliation W f m . The construction is described as follows. Let x 0 be a fixed point of f . For any . Consider a probability measure η 0 supported on V 0 with density proportional to ρ(·). For n > 0, define By the Krylov-Bogoljubov Theorem, {µ n ; n ≥ 0} is weakly compact and any of its limits is f -invariant. Let µ be an accumulation point of {µ n ; n ≥ 0}. This is the measure that we are looking for. Foliation W f m is absolutely continuous with respect to µ. We refer to [PS83] or [GG08] for the proof. The proof of [GG08] requires some minimal modifications that are due to the higher dimension of W f m . Since the foliation W f m is conjugate to the linear foliation W L m we have that for any open ball B, where W f m (y, R) is a ball of radius R inside of the leaf W f m (y). Together with absolute continuity, this guarantees that µ almost every point is transitive. See [GG08], Section 4.3, Step 3 for the proof. We stress that we do not need to know that µ has full support in that argument.
It is left to show that the conjugacy h is C 1+ν -differentiable in the direction of V f m at µ almost every point. For this we need to argue that V f m is absolutely continuous with respect to µ.
The foliation W f (m, k) is Lipschitz inside of a leaf of W f (again we refer to [LY85], Section 4.2).
So V f m is absolutely continuous with respect to the Lebesgue measure on a leaf of W f m while W f m is absolutely continuous with respect to µ. Therefore V f m is absolutely continuous with respect to µ.
6.6. Induction step 1 revisited. To carry out the proof of Lemma 6.6 assuming Property A only, we shrink the neighborhood U even more. In addition to (29) and (30), we require f ∈ U to have a narrow spectrum. Namely, and the analogous condition on the contraction rates α j ,α j . The following condition that we will actually use is obviously a consequence of the above one.
∀i < k and ∀m, i < m ≤ k, ρ This inequality can be achieved by shrinking the size of U since β j andβ j get arbitrarily close to λ j , j = 1, . . . k.
Remark. Condition (43) greatly simplifies the proof of Lemma 6.6. We have yet another, longer, proof (but based on the same idea) of Lemma 6.6 that works for any f with Property A in U as defined in Section 6.1. It will not appear here.
We start the proof as in Section 6.4. The first place where we use Property A ′ is the proof of Lemma 6.10. So we reprove induction step 1 with Property A only assuming that we have got everything that preceded Lemma 6.10. With Property A, the proof of Lemma 6.10 still goes through, although instead of (39), we get Thus we still have Lemma 6.10 and the upper bound (41) but not the lower bound d f i ([z, sh(z)], sh(z)) > δ. This is the reason why we cannot proceed with the proof of the induction step as at the end of Section 6.4.
Proof of the induction step. As before, we need to show that U subfoliates foliation W f (i + 1, m).
. This way we define an ε-"rectangle" R = R(x,x, y,ȳ) with base point x, vertical size d f m (x,x) = ε, and horizontal size d f i (x, y) =ε. Remark. Notice that we measure vertical size in a way different from the one in Section 6.4. It is clear that this "rectangle" is uniquely defined by its "diagonal" (x,ȳ) (Figure 9 is the picture of "rectangle" with diagonal (x n , x n+1 )). Sometimes we will use the notation R(x,ȳ). Note that by Lemma 6.10, O(x, y) does not depend on x and ε. It also guarantees that the piece of U (x) between x andȳ lies "inside" of R(x,ȳ). The horizontal sizeε might happen to be equal to zero.
Next we define a set of base points X ε such that U (x), x ∈ X ε , has big Hölder slope inside of corresponding ε-rectangle, with some δ satisfying inequality ρ > δ > log β i / logβ m .
Let µ be the measure constructed in Section 6.5. Recall that µ-almost every point is transitive. The foliation W f (i, m) is absolutely continuous with respect to µ. The latter can be shown in the same way as absolute continuity of V f m is shown in Section 6.5.
We consider two cases.
The idea now is to iterate a rectangle with base point in X εn and vertical size ε n until the vertical size is approximately 1. Since the Hölder slope of the initial rectangle was big, it will turn out that the horizontal size of the iterated rectangle is extremely small. This argument will show that for a set of base points of positive measure, the horizontal size of rectangles is equal to zero. Hence the leaves of U lie inside of the leaves of W f (i + 1, m).
Given n, let N = N (n) be the largest number such that 1 Cβ N m ε n < 1 (constant C here is from Definition (30)). Take x ∈ X εn and the corresponding ε n -rectangle R(x, y,x,ȳ) and consider its image R(f N (x), f N (y), f N (x), f N (ȳ)). The choice of N provides a lower bound on the vertical size, while the horizontal size can be estimated as follows: Rather than continuing to look at the rectangle R(f N (x), f N (y), f N (x), f N (ȳ)), we will now consider the rectangleR(f N (x)) with base point f N (x) and fixed vertical size 1/β m . Lemma 6.10, together with the estimate on the vertical size of R(f N (x), f N (y), f N (x), f N (ȳ)), guarantees that horizontal size ofR(f N (x)) is less Thus, for every x ∈ f N (X εn ), the horizontal size ofR(x) =R(x, z,x,z) is less than C 1+δ β i /β δ m N . Note that β i /β δ m N → 0 as n → ∞ since β i /β δ m < 1 and N → ∞ as n → ∞.
Let X = lim n→∞ f N (X εn ). Since any x ∈ X also belongs to f N (X εn ) with arbitrarily large N we conclude thatR(x) has zero horizontal size, i.e., x = z. Hence by Lemma 6.10 we conclude that the piece of U (x) from x toz lies inside of It is a simple exercise in measure theory to show that Finally recall that µ-almost every point is transitive, ({f j (x), j ≥ 1} = T d ). Hence by taking a transitive point x ∈ X and applying a straightforward approximation argument, we get that ∀y U (y) ⊂ W f (i + 1, m)(y). Case 2. lim ε→0 µ(X ε ) = 0.
In this case, the idea is to use the assumption above to find a leaf U (x) which is "flat", i.e., arbitrarily close to W f (i, m − 1)(x). Since the leaf U (x) has to "feel" the measure µ, we need to take it together with a small neighborhood. The choice of this neighborhood is done by multiple applications of the pigeonhole principle.
Given a pointȳ ∈ U (x), denote by U xȳ the piece of U (x) between x andȳ. As before, by R(x,ȳ) we denote the rectangle spanned by x andȳ. Recall that HS (R(x,ȳ)) and V S (R(x,ȳ)) stand for the horizontal and vertical sizes of R(x,ȳ). We will also need to measure the sizes of U xȳ . Let HS(U xȳ ) = HS(R(x,ȳ)) and V S(U xȳ ) = V S (R(x,ȳ)).
Iterating Pigeonhole Principle. Divide T d into finite number of tubes T 1 , T 2 , . . . T q foliated by U such that any connected component of U (x)∩T j , j = 1, . . . q, has vertical size between S 0 and S 1 . The numbers S 0 and S 1 are fixed, 0 < S 0 < S 1 . We also require every tube T j to be W f (i, m − 1)-foliated so it can be represented as where Transv is a plaque of U and Plaque(y) are plaques of W f (i, m − 1). Given a small number τ > 0, we can find an ε > 0 such that µ(X ε ) < τ . Then by the pigeonhole principle we can choose a tube T j such that µ(T j ) = 0 and The tube T j can be represented as T j = z∈Tj W (z), whereT j is a transversal to W f (i, m) and W (z), z ∈T j , are connected plaques of W f (i, m). By absolute continuity, whereμ is the factor measure onT j and µ W (z) is the conditional measure on W (z). Applying the pigeonhole principle again, we choose W = W (z) such that Recall that µ W (W ) = 1 by definition of the conditional measure and µ W is equivalent to the induced Riemannian volume on W by the absolute continuity of W f (i, m). The plaque W is subfoliated by plaques of U of sizes between S 0 and S 1 . Unfortunately, we do not know if U is absolutely continuous with respect to µ W . So we construct a finite partition of W into smaller plaques of W f (i, m) which are thin U -foliated tubes.
To construct this partition, we switch to h(W ), which is a plaque of W g (i, m) subfoliated by the plaques of h(U ) = V g m . The partition {T 1 ,T 2 , . . .T p } will consist of V g m -tubes inside of h(W ) that can be represented as whereT j is a transversal to V g m inside of h(W ) and V (z) are plaques of V g m . For every j = 1, . . . p, choose z j ∈T. Then the tubeT j can also be represented as whereP j (y) ⊂ W g (i, m − 1)(y) are connected plaques.
Recall that V g m is Lipschitz inside of W g (i, m). Hence for any ξ > 0 it is possible to find a partition {T 1 ,T 2 , . . .T p }, p = p(ξ), such that where B j (C 1 ξ) and B j (C 2 ξ) are balls inside of (W g (i, m − 1)(y), induced Riemannian distance) of radiiC 1 ξ andC 2 ξ respectively. The constantsC 1 andC 2 are independent of ξ. Since we are working in a bounded plaque h(W ) they also do not depend on any other choices but S 1 .
In the sequel we will need to take ξ to be much smaller than ε. Now we pool this partition back into a partition of W .
Although we use the same notation for this partition, it is clearly different from the initial partition of T d . Each tube T j can be represented as where Figure 11. We construct the partition {T 1 , T 2 , . . . T p } as a pullback of the partition of h(W ) by V g m -tubes. The foliation V g m is Lipschitz and h is continuously differentiable along W f (i, m − 1). This guarantees that the "width" of a tube T j is of the same order as we move along T j (46). Hence µ W is "uniformly distributed" along T j . By Lemma 6.7, h is continuously differentiable along W f (i, m−1). Moreover, the derivative depends continuously on the points in W . Hence property (44) persists: The constants C 1 and C 2 differ fromC 1 andC 2 by a finite factor due to the bounded distortion along W f (i, m − 1) by the differential of h.
Applying the pigeonhole principle for the last time, we find T ∈ {T 1 , T 2 , . . .
Take a plaque U xȳ inside of T. By construction, Estimating horizontal size of U xȳ from below. We have constructed U xȳ such that a lot of points in the neighborhood of U xȳ T lie outside of X ε . The corresponding ε-rectangles R(x) have vertical size greater than ε δ . It is clear that we can use this fact to show that V S(U xȳ ) is large. Choose First we estimate the number of rectangles N .
Lemma 6.11. The holonomy map Hol : We postpone the proof until the end of the current section.
Then according to the lemma above, d f m (x j−1 ,x j ) ≤ C Hol V S(R(x j−1 , x j )) ρ = C Hol ε ρ , j = 1, . . . N, which allows us to estimate N Along with the rectangles R(x j , x j+1 ), let us consider sets A(x j , x j+1 ) ⊂ T, j = 0, . . . N − 1, given by the formula where P (y) are the plaques of W f (i, m − 1) from the representation (45) for T. The sets A(x j , x j+1 ) have the same vertical size. The following property of these sets is a direct consequence of (46) and the fact that µ W is equivalent to the Riemannian volume on W .
The constant C univ depends on C 1 , C 2 and size of W , but is independent of ε and ξ. Let It follows from (47) that either For concreteness, assume that the first possibility holds. The bounds (49) allow us to estimate the number N 1 of sets Here ⌊N/2⌋ is the total number of sets A(x j , x j+1 ) in A 1 and ⌊C univ τ N ⌋ is the maximal possible number of sets A(x j , x j+1 ) in A 1 ∩ X ε . Clearly we can choose τ and ε accordingly so N 1 ≥ N/3. For every A(x j , x j+1 ) as above, fix q j ∈ A(x j , x j+1 ), q j / ∈ X ε , and consider rectangle R(q j ) of vertical size ε. Then Consider two rectangles R(q j ) and R(qj) as above. Since |j −j| ≥ 2, they do not "overlap" vertically if ξ is sufficiently small (although this is not important to us). They might happen to "overlap" horizontally as shown on the Figure 12 but the size of the overlap cannot exceed the diameter of the tube T, which, according to (46), is bounded by C 2 ξ.
The above considerations result in the following estimate: where C H is the Lipschitz constant of the holonomy map along W f (i + 1, m). We used estimate on N 1 and estimate (48) on N . Finally, recall that δ − ρ < 0, while ξ can be chosen arbitrarily small independently of ε (and hence N ). Hence by choosing ε small, we can find U xȳ with arbitrarily big horizontal size, which contradicts to the uniform upper bound (41) that follows from compactness. Hence Case 2 is impossible.
Remark. Note that we do not need to take τ arbitrarily small. The constant τ just needs to be small enough to provide the estimate on N 1 .
Proof of Lemma 6.11. Take points x and y ∈ V f m (x) such that By Lemma 6.9, there exist constants c 1 and c 2 such that T Figure 12. This picture illustrates the key estimate (50). Since the holonomy along W f (i+1, m) is Lipschitz, the horizontal size of U x0xN can be estimated from below by the sum of horizontal sizes of "flat" rectangles with base points q j ∈ A 1 ⊂ T, j = 1, . . . N 1 . They might overlap horizontally as shown, but the overlap is of order ξ ≪ ε.
Moreover, since c 1 and c 2 depend only ond(x, y) (see the remark after the proof of Lemma 6.9), they can be chosen independently of x and y as long as x and y satisfy (51). Take x, y ∈ T (a) close to each other. Let N be the smallest integer such that and, obviously, Hence by taking in (52)x = f N (Hol(x)) andỹ = f N (Hol(y)), we get On the other hand, Combining (53), (54), (55) and (56), we finish the proof , Hol(y)) ρ . We used (43) for the last equality. 7. Proof of Theorem C 7.1. Scheme of the proof of Theorem C. The way we choose the neighborhood U is the same as in Theorem A. We look at the L-invariant splitting L , E wu L are eigendirections with eigenvalues λ −1 < λ and E ss L ⊕ E su L is the Anosov splitting of g. We choose U in such a way that for any f ∈ U the invariant splitting survives, and f is partially hyperbolic in the strongest sense (30) with respect to the splitting (57).
We are ready to construct the conjugacy h : . The homeomorphism h maps the central foliation into the vertical foliation and the foliation S into the horizontal foliation.
Remark. Notice that at this point we do not know if h is a C 1+ν -diffeomorphism, although h c and h 0 are C 1+ν -differentiable.
). Moreover, it is clear from the definition of S that the restriction of this projection to W ws f (x) is an isometry with respect to the distance d ws . According to the formula for the first component of h, we compose this projection with h c , which is an isometry when restricted to the leaf W ws f (pr(x)) by the definition of d ws . The diffeomorphism h c straightens the weak stable foliation into a foliation by straight lines W ws L . Hence h(W ws f ) = W ws L and h is an isometry as a map (W ws f (x), d ws ) → (W ws L (h(x)), Riemannian metric). Thus h is C 1+ν along W ws f . Everything above can be repeated for the weak unstable foliation. Applying the Regularity Lemma, we get the desired statement.
Proof. The restriction of h to S 0 is just h 0 . The restriction of h to some other leaf S(x) can be viewed as composition of the holonomy H c f , h 0 and the holonomy H c L . Hence this restriction is C 1+ν -differentiable as well. We need to make sure that the derivative of h along S is Hölder-continuous on T 4 . For this we need only show that the derivative of H c f : S(x) → S 0 depends Hölder-continuously on x. This assertion will become clear in the proof of Lemma 7.3. Now by the Regularity Lemma, we conclude that h is a C 1+ν -diffeomorphism. LetL = h • f • h −1 . Clearly the foliations W ws L and W wu L areL-invariant. By construction, h and h −1 are isometries when restricted to the leaves of the weak foliations. Recall that f stretches by a factor λ the distance d wu on W wu f and contracts by a factor λ −1 the distance d ws on W ws f . Hence, if we consider the restriction ofL on a fixed vertical two-torus W c L (x) → W c L (L(x)), then it acts by a hyperbolic automorphism A.
Also, it is obvious from the construction of h that the factor map ofL on a horizontal two-torus is g. These observations show thatL is of the form L = (Ax + ϕ(y), g(y)). (4) Note that we do not have to additionally argue that ϕ is smooth since we know thatL is a C 1+ν -diffeomorphism.
Remark. An observant reader would notice that our choice of h and henceL is far from being unique. The starting point of the construction of h is the torus S 0 .
Although we have chosen a concrete S 0 , in fact, the only thing we need from S 0 is transversality to W c f . This is not surprising. Many diffeomorphisms of type (4) are C 1 -conjugate to each other. In the linear case this is controlled by the invariants (11).
In the rest of this section we prove Lemmas 7.2 and 7.3. 7.2. A technical Lemma. Before we proceed with proofs of Lemmas 7.2 and 7.3, we establish a crucial technical lemma which is a corollary of Lemma 7.1.
Lemma 7.6. Fix x ∈ T 4 and y ∈ W c L (x). Let v be a vector connecting x and y inside of W c L (x). Then In other words, the foliation U σ is invariant under translations along W c L , σ = ss, su.
Proof. For concreteness, we take σ = ss. The proof in the case where σ = su is the same.
First let us assume that y ∈ W ws L (x). This allows us to restrict our attention to the stable leaf W s L (x), since U ss (x) and U ss (y) lie inside of W s L (x). Pick a point z ∈ U ss (x) and letz = W ws L (z) ∩ U ss (y). We only need to show that d(x, y) = d(z,z), where d is the Riemannian distance along weak stable leaves. The simple idea of the proof of Claim 1 from [GG08] works here. We briefly outline the argument.
