LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS ∗ ∗∗ ∗∗∗ ANDREY GOGOLEV , BORIS KALININ , VICTORIA SADOVSKAYA 2 1 WITH AN APPENDIX BY RAFAEL DE LA LLAVE 0 2 n Abstract. We consider an irreducible Anosov automorphism L of a torus Td a suchthat no three eigenvalues havethe same modulus. We show that L is locally J rigid, that is, L is C1+Hölder conjugate to any C1-small perturbation f such that 7 the derivative D fn is conjugate to Ln whenever fnp = p. We also prove that 1 p toral automorphisms satisfying these assumptions are generic in SL(d,Z). Ex- ] amples constructed in the Appendix show importance of the assumption on the S eigenvalues. D . h t a m 1. Introduction [ Hyperbolic dynamical systems have been one of the main objects of study in 2 smooth dynamics. Basic examples of such systems are given by Anosov automor- v phisms of tori: for a hyperbolic matrix F in SL(d,Z) the map F : Rd → Rd projects 4 9 toanautomorphism ofthe torusTd = Rd/Zd. Moregenerally, adiffeomorphism f of 9 a compact Riemannian manifold M is called Anosov if there exist a decomposition 2 . of the tangent bundle TM into two f-invariant continuous distributions Es,f and 9 Eu,f, and constants C > 0, λ > 0, such that for all n ∈ N, 0 0 kDfn(v)k ≤ Ce−λnkvk for all v ∈ Es,f, 1 : v kDf−n(v)k ≤ Ce−λnkvk for all v ∈ Eu,f. i X The distributions Es,f and Eu,f are called the stable and unstable distributions of f. r a Structural stability is a fundamental property of hyperbolic systems. If g is an Anosov diffeomorphism and f is sufficiently C1 close to g, then f is also Anosov and is topologically conjugate to g, i.e. there is a homeomorphism h of M such that g = h−1 ◦f ◦h. In this paper we study regularity of the conjugacy h. It is well known that in general h is only Hölder continuous. A necessary condition for it to be C1 is that the derivatives of the return maps of f and g at the corresponding periodic points Date: January 18, 2012. ∗ Supported in part by NSF grant DMS-1001610. ∗∗ Supported in part by NSF grant DMS-0701292. ∗∗∗ Supported in part by NSF grant DMS-0901842. 1 LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 2 are conjugate. Indeed, differentiating gn = h−1◦fn◦h at a periodic point p = fn(p) gives D gn = (D h)−1 ◦D fn ◦D h. p p h(p) p A diffeomorphism g is said to be locally rigid if for any C1-small perturbation f this condition is also sufficient for the conjugacy to be a C1 diffeomorphism. The problem of local rigