SMOOTH CONJUGACY OF ANOSOV DIFFEOMORPHISMS ON HIGHER DIMENSIONAL TORI

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1 SMOOTH CONJUGACY OF ANOSOV DIFFEOMORPHISMS ON HIGHER DIMENSIONAL TORI ANDREY GOGOLEV Abstract. Let L be a hyperbolic automorphism o T d, d 3. We study the smooth conjugacy problem in a small C 1 -neighborhood U o L. The main result establishes C 1+ν regularity o 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 satisy a natural transitivity assumption on certain intermediate oliations. We elaborate on the example o de la Llave o two Anosov systems on T 4 with the same constant periodic eigenvalue data that are only Hölder conjugate. We show that these examples exhaust all possible ways to perturb C 1+ν conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority o reducible toral automorphisms as well as to certain product dieomorphisms o T 4 C 1 -close to the original example. Contents 1. Introduction and statements Positive answers When the coincidence o periodic data is not suicient Additional moduli o C 1 conjugacy in the neighborhood o the counterexample o de la Llave Organization o the paper and a remark on terminology Acknowledgements 7 2. The counterexample on T Proo o Theorem B On the Property A Transitivity versus minimality Examples o dieomorphisms that satisy Property A An example o an open set o dieomorphisms that possess Property A Proo o Theorem A Scheme o the proo o Theorem A Proo o the integrability lemmas Weak unstable lag is preserved: proo o Lemma Induction step 1: the conjugacy preserves oliation V m Induction step 2: proo o Lemma 6.7 by transitive point argument Induction step 1 revisited Proo o Theorem C Scheme o the proo o Theorem C A technical Lemma Smoothness o central holonomies 42 1

2 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 2 8. Proo o Theorem D Scheme o the proo o Theorem D Smoothness along the central oliation: proo o Proposition Reerences Introduction and statements Consider an Anosov dieomorphism o a compact smooth maniold. Structural stability asserts that i a dieomorphism g is C 1 close to then and g are topologically conjugate. The conjugacy h is unique in the neighborhood o identity. h = g h It is known that h is Hölder continuous. There are simple obstructions or h to be smooth. Namely, let x be a periodic point o, p (x) = x then g p (h(x)) = h(x) and i h were dierentiable then D p (x) = (Dh(x)) 1 Dg p (h(x))dh(x) i. e. D p (x) and Dg p (h(x)) are conjugate. We see that every periodic point carries a modulus o smooth conjugacy. Suppose that or every periodic point x, p (x) = x, dierentials o return maps D p (x) and Dg p (h(x)) are conjugate then we say that periodic data (p. d.) o and g coincide. Question 1. Suppose that p. d. coincide, is h dierentiable? I it is then how smooth is it? 1.1. Positive answers. We describe situations when p. d. orm ull set o moduli o C 1 conjugacy. The only surace that supports Anosov dieomorphisms is two dimensional torus. For Anosov dieomorphisms o T 2 the complete answer was given by de la Llave, Marco and Moriyón. Theorem ([LMM88], [L92]). Let and g be C r, r > 1, Anosov dieomorphisms o T 2 that are topologically conjugate, h = g h. Suppose that p. d. coincide. Then h is C r ε where ε > 0 is arbitrarily small. De la Llave [L92] also observed that the answer is negative or Anosov dieomorphisms o T d, d 4. He constructed two dieomorphisms with the same p. d. which are only Hölder conjugate. We describe this example in Section 2. In dimension three the only maniold that supports Anosov dieomorphisms is three dimensional torus. Moreover, all Anosov dieomorphisms o T 3 are topologically conjugate to the linear automorphisms o T 3. Nevertheless the answer to the Question 1 is not known. Conjecture 1. Let and g be C r, r > 1, Anosov dieomorphisms o T 3 that are topologically conjugate, h = g h. Suppose that p. d. coincide. Then h is at least C 1.

3 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 3 There are partial results that support this conjecture. Theorem ([GG08]). Let L be a hyperbolic automorphism o T 3 with real eigenvalues. Then there exists a C 1 -neighborhood U o L such that any and g in U having the same p. d. are C 1+ν conjugate. Theorem ([KS07]). Let L be a hyperbolic automorphism o T 3 that has one real and two complex eigenvalues. Then any suiciently 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 big progress has been made (see [L02], [KS03], [L04], [F04], [S05], [KS07]) in the case when stable and unstable oliations carry invariant conormal structures. To ensure existence o these conormal structures one has at least to assume that every periodic orbit has only one positive and one negative Lyapunov exponent. This is a very restrictive assumption on p. d. In contrast to above we will study smooth conjugacy problem in proximity o a hyperbolic automorphism L : T d T d with simple spectrum. Namely, with exception o Theorem B we will always assume that the eigenvalues o L are real and have dierent absolute values. For the sake o notation we assume that the eigenvalues o L are positive. This is not restrictive. Let l be the dimension o the stable subspace o L and k be the dimension o the unstable subspace o L, k + l = d. Consider L-invariant splitting T T d = F l F l 1... F 1 E 1 E 2... E k 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 o L. Precise choice o U is described in Section 6.1. Theory o partially hyperbolic dynamical systems guarantees that or any in U the invariant splitting survives (e. g. see [Pes04]) T T d = F l F l 1... F 1 E 1 E 2... E k. We will see these one dimensional invariant distributions integrate uniquely to oliations U l, U l 1,... U 1, V 1, V 2,... V k. Given a oliation F on T d and an open set B deine F(B) = F(y). y B We will be assuming the ollowing property o Property A. For every x T d and every open ball B x U l 1 (B) = U l 2 (B) =... = U 1 (B) = V 1 (B) = V 2 (B) =... = V k 1 (B) = Td. We discuss this property in Section 4.1. Theorem A. Let L be a hyperbolic automorphism o T d, d 3, with simple real spectrum. Assume that characteristic polynomial o L is irreducible over Z. There exists a C 1 -neighborhood U Di r (T d ), r 2, o L such that any U satisying A and any g U with the same p. d. are C 1+ν conjugate. (A)

4 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 4 Remark. We will see in Section 4.1 that irreducibility o characteristic polynomial o L is necessary or to satisy A. Formally, we could have omitted the irreducibility assumption above. Theorem B below shows that irreducibility o L is a necessary assumption or the conjugacy to be C 1. We believe that Theorem A holds when L is irreducible without assuming that satisies A. Remark. Number ν is a small positive number. It is possible to estimate ν rom below in terms o eigenvalues o L and the size o U. Remark. Obviously analogous result holds on inite actors o tori. But we do not know how prove it on nilmaniolds. The problem is that or an algebraic Anosov automorphism o a nilmaniold various intermediate distributions may happen to be non-integrable. Theorem A is a generalization o the theorem rom [GG08] quoted above. Our method does not lead to higher regularity o the conjugacy (see the last section o [GG08] or an explanation). Nevertheless we conjecture that the situation is the same as in dimension two. Conjecture 2. In the setup o Theorem A one can actually conclude that and g are C r ε conjugate, where ε is an arbitrarily small positive number. Simple examples o dieomorphisms that possess Property A include = L and any U when max(k, l) 2 (see Section 4.1). In addition we construct a C 1 - open set o Anosov dieomorphisms o T 5 and T 6 close to L that have Property A. It seems that this construction can be extended to arbitrary dimension. We describe this open set when l = 2 and k = 3. Given U denote by D wu the derivative o along V 1. Choose U in such a way that x x 0 D wu (x) > D wu (x 0 ), where x 0 is a ixed point o. Then any dieomorphism suiciently C 1 close to possess Property A When the coincidence o periodic data is not suicient. First let us briely describe the counterexample o de la Llave. Let L : T 4 T 4 be an automorphism o the product type L(x, y) = (Ax, By), (x, y) T 2 T 2, (1) where A and B are Anosov automorphisms. Let λ, λ 1 be the eigenvalues o A and µ, µ 1 be the eigenvalues o B. We assume that µ > λ > 1. Consider perturbations o the orm L = (Ax + ϕ(y), By), (2) where ϕ : T 2 R 2 is a C 1 -small C r, r > 1, unction. Obviously p. d. o L and L coincide. We will see in Section 2 that majority o perturbations (2) are only Hölder conjugate to L. The ollowing theorem is a simple generalization o this counterexample. Theorem B. Let L : T d T d be a hyperbolic automorphism. Assume that characteristic polynomial o L actors over Q. Then there exist C dieomorphisms L : T d T d and ˆL : T d T d arbitrarily C 1 -close to L with the same p. d. such that the conjugacy between L and ˆL is not Lipschitz.

5 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 5 Remark. In majority o cases one can take ˆL = L. The need to take L and ˆL both to be dierent rom L appears, or instance, when L(x, y) = (Ax, Ay). It was shown in [L02] that p. d. orm complete set o moduli or smooth conjugacy problem to L. This is a remarkable phenomenon due to invariance o conormal structures on stable and unstable oliations. Nevertheless we still have a counterexample i we go a little bit away rom L. Next we study smooth conjugacy problem in the neighborhood o (1) assuming that µ > λ > 1. We show that perturbations (2) exhaust all possibilities. Beore ormulating 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 Anosov dieomorphism L(x, y) = (Ax, g(y)), (x, y) T 2 T 2, (3) where g is an Anosov dieomorphism suiciently C 1 -close to B so that L can be treated as a partially hyperbolic dieomorphism with automorphism A acting in the central direction. Consider perturbations o the orm L = (Ax + ϕ(y), g(y)). (4) As beore, it is obvious that p. d. o L and L coincide. In Section 8 we will see that L and L with non-linear 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 Di r (T 4 ), r 2, o L such that any U that has the same p. d. as L is C 1+ν, ν > 0, conjugate to a dieomorphism L o type (4) Additional moduli o C 1 conjugacy in the neighborhood o the counterexample o de la Llave. Let L be given by (1) with µ > λ > 1 and let U be a small C 1 -neighborhood o L. It is ruitul to think o dieomorphisms rom U as o partially hyperbolic dieomorphisms with two dimensional central oliations. Consider, g U, h = g h. According to the celebrated theorem o Hirsch, Pugh and Shub [HPS77] the conjugacy h maps the central oliation o into the central oliation o g. Assume that p. d. o and g are the same. Then we show that h is C 1+ν along the central oliation. As described above it can still happen that h is not a C 1 -dieomorphism. This means that the conjugacy is not dierentiable in the direction transverse to the central oliation. The geometric reason or this is mismatch between strong stable (unstable) oliations o and g the conjugacy h does not map strong stable (unstable) oliation o into strong stable (unstable) oliation o g. Motivated by this observation we introduce additional moduli o C 1 -dierentiable conjugacy. Roughly speaking these moduli measure the tilt o strong stable (unstable) leaves when compared to the model (1). We deine these moduli precisely. Let WL ss, W L ws, W L wu su and WL be the oliations by straight lines along the eigendirections with eigenvalues µ 1, λ 1, λ and µ respectively. For any U these invariant oliations survive. We denote them by W ss, W ws, W wu and W su. Also we write W s and W u or two dimensional stable and unstable oliations. Let h be the conjugacy to the linear model, h = L h. Then h (W σ ) = W σ L, σ = s, u, ws, wu. (5)

6 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 6 Fix orientation o W σ L, σ = ss, ws, wu, su. Then or every x T4 there exists a unique orientation preserving isometry I σ (x) : W σ L (x) R, Iσ (x) = 0, σ = ss, ws, wu, su. Deine Φ u : T4 R R by the ormula Φ u (x, t) = I wu( I su (x) 1 (t) )( h (W su (h 1 (x))) WL wu (I su (x) 1 (t)) ). The geometric meaning is transparent and illustrated on Figure 1. Image o strong unstable maniold h (W su(x)) can be viewed as a graph o unction Φu (x, ) over WL su(x). Analogously we deine Φs : T4 R R. WL wu( x) 01 h (W su (h 1 (x)) Φ u (x, t) x t x 01 W su L (x) Figure 1. Geometric meaning o Φ u. Here x = Isu (x) 1 (t). Clearly Φ s/u are moduli o C 1 conjugacy. Indeed, assume that and g are C 1 conjugate by h. Then h(w su su ) = h(wg ) and h(w ss ss ) = h(wg ) since strong stable and unstable oliations are characterized by the speed o convergence which is preserved by C 1 conjugacy. Hence Φ s/u = Φ s/u g. It is possible to choose a subamily o these moduli in an eicient way. We say that and g rom U have the same strong unstable oliation moduli i or t 0 such that x T 4, Φ u (x, t) = Φ u g (x, t) (6) x T 4 and I = (a, b) R such that t I Φ u (x, t) = Φ u g (x, t). (7) Deinition o strong stable oliation moduli is analogous. Theorem D. Given L as in (1) with µ > λ > 1 there exists a C 1 -neighborhood U Di r (T 4 ), r 2 o L such that i, g U have the same p. d. and the same strong unstable and strong stable oliation moduli. Then and g are C 1+ν conjugate. Remark. In this case C 1+ν -dierentiability is in act the optimal regularity Organization o the paper and a remark on terminology. In Section 2 we describe the counterexample o de la Llave in a way that allows us to generalize it to Theorem B in Section 3. Sections 2 and 3 are independent o the rest o the paper.

7 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 7 In Sections 4 and 5 we discuss Property A and construct examples o dieomorphisms that satisy Property A. These sections are sel-contained. Section 6 is devoted to the proo o our main result, Theorem A. It is selcontained but in number o places we reer to [GG08] where three dimensional version o Theorem A was established. Theorem C is proved in Section 7. It is independent o the rest o the paper with an exception o a reerence to Proposition 10. Proo o Theorem D appears in Section 8 and relies on some technical results rom [GG08]. Throughout the paper we will be proving that various maps are C 1+ν -dierentiable. This should be understood in the usual way: the map is C 1 dierentiable and the derivative is Hölder continuous with some positive exponent ν. Number ν is not the same in dierent statements. When we say that a map is C 1+ν -dierentiable along oliation F we mean that restrictions o the map to the leaves o F are C 1+ν -dierentiable and the derivative is a Hölder continuous unction on the maniold, not only on the lea Acknowledgements. The author is grateul to Anatole Katok or numerous discussions, advice, and or introducing him to this problem. Many thanks go to Misha Guysinsky and Dmitry Scheglov or useul discussions. The author also would like to thank the reerees or providing helpul suggestions and pointing out errors. It was pointed out that tubular minimality o a oliation is equivalent to its transitivity. All these led to a better exposition. 2. The counterexample on T 4 Here we describe the example o de la Llave o two Anosov dieomorphisms o T 4 with the same p. d. that are only Hölder conjugate. Understanding o the example is important or the proo o Theorem B. Recall that we start with an automorphism L : T 4 T 4 L(x, y) = (Ax, By), (x, y) T 2 T 2, where A and B are Anosov automorphisms, Av = λv, Aṽ = λ 1 ṽ, Bu = µu, Bũ = µ 1 ũ. We assume that µ λ > 1. To simpliy computations we consider a special perturbation o the orm L = (Ax + ϕ(y)v, By). We look or the conjugacy h o the orm h(x, y) = (x + ψ(y)v, y). (8) The conjugacy equation h L = L h transorms into a cohomological equation on ψ ϕ(y) + ψ(by) = λψ(y). (9) Let us solve or ψ using the recurrent ormula We get a continuous solution to (9) ψ(y) = λ 1 ϕ(y) + λ 1 ψ(by). ψ(y) = λ 1 k 0 λ k ϕ(b k y). (10) Hence the conjugacy is indeed given by the ormula (8).

8 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 8 In the ollowing proposition we denote by subscript u the partial derivative in the direction o u. Proposition 1. Assume that µ > λ > 1. Then unction ψ is Lipschitz in the direction o u i and only i ( µ ) k ϕu (B k y) = 0, (11) λ k Z i. e. the series on the let converge in the sense o distribution convergence and the limit is equal to zero. Proo. First assume (11). Let us consider series (10) as series o distributions that converge to ψ. Then as a distribution ψ u is obtained by dierentiating (10) termwise. ψ u = λ 1 k 0 λ k µ k ϕ u (B k ). (12) Applying (11) we get ψ u = λ 1 k<0 λ k µ k ϕ u (B k ). Since µ > λ the above series converge and the distribution is regular. Hence ψ is dierentiable in the direction o u. Now assume that ψ is u-lipschitz. By dierentiating (9) we get cohomological equation on ψ u ϕ u (x) + µψ u (By) = λψ u (y) that is satisied on a B-invariant set o ull measure. We solve it using the recurrent ormula ψ u (y) = 1 µ ϕ u(b 1 y) + λ µ ψ u(b 1 y). Hence ψ u = λ 1 k<0 λ k µ k ϕ u (B k ). (13) 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). I µ = λ then the argument above works only in one direction. We will see that in this case L and L do not provide a counterexample since p. d. are dierent. Proposition 2. Assume that µ = λ. Then (11) is a necessary assumption or ψ to be Lipschitz in the direction o u. Proo. As in the proo o Proposition 1, viewed as distribution, ψ u is given by ψ u = λ 1 k 0 ϕ u (B k ). (14) Assume that ψ is u-lipschitz then analogously to (13) we get ψ u = λ 1 ϕ u (B k ) + ψ(b N ). (15) N k<0

9 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 9 Note that in the sense o 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). ψ u = λ 1 k<0 ϕ u (B k ). (16) By rewriting condition (11) in terms o Fourier coeicients o ϕ one can see that it is an ininite codimension condition. Moreover, one can easily construct unctions that do not satisy (11). One only need to make sure that some Fourier coeicients o the sum (11) are non-zero. For instance, or any ε > 0 and positive integer p unction ϕ(y) = ϕ(y 1, y 2 ) = ε sin(pπy 1 ) (17) will serve the purpose. Thus corresponding L is not C 1 conjugate to L. Note that L maybe chosen arbitrarily close to L. Remark. Perturbations o the general type (2) can be treated analogously by decomposing φ = φ 1 v + φ 2 ṽ. Remark. Notice that the assumption µ λ > 1 is crucial in this construction. Remark. By choosing appropriate λ and µ one can get any desired regularity o the conjugacy (see [L92] or details). For example, i µ 2 > λ > µ > 1 then the conjugacy is C 1 but not C 2. From now on let us assume that µ = λ. As we have remarked in the introduction L and L do not provide a counterexample. Indeed, the derivative o L in the basis {v, u, ṽ, ũ} is λ ϕ u 0 ϕũ 0 λ λ λ 1 Let x be a periodic point, L p (x) = x. Then the derivative o the returm map at x is λ p λ p 1 y O(x) ϕ u(y) 0 0 λ p λ p 0. (18) λ p We see that it is likely to have a Jordan block while L is diagolizable. Hence L and L have dierent p. d. It is still easy to cook up a counterexample in the neighborhood o L. Let ˆL = (Ax + ξ(y)v, By) an let be the conjugacy between L and ˆL Proposition 3. Condition h(x, y) = (x + ψ(y)v, y) (ξ ϕ) u (B k y) = 0, k Z is necessary or ψ to be Lipschitz in the direction o u.

10 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 10 The proo is exactly the same as the one o Proposition 2. Take ϕ that does not satisy (11) as beore and take ξ = 2ϕ. Then obviously the condition o Proposition 3 is not satisied. Hence h is not Lipschitz. By looking at (18) it is obvious that our choice o ξ guarantees that Jordan normal orms o the derivatives o the return maps at periodic points o L and ˆL are the same. Remark. Due to the special choice o ξ it was easy to ensure that p. d. o L and ˆL are the same. We could have taken a dierent and somewhat more general approach. It is possible to show that or many choices o ϕ the sum that appears over the diagonal in (18) is non-zero or every periodic point x. All correspoding dieomorphism will have the same p. d. with a Jordan block at every periodic point. 3. Proo o Theorem B Here we consider L : T d T d with reducible characteristic polynomial. We show how to construct L and ˆL with the same p. d. which are not Lipschitz conjugate. Assume that all real eigenvalues o L are positive. Otherwise we would consider L 2. Let M : R d R d be the lit o L. And let {e 1, e 2,... e d } be the canonical basis so that T d = R d /span Z {e 1, e 2,... e d }. It is well known that characteristic polynomial o M actors over Z into the product o polynomials irreducible over Q. P (x) = P 1 (x)p 2 (x)... P r (x), r 2. Let λ be the eigenvalue o M with the smallest absolute value which is greater than one. Without loss o generality we assume that P 1 (λ) = 0. Let V i be the invariant subspace that corresponds to the roots o P i. Then dim V i = deg P i and it is easy to show that V i = Ker(P i (M)). Matrices o 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 matrix o M in this basis has integer entries and is o a block diagonal orm with blocks corresponding to invariant subspaces V i, i = 1,... r. We consider projection o M to T d = R d /span Z {ẽ 1, ẽ 2,... ẽ d }. Denote by N the induced map on T d. We have the ollowing commutative diagram where π is a inite-to-one projection. R d M R d T d N T d π π T d L T d Notice that N has the orm N(x, y) = (Ax, By), (x, y) T deg P 1 T d deg P 1. Let µ be an eigenvalue o B. By construction λ, λ µ, is an eigenvalue o A. With certain care the construction o Section 2 can be applied to N. We have to distinguish the ollowing cases.

11 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 11 (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. Assume that λ < µ. Then we take ˆL = L. In the irst case construction o Section 2 applies straightorwardly. We use unction o the type (17) to produce Ñ. Now we only need to make sure that Ñ can be projected to a map L : T d T d. Since π is a inite-to-one covering map this can be achieved by choosing suitable p in (17). Other cases require heavier calculations but ollow the same scheme o Proposition 1. We outline the construction in the case 4 that can appear, or instance, i A and B are hyperbolic automorphisms o our 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 bases {v 1, v 2 } and {u 1, u 2 } respectively. We are ollowing the construction rom the previous section. Let Ñ(x, y) = (Ax + ϕ(y) v, By) de = (Ax + ϕ 1 (y)v 1 + ϕ 2 (y)v 2, By). Then we look or the conjugacy in the orm h(x, y) = (x + ψ(y) v, y) de = (x + ψ 1 (y)v 1 + ψ 2 (y)v 2, y). The conjugacy equation h Ñ = N h transorms into ϕ(y) v + ψ(by) v = λ R A ψ(y). (19) Solving or ψ gives ψ(y) = λ k 1 R k 1 A ϕ(b k y), k 0 which we would like to dierentiate in the directions u 1 and u 2. We use the ormula ( ) ( ) ϕ1 (By) ϕ(by) u = u1 ϕ 1 (By) u2 (ϕ1 ) = µ u1 (ϕ 1 ) u2 (By)R ϕ 2 (By) u1 ϕ 2 (By) u2 (ϕ 2 ) u1 (ϕ 2 ) B = ϕ u (By)R B u2 to get that as a distribution ψ u = k 0 λ k 1 µ k R k 1 A ϕ u (B k )R k B. Now we assume that ψ is Lipschitz and we dierentiate (19) in the directions u 1 and u 2 ϕ u(y) + µ ψ u(by)r B = λ R A ψ u(y). Hence by the recurrent ormula ψ u = k<0 λ k 1 µ k R k 1 A ϕ u (B k )R k B. Combining the expressions or ψ u we get λ k µ k R k A ϕ u(b k )RB k = 0. k Z

12 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 12 Using Fourier decomposition one can ind unctions ϕ that do not satisy the condition above. One also needs to make sure that the choice o ϕ allows to project Ñ down to L. We omit this analysis since it is routine. This is a contradiction and thereore ψ (and hence h) is not Lipschitz. I λ = µ but λ µ then the scheme above still works. Obviously extra Jordan blocks do not appear in the normal orms at periodic points o L. Finally, the case λ = µ must be treated separately. We use the same trick as in Section 2 to ind L and ˆL with the same p. d. that are only Hölder conjugate. The trick works well in the case o complex eigenvalues as well. We omit the details. 4. On the Property A 4.1. Transitivity versus minimality. Here we discuss Property A. Let F be a oliation o a compact maniold M. As usually F(x) stands or the lea o F that contains x and F(x, R) stands or the ball o radius R centered at x inside o F(x). Deinition 1. Foliation F is called minimal i every lea o F is dense in M. Deinition 2. Foliation F is called transitive i these exists a lea o F that is dense in M. Deinition 3. Foliation F is called tubularly minimal i or every x and every open ball B x F(y) = M. y B Property A simply requires oliations U l 1, U l 2,... U 1, V 1, V 2,... V k 1 to be tubularly minimal. Property A. Foliations U l 1, U l 2,... U 1, V 1, V 2,... V k 1 are minimal. (A ) Proposition 4. Foliation F is transitive i and only i it is tubularly minimal. 1 Proo. Transitivity obviously implies tubular minimality. Assume that F is tubularly minimal. Let {B n, n 1} be a countable basis or the topology o M. By the deinition o tubular minimality sets F(B n ) are open and dense in M. Hence by Baire category theorem we have the set B = n 1 F(B n ) is non-empty. For every x B the lea F(x) is dense in M. Remark. We deine Property A in terms o tubular minimality as opposed to transitivity because we need denseness o the tubes to carry out the proo o Theorem A. A priori, transitivity is weaker than minimality. Hence, a priori, Property A is weaker than Property A. I in Theorem A we require to satisy A instead o A then the induction procedure that we use (induction step 1) is much simpler. Proo o the induction step 1 assuming only Property A requires much more lengthy and delicate argument. It is not clear to us what is the relation between Properties A and A. They may happen to be equivalent. Thus irst we provide a proo o Theorem A assuming that 1 We would like to thank the reeree or pointing out this act.

13 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 13 has Property A. Then we present a separate proo o induction step 1 (namely Lemma 6.6) that uses only Property A. Minimality o a oliation can be characterized similarly to tubular transitivity. Proposition 5. Foliation F is minimal i and only i or every x and every open ball B x F(y) = M. y B The proo is simple so we omit it. As a corollary we get that oliation F is minimal i and only i or every x and every open ball B x there exists a number R such that F(y, R) = M. (20) y B This is the property which we will actually use in the proo o the induction step Examples o dieomorphisms that satisy Property A. Proposition 6. Assume that L is irreducible. Then oliations U L j, V L i, j = 1... l, i = 1... k are minimal. Proo. Denote by F one o the oliations under consideration. Since F is a oliation by straight lines the closure o a lea F(x) is a subtorus o T d. This subtorus lits to a rational invariant subspace o R d. The invariant subspace corresponds to a rational actor o the characteristic polynomial o L while we have assumed that it is irreducible over Q. Hence the invariant subspace is the whole R d and the subtorus is the whole T d. Hence the conclusion o Theorem A holds at least or = L. We will see in Section 6.1 that or any U oliations U 1 and V 1 are minimal. Hence the conclusion o Theorem A holds or any U i max(k, l) 2. It is easy to construct L that satisies A when k = 3 and l = 2 since we only have to worry about the oliation V 2. We let = s L where s is any small shit along V 2. Clearly V 2 = V 2 L and hence satisies A. Question about robust minimality o oliations U l 1, U l 2,... U 1, V 1, V 2,... V k 1 arises naturally. Robust minimality o strong stable and strong unstable oliations o partially hyperbolic systems received some attention in the literature due to its intimate connection with robust transitivity. See [Ma78] and more recent papers [BDU02], [PS06], where robust minimality o the ull expanding oliation is established under some assumptions. We do not have this luxury in our setting: expanding oliations that we are intrested in suboliate ull unstable oliation. A representative problem here is the ollowing. Question 2. Let L : T 3 T 3 be a hyperbolic linear automorphism with real spectrum λ 1 < 1 < λ 2 < λ 3. Consider one dimensional strong unstable oliation. Is it true that this oliation is robustly minimal? In other words, is it true that or any suiciently C 1 -close to L the strong unstable oliation o is minimal? In addition to the simple examples above we construct a C 1 -open set o dieomorphisms that possess Property A in the next section. The ollowing statement can be obtained by applying the construction and the arguments o the next section in the setup o Question 2.

14 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 14 Proposition 7. Let L be as in Question 2. Then there exists a C 1 -open set U C 1 - close to L such that or every U the strong unstable oliation o is transitive. 5. An example o an open set o dieomorphisms that possess 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 o L chosen as in Section 6.1. Recall that D wu stands or the derivative o U along V 1. Choose U in such a way that x x 0 D wu (x) > D wu (x 0 ), (21) where x 0 is a ixed point o. Proposition 8. There exists a C 1 -neighborhood Ũ o such that any dieomorphism g Ũ has Property A. Remark. Similar example can be constructed on T 6 with l = 3, k = 3. We only need to do the trick described below or both stable and unstable maniolds o the ixed point x 0. Beore proving the proposition let us briely explain the idea behind the proo. We know that U g 1 and V g 1 are minimal. Hence we only need to show that oliation V g 2 is tubularly minimal i. e. or every x T5 and every open ball B x V g 2 (y) = T5. (22) y B To illustrate the idea we take g = and x = x 0. We work on the universal cover R 5 with lited oliations. Let T de = y B V 2 (y) R5, (23) which is an open tube. We show that T contains arbitrarily long connected pieces o the leaves o V 1 as shown on Figure 2. It would ollow that T is dense in T 5. Indeed, oliation V 1 is not just minimal but uniormly minimal: or any ε > 0 there exists R > 0 such that z T 5 V 1 (z, R) is ε-dense in T5. This property ollows rom the act that V L 1 is conjugate to the linear oliation V1. Pick y 0 B V 1 (x 0) close to x 0. Let x V 2 (x 0) be a point ar away in the tube T and y = V 1 (x) V 2 (y 0). To show that T contains arbitrarily long pieces o leaves o V 1 we prove that d 1 (x, y) (recall that d i is the Riemannian distance along V i ) is unbounded unction o x. We make use o the aine structure on V 1. We reer to [GG08] or the deinition o aine distance-like unction d 1. Recall crucial properties o d 1 (D1) d 1 (x, y) = d 1 (x, y) + o(d 1 (x, y)), (D2) d 1 ((x), (y)) = D wu(x) d 1 (x, y), (D3) K > 0 C > 0 such that 1 C d 1 (x, y) d 1 (x, y) C d 1 (x, y) whenever d 1 (x, y) < K.

15 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 15 V 1 (x) y x V 2 (y 0 ) V 2 (x 0 ) T y 0 x 0 V 1 (x 0 ) B Figure 2. Tube T contains arbitrarily long pieces o leaves o V 1. By property (D3) it is enough to show that d 1 (x, y) is unbounded. Given x as above pick N large so that the ratio d 1 ( N (x), N (y))/ d 1 (x 0, N (y 0 )) is close to 1 as shown on the Figure 3. It is possible since V 2 contracts exponentially aster than V 1 under the action o 1. It is not hard to see that given a large number n we can pick x (and N correspondingly) ar enough rom x 0 so that at least n points rom the orbit {x, 1 (x),... N (x)} lie outside o B. For such a point z = i (x) that is not in B D wu (z) D wu (x 0 ) + δ, where δ > 0 depends only on the size o B. Using (D2) we get d 1 (x, y) N d 1 (x 0, y 0 ) = i=1 D wu( i (x)) D wu(x d 1 ( N (x), N (y)) 0) d 1 (x 0, N (y 0 )) ( D wu (x ) n 0) + δ d 1 ( N (x), N (y)) d 1 (x 0, N (y 0 )) D wu (x 0) which is an arbitrary large number. Hence d 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 or any g U the ollowing alternative holds. Either V g 2 is conjugate to the linear oliation V2 L or there exist a dense set Λ such that or any x Λ and any B x the tube V 2 (y) R5 y B contains arbitrarily long connected pieces o the leaves o V g 1. Proo o Proposition 8. The argument is more delicate than the one presented above since we do not know that the minimum o the derivative is achieved at x 0. (24)

16 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 16 y 0 V 1 (x 0 ) V 2 (y 0 ) V 1 (x) y x 0 V 2 (x 0 ) x N (y 0 ) N (y) N x 0 N (x) 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 be a bigger ball. Condition (21) guarantees that we can choose them in such a way that m 0 < D wu (x 0 ) < sup with m 0, m 1 and M satisying D wu x B 0 (x) < m 1 < M < min D wu x / B 1 Mm q 1 0 m q > 1, (25) 1 where q is an integer that depends only on the size o U and the size o B 1. Ater that we choose Ũ U so the ixed point o g (that corresponds to x 0) is inside o B 0 and the property above persists. Namely, g Ũ m 0 < in Dg wu x B 0 (x) < sup Dg wu x B 0 (x) (x) < m 1 < M < min x/ B 1 D wu (x). (26) Note that provided that is suiciently C 1 -close to L and the ball B 1 is small enough any piece o a lea o V g 2 outside o B 1 that starts and ends on the boundary o B 1 cannot be homotoped into 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 o pieces o leaves o V g 2 outside o B 1 with endpoints on the boundary o B 1. Obviously, there is also an upper bound r on the lengths o pieces o leaves o V g 2 inside B 1.

17 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 17 V g 2 B 1 I 1 B 1 x 0 V g 2 ( x 0 ) (a) (b) Figure 4. (a) does not occur i B is suiciently small; (b) choice o I 1. It is enough to check (22) or a dense set Λ o points x T 5. We take Λ to be a subset o the set o periodic points o g Λ = {p : D wu n(p) (p) m n(p) 1 }, (27) where n(p) stands or the period o p. Set Λ consists o periodic points that spend large but ixed percentage o time inside o B 0. It is airly easy to show that Λ is dense in T 5. The proo is a trivial corollary o speciication property (e. g. see [KH95]). So we ix x 0 Λ, a small ball B centered at x 0 and y 0 B V g 1 (x 0) close to x 0. Our goal now is to ind x V g 2 ( x 0) ar in the tube T deined by (23) or which we can carry out estimates similar to (24). We will be working with pieces o leaves o 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 or any piece I, I = R, we have g q (I) > 2R + r. (28) Notice that q can be chosen to be independent o g and depends only on β 2, R and r. Pick I 1 V g 2 ( x 0), I 1 = R, I 1 B 1 =, as close to x 0 as possible i x 0 B 1 (see Figure 4b) or passing through x 0 i x 0 / B 1. Given I i, i 1 we choose I i+1 q (I i ), I i+1 = R, I i+1 B 1 =. Condition (28) guarantees that such choice is possible. We ix N large and take x I Nq V g 2 ( x 0). Let y = V g 1 (x) V g 2 (y 0) as beore. Construction o the sequence {I i, i 1} ensures that points qi (x), i = 0,... N 1, are outside B 1. This act together with (26) and (27) allows to carry out the

18 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 18 ollowing estimate Nq d 1 (x, y) d 1 ( x 0, y 0 ) = i=1 Dg wu (g i (x)) D wu g (g i ( x 0 )) d 1 ( Nq (x), Nq (y)) d 1 ( x 0, Nq (y 0 )) M N m N(q 1) 0 m Nq d 1 ( Nq (x), Nq (y)). 1 d 1 ( x 0, Nq (y 0 )) The aine-like distance ratio on the right is bounded away rom 0 independently o N since Nq (x) I 1 while the coeicient in ront o it is arbitrarily large according to (25). Hence d g 1 (x, y) is arbitrarily large and the projection o tube T is dense in T Proo o Theorem A For reasons explained in Section 4 we irst prove Theorem A assuming that has Property A. The only place where we use A is the proo o Lemma 6.6. In Section 6.6 we give another proo o Lemma 6.6 that uses Property A only Scheme o the proo o Theorem A. Recall the notation rom 1.1 or the L-invariant splitting T T d = F l F l 1... F 1 E 1 E 2... E k along the eigendirections with corresponding eigenvalues µ l < µ l 1 <... < µ 1 < 1 < λ 1 < λ 2 <... < λ k. We choose neighborhood U in such a way that or any in U the invariant splitting survives T T d = F l F l 1... F 1 E 1 E 2... E k, with (F i, F i ) < π 2, (E j, E j ) < π, i = 1,... l, j = 1,... k (29) 2 and is partially hyperbolic in the strongest sense: there exist C > 0 and constants α l < α l 1 < α l 1 <... < α 1 < α 1 < 1 < β 1 < β 1 <... < β k independent o the choice o in U such that or n > 0 D( n )(x)(v) Cαl n v, v F l (x), 1 C αn l 1 v D( n )(x)(v) Cαl 1 v, n 1 C αn 1 v D( n )(x)(v) Cα1 n v, 1 C β 1 n v D( n )(x)(v) Cβ1 n v, v F l 1 (x), v F 1 (x), v E 1 (x), 1 C β k n v D( n )(x)(v), v E k (x). (30) Equivalently the Mather spectrum o does not contain 1 and has d connected components.

19 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 19 We show that the choice o U guarantees unique integrability o intermediate distributions. From now on or the sake o concreteness we work with unstable distributions and oliations. For a given U let E (i, j) = E i E i+1... E j, i j. Lemma 6.1. For any in U distribution E (1, 1), E (1, 2),... E (1, k) are uniquely integrable. This is a direct corollary o Hirsch, Pugh and Shub theorem but we will present a direct proo. Let W 1 W 2... W k be the corresponding lag o weak unstable oliations. The last oliation in the lag is the unstable oliation W = W k. Lemma 6.2. For any in U and i j distribution E(i, j) is uniquely integrable. Denote by W (i, j), i j, the integral oliation o E (i, j). Also recall that we denote by V 1, V 2,... V k the integral oliations o E 1, E 2,... E k correspondingly. Notice that V i = W (i, i) and W i = W (1, i), i = 1,... k. Now we consider and g as in Theorem A, h = g h. The conjugacy h maps unstable (stable) oliation o into unstable (stable) oliation o g. Moreover, h preserves the whole lag o weak unstable (stable) oliations. Lemma 6.3. Fix an i = 1,... k. Then h(w i ) = W g i. Remark. Proo o this lemma does not use the assumption on p. d. We only need and g to be in U. Lemmas 6.1, 6.2 and 6.3 can be proved under a milder assumption. Instead o requiring and g to be in U we can require an Alternative assumption: and g are partially hyperbolic in the strongest sence (30) with the rate constants satisying µ l < α l < α l 1 < µ l 1 < α l 1 <... < β k 1 < λ k 1 < β k 1 < β k < λ k. ( ) We think that ( ) is actually automatic rom (30). Remark. To carry out proos o Lemmas above under the Alternative assumption one needs to transer the picture to the linear model by the conjugacy and use inequalities ( ) or growth arguments. This way one uses quasi-isometric oliations by straight lines o the linear model instead o oliations o which are a priori not known to be quasi-isometric. Conjecture 3. Suppose that is homotopic to L and partially hyperbolic in the strongest sense (30) then the rate constants satisy ( ). Remark. The proo o Lemmas 6.1, 6.2 and 6.3 is the only place where we really need and g to be in U. So in Theorem A the assumption that, g U can be substituted by the alternative assumption. Lemma 6.4. A lea W 1 (x) is dense in Td Proo. By Lemma 6.3 we have that the conjugacy between L and takes the oliation W1 L into the oliation W 1. According to Proposition 6 leaves o W 1 L are dense. Hence leaves o W 1 are dense.

20 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 20 Next we describe the inductive procedure which leads to smoothness o h along the unstable oliation. Induction base. We know that h takes W 1 into W g 1. Lemma 6.5. Conjugacy h is C 1+ν -dierentiable along W 1 i. e. restrictions o h to the leaves o W 1 are dierentiable and the derivative is Cν unction on T d. Provided that we have Lemma 6.4 the proo o Lemma 6.5 is the same as the proo o Lemma 5 rom [GG08]. Induction step. The induction procedure is based on the ollowing lemmas. Lemma 6.6. Assume that h is C 1+ν -dierentiable along W m 1 h(v g i ), i = 1,... m 1, 1 < m k. Then h(v m) = Vm. g and h(vi ) = Lemma 6.7. Assume that h(v m) = V g m or some m = 1,... k. Then h is C 1+ν - dierentiable along V m. We also use a regularity result due to Journé. Regularity Lemma ([J88]). Let M j be a maniold and Wj s, W j u be continuous transverse oliations with uniormly smooth leaves, j = 1, 2. Suppose that h : M 1 M 2 is a homeomorphism that maps W1 s into W2 s and W1 u into W2 u. Moreover, assume that the restrictions o h to the leaves o these oliations are uniormly C r+ν, r N, 0 < ν < 1. Then h is C r+ν. Remark. There are two more methods o proving analytical results o this lavor besides Journé s. One is due to de la Llave, Marco, Moriyón and the other one is due to Hurder and Katok (see [KN08] or a detailed discussion and proos). We remark that we really need Journé s result since the alternative approaches require oliations to be absolutely continuous while we apply the Regularity Lemma to various oliations that do not have to be absolutely continuous. Now the inductive scheme can be described as ollows. Assume that h is C 1+ν along W m 1 or some m k and h(vi ) = h(v g i ), i = 1,... m 1. By Lemma 6.6 we have that h(vm) = Vm g and by Lemma 6.7 h is C 1+ν along Vm. Fix a lea Wm(x). Leaves o W m 1 and V m suboliate Wm(x) and it is clear that the Regularity Lemma can be applied or h : Wm(x) Wm(h(x)). g Hence we get that h is C 1+ν on every lea o Wm. Hölder continuity o the derivative o h in the direction transverse to Wm is direct consequence o Hölder o the derivatives along W m 1 and Vm. We conclude that h is C 1+ν -dierentiable along Wm. By induction we get that h is C 1+ν -dierentiable along the unstable oliation and analogously along the stable oliation. We inish the proo o the Theorem A by applying the Regularity Lemma to stable and unstable oliations Proo o the integrability lemmas. In the proos o Lemmas 6.1 and 6.2 we work with lits o maps, distributions and oliations to R d. We use the same notation or lits as or the objects themselves. Proo o Lemma 6.1. Fix i < k. We assume that the distribution E (1, i) is not integrable or it is integrable but not uniquely. In any case it ollows that we can ind distinct points a 0, a 1,... a m such that

21 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 21 (1) {a 1, a 2,... a m } W (a 0 ), (2) there are smooth curves τ j : [0, 1] W (a 0 ), j = 1,... m, such that τ j (0) = a j 1, τ j (1) = a j and τ j E p(j), where p(j) i, (3) there is a smooth curve τ : [0, 1] W (a 0 ) such that τ(0) = a 0, τ(1) = a m and τ j E q or some q > i. Let τ be a piecewise smooth curve obtained by concatenating τ 1, τ 2,... τ m 1 and τ m. From the second property above and (30) we get the ollowing rough estimate Similarly n 0 length( n ( τ)) β n i length( τ). (31) n 0 length( n (τ)) β n i+1length(τ). (32) Denote by d(, ) the usual distance in R d. It ollows rom the assumption (29) that any curve γ : [0, 1] R d tangent to the distribution Eq is quasi-isometric: In particular c > 0 such that length(γ) c d(γ(0), γ(1)). n 0 d( n (a 0 ), n (a m )) 1 c length( n (τ)). (33) Inequalities (31), (32) and (33) sum up to a contradiction. Proo o Lemma 6.2. The theory o partial hyperbolicity guarantees that distributions E (i, k), i = 1,... k, integrate uniquely to oliations W (i, k). Let us ix i and j, i < j, and deine W (i, j) = W (1, j) W (i, k). Obviously W (i, j) is an integral oliation or E (i, j). Unique integrability o E (i, j) is a direct consequence o the unique integrability o E (1, j) and E (i, k) Weak unstable lag is preserved: proo o Lemma 6.3. Proo. We continue working on the universal cover. Pick two points a and b, a W i (b). Since h (x + m) = h (x) + m, m Z d (34) we have that d(h(x), h(y)) c 1 d(x, y) or any x and y such that d(x, y) 1. Hence or any n > 0 d(g n (h(a)), g n (h(b))) = d(h( n (a)), h( n (b))) c 2 d( n (a), n (b)) c 2 c 3 β n i, where c 2 and c 3 depend on d(a, b). This inequality guarantees that h(a) W g i (h(b)). Since the choice o a and b was arbitrary we conclude that h(w i ) = W g i Induction step 1: the conjugacy preserves oliation V m. We prove Lemma 6.6 which is the key ingredient in the proo o Theorem A. The proo is based on our idea rom [GG08] but we take a rather dierent approach in order to deal with high dimension o W. We provide a complete proo almost without reerring to [GG08]. Nevertheless we strongly encourage the reader to read Section 4.4 o [GG08] irst. The goal is to prove that h(v m) = V g m. So we consider oliation U = h 1 (V g m). As or usual oliations U(x) stands or the lea o U passing through x and U(x, R) stands or the local lea o size R. A priori, the leaves o U are just Hölder continuous curves. Hence the local lea needs to be deined with certain care. One way is to consider the lit o U and deine the lit o local lea U(x, R) as connected component o x o the intersection U(x) B(x, R). We prove Lemma 6.6 by induction.

22 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 22 Induction base. We will be working on m-dimensional leaves o W m. By Lemma 6.3 U suboliate W m. In other words or any x T d U(x) W m(x). Induction step. Suppose that U suboliate W (i, m) or some i < m. Then U suboliate W (i + 1, m). By induction we get that U suboliate W (m, m) = V m. Hence U = V m. First let us prove several auxiliary claims. Note that all oliations that we are dealing with are oriented and the orientation is preserved under the dynamics. Denote by d j and dg j the induced distances on the leaves o Vj and V g j correspondingly, j = 1,... k. Lemma 6.8. Consider a point a T d. Pick a point b U(a) and let b = V i (b) W (i + 1, m)(a). Assume that b b. Pick a point c V i (a) and let d = U(c) W (i, m 1)(b), d = V i (d) W (i + 1, m)(c). Then d d and the orientations o the pairs (b, b) and (d, d) in V i are the same. The statement o the lemma when i = 1 and m = 3 is illustrated on Figure 5. Remark. Since by the induction hypothesis h(w (i, m 1)) = W g (i, m 1) we see that the lea U(a) intersects each lea W (i, m 1)(x), x W (i, m)(a) exactly once. Proo. Let e = V i (b) W (i+1, m)(d) and ẽ = V i (b) W (i+1, m)( d). Obviously (e, ẽ) has the same orientation as (d, d) and also has advantage o lying on the lea V i (b). Thereore we orget about (d, d) and work with (e, ẽ). W (2, 3)(a) U(a) W (2, 3)(c) U(c) b 01 b 01 V 1 (b) ẽ d e d W 2 (b) a V 1 (a) 01 c V 1 (a) Figure 5. Illustration to Lemma 6.8 when i = 1 and m = 3. We use aine structure on the expanding oliation V i. Namely we work with aine distance-like unction d i. We reer to [GG08] or the deinition. There we

23 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 23 deine aine distance-like unction on weak unstable oliation. The deinition or oliation V i is the same with obvious modiications. Recall crucial properties o d i (D1) d i (x, y) = d i (x, y) + o(d i (x, y)), (D2) d i ((x), (y)) = D i (x) d i (x, y), where D i (D3) K > 0 C > 0 such that whenever d i (x, y) < K. 1 C d i (x, y) d i (x, y) C d i (x, y) is the derivative o along V i. Assume that (e, ẽ) has orientation opposite to (b, b) or e = ẽ. For the sake o concreteness we assume that these points lie on V i (b) in the order b, b, ẽ, e. All other cases can be treated similarly. Then d i (b, e) d i (b, ẽ) > d i (b, ẽ) d i (b, b). Remark. Notice that d i (b, ẽ) d i (b, b) d i ( b, ẽ) since d i is neither symmetric nor additive. Distance d i is given by an integral o a certain density with normalization deined by the irst argument. As long as the irst argument (point b in the above inequality) is the same all natural inequalities hold. Applying (D2) we get that n > 0 d i ( n (b), n (e)) d i ( n (b), n (ẽ)) d i ( n (b), n ( b)) = c 1 > 1 where c 1 does not depend on n. distance N : n > N By property (D1) we can switch to the usual d i ( n (b), n (e)) d i ( n ( b), n (ẽ)) > c 2 > 1 (35) where c 2 does not depend on n. Under the action o 1 strong unstable leaves o W (i + 1, m) contract exponentially aster then weak unstable leaves o V i. Thus we get that d i ε > 0 N : n > N ( n (a), n (c)) d i ( n ( b), n (ẽ)) 1 < ε. (36) Point h(e) W g (i+1, m)(h(c)). Indeed, notice that e = V i (b) W (i + 1, m)(d) = V i (b) W (i + 1, m 1)(d) (i i = m 1 than we have e = d). Thus h(e) = h(v i (b) W (i + 1, m 1)(d)) = V g i (h(b)) W g (i + 1, m 1)(h(d)) = V g i (h(b)) W g (i + 1, m)(h(d)) = V g i (h(b)) W g (i + 1, m)(h(c)), where the last equality is justiied by the act that h(d) Vm(h(c)). g We know also that h(b) W g (i + 1, m)(h(a)). Hence, analogously to (36), we have ε > 0 N : n > N d g i (g n (h(a)), g n (h(c))) d g i (g n (h(b)), g n (h(e))) 1 < ε. (37) On the other hand, we know that h is continuously dierentiable along V i. Hence

24 SMOOTH CONJUGACY OF ANOSOV SYSTEMS 24 d g i ε > 0 N : n > N (g n (h(a)), g n (h(c))) d i ( Dh( i n (a)) n (a), n (c)) < ε and d g i (g n (h(b)), g n (h(e))) d g i ( n (b), n D i (e)) h( n (a)) < ε. (38) Thereore rom (37) and (38) we have d i ε > 0 N : n > N ( n (a), n (c)) d i ( n (b), n (e)) 1 < ε, which we combine with (36) to get ε > 0 N : n > N We have reached a contradiction with (35) d i ( n (b), n (e)) d i ( n ( b), n (ẽ)) 1 < ε. Remark. By the same argument one can prove that i b = b then d = d. Lemma 6.9. Consider a weak unstable lea W m 1 (a) and b V m(a), b a. For any y W m 1 (a) let y = W m 1 (b) V m(y). Then c 1, c 2 > 0 such that y W m 1 (a) c 1 > d m(y, y ) > c 2. Proo. We will be working on the universal cover R d. We abuse the notation slightly by using the same notation or the lited objects. Note that the leaves on R d are connected components o preimages by the projection map o the leaves on T d. Let h be the conjugacy with the linear model, h = L h. Lemma 6.3 holds or h : h (W m 1 ) = W m 1. L Leaves Wm 1(h L (a)) and Wm 1(h L (b)) are parallel hyperplanes. Thus the lower bound ollows rom the uniorm continuity o h. It ollows rom (34) that h 1 Id is bounded. Hence we can ind positive R that depends only on size o U such that and Then, obviously, W m 1 (a) T ube a W m 1 (b) T ube b de = x B(a,R) W L m 1(x) de = x B(b,R) W L m 1(x). d m(y, y ) sup{d m(x, x ) x T ube a, x T ube b V m(x)}. Assumption (29) guarantees that Em is uniormly transversal to T Wm 1 L = E1 L E2 L... Em 1. L Thus the supremum above is inite. Remark. Given two points a, b R d let ˆd(a, b) = distance(w L m 1(h (a)), W L m 1(h (b))). It is clear rom the proo that constants c 1 and c 2 can be chosen in such a way that they depend only on ˆd(a, b). Remark. In the proo above we do not use the act that both W m 1 and V m are expanding. We only need them to be transversal. Thus, i we substitute weak unstable oliation W m 1 by some weak stable oliation F, the statement still holds.