Since H −1 (z) ∈ W ss f (x), H −1 (z) ∈ W ss f (y), and strong stable leaves contract exponentially faster than weak stable leaves, we have On the other hand, since the derivative of H along W ws f is continuous, the ratios d(L n (z), L n (z)) d (H −1 (L n (z)), H −1 (L n (z))) and d(L n (x), L n (y)) d (H −1 (L n (x)), H −1 (L n (y))) are arbitrarily close when n → +∞. Together with (60), this shows that the constant c from (59) is arbitrarily close to 1. Hence c = 1.
Finally, recall that for any x the leaf W ws L (x) is dense in W c L (x). Hence by continuity, we get the statement of the lemma for any y ∈ W c L (x).
Lemma 7.6 leads to some nontrivial structural information about f which is of interest on its own.
First, we map W su f (x) into a Hölder-continuous curve U su (H(x)) ⊂ W u L (H(x)) and then we project it on W su L (H(x)) along the linear foliation W wu L , as shown on the Figure 14.
Lemma 7.9. For any x ∈ T 4 , the map H x is C 1+ν -differentiable. Figure 14.  ≤ c 1 c 2 c 3 c 4 d su f (y, z). The first and fourth inequalities hold since W su L and W u L are quasi-isometric. The second inequality holds with a universal constant c 2 due to the uniform transversality of W wu L and W su f . Inequalities 3 and 6 are obvious. The fifth inequality holds since d su f (y, z) ≥ 1 and the lift of the conjugacy satisfies Here we slightly abuse notation by denoting the lift and the map itself by the same letter. Now we need to show that H x is Lipschitz if y and z are close on the leaf. Notice that H x is the composition of H y and the holonomy H wu L : W su L (H(y)) → W su L (H(x)), which is just a translation. Hence, to show that H x is Lipschitz at y we only need to show that H y is Lipshitz at y.
So we fix x and y on W su L (x) close to x and show that d su whenever d su f (x, y) < K. Consider the Hölder-continuous functions D su f (·) and D su L (H(·)). The assumption on the p. d. of f and L guarantee that the products of these derivatives along periodic orbits coincide. Thus we can apply the Livshitz Theorem and get the Hölder-continuous positive transfer function P such that .
Choose the smallest N such that d su f (f N (x), f N (y)) ≥ 1. Theñ · c 1 c 2 c 3 c 4 .
The function P is uniformly bounded away from zero and infinity. Hence, together with (D3), this shows that H x is Lipschitz at x uniformly in x and hence is uniformly Lipschitz.
Next we apply the transitive point argument. Consider the SRB measure µ u which is the equilibrium state for the potential minus the logarithm of the unstable jacobian of f . It is well known that W u f is absolutely continuous with respect to µ u . On a fixed leaf of W u f , the foliation W su f is absolutely continuous with respect to the Lebesgue measure on the leaf (for proof see [LY85], Section 4.2; they prove that the unstable foliation is Lipschitz with center-unstable leaves, but the proof goes through for strong unstable foliation within unstable leaves). Hence W su f is absolutely continuous with respect to µ u .
We know that H x is Lipschitz and hence almost everywhere differentiable on W su f (x). It is clear from the definition that H x is differentiable at y if and only if H y is differentiable at y. Thus it makes sense to speak about differentiability at a point on a strong unstable leaf without referring to a particular map H x . The absolute continuity of W su f allows to conclude that H x is differentiable at x for µ u -almost every x.
Since µ u is ergodic and has full support we can consider a transitive pointx such that Hx is differentiable atx. Now C 1 -differentiability of H x for any x ∈ T 4 can be shown by an approximation argument: we approximate the target point by iterates ofx. The argument is the same as the proof of Step 1, Lemma 5 from [GG08] with minimal modifications, so we omit it. This argument shows even more, namely, D(H x )(x) = P (x) P (x) D(Hx)(x).
Note that D(H x )(y) = D(H y )(y). Hence H x maps the Lebesgue measure on the leaf W su f (x) into an absolutely continuous measure, dy → P (y) P (x) dLeb. Recall that P is Hölder-continuous. Hence H x is C 1+ν -differentiable.
Proof of Lemma 7.7. We work in a ball B inside of the leafW f (x) that contains T 1 (x) andT 2 (H wu f (x)). Recall that B is subfoliated by W c f and W su f . We apply the conjugacy map H to the ball B. It maps W su f and W c f into U su and W c L , respectively. We construct a shift map sh : H(B) →W L (H(x)) in such a way that, for any z, the leaf W c L (z) is sh-invariant and the action of sh on the leaf is a rigid translation.
Clearly sh(U su (H(x))) = W su L (H(x)). Moreover, by Lemma 7.6, sh(U su ) = W su L . The shift sh is designed such that the composition sh • H maps the foliation W c f into W c L and the foliation W su f into W su L . According to Lemma 7.1, sh • H is C 1+ν -differentiable along W c f . Also notice that the restriction of sh • H to a strong unstable leaf W su f is nothing but H y composed with constant parallel transport along W wu L . Recall that H y is C 1+ν -differentiable by Lemma 7.9. Hence, by the Regularity Lemma, we conclude that sh • H is C 1+ν -diffeomorphism.

H(B) H(x)
ThereforeT 1 = sh • H(T 1 (x)) andT 2 = sh • H(T 2 (H wu f (x))) are smooth curves inside of H(B) and the holonomy map H wu f can be represented as a composition as shown on the commutative diagram The holonomy H wu L is smooth since W wu L is a foliation by straight lines. Hence H wu f is C 1+ν -differentiable.
Remark. Notice that this argument completely avoids dealing with the geometry of transversals, i.e., their relative position to the foliations.
Proof of Lemma 7.8. We use exactly the same argument as in the previous proof. Notice that the picture is not completely symmetric compared to the picture in Lemma 7.7 since we are dealing with the weak unstable holonomy. Nevertheless the argument goes through by looking at transversalsT 1 (x) andT 2 (H wu f (x)) on the leaf ofW f . The shift map must be constructed in such a way that it maps U ss into W ss L .
Proof of Lemma 7.3. In this proof we exploit the same idea of composing H with some shift map. We fix S 1 = S(x 1 ) ∈ S which is, a priori, just an embedded topological torus. We assume that x 1 ∈ W wu f (x 0 ). It is easy to see that this is not restrictive.
Foliate S 0 and S 1 byT 0 ,T 0 andT 1 ,T 1 , respectively, by taking intersections with leaves ofW f andW f . To prove the lemma we only have to show that the leaves of T 1 andT 1 are C 1+ν -differentiable curves.
We restrict our attention to a leaf ofW f . Construct the shift map sh in the same way as in Lemma 7.7. Fix an x ∈ S 0 and letT 0 = sh • H(T 0 (x)),T 1 = sh • H(T 1 (H wu f (x))). T 0 is a C 1+ν -curve since sh • H is C 1+ν -diffeomorphism. By the definition of S 1 , ∀y ∈T 0 d wu (y, H wu f (y)) = d wu (x, H wu L (x)). Recalling the definition of d wu , we see that the conjugacy H acts as an isometry on a weak unstable leaf. Obviously sh is an isometry when restricted to a weak unstable leaf as well. Therefore where d is the Riemannian distance along W wu L . HenceT 1 is smooth as a parallel translation ofT 0 . We conclude thatT 1 (H wu f (x)) = (sh • H) −1 (T 1 ) is C 1+ν -curve. Repeating the same argument forT 0 (x) andT 1 (H wu f (x)), we can show that T 1 (H wu f (x)) is C 1+ν -curve. Hence the lemma is proved.
8. Proof of Theorem D 8.1. Scheme of the proof of Theorem D. We choose U in the same way as in 7.1. The only difference is that L is given by (1) not by (3). Given f ∈ U we denote by W c f the two-dimensional central foliation. Take f and g in U. Then they are conjugate, h • f = g • h.
transitivity, we only need to cover T 4 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 the Birkhoff Ergodic Theorem, where I is the 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.
LetB ⊂ B be a slightly smaller ball and let W c (B) = x∈B W c f (x). Since weak unstable leaves are dense in the corresponding central leaves it is possible to find R > 0 such that Applying the standard Hopf argument, for µ-a. e. x, the function E(ψ|I) is constant on W (x, R). Now the absolute continuity of W wu f together with the above observations show that E(ψ|I)(x) > 0 for µ − a. e. x ∈ W c (B).
Repeat the same argument to get ∀n E(ψ|I)(x) > 0 for µ − a. e. x ∈ W c (f n (B)). According to (61) this means that µ-a. e. x visits B ′ infinitely many times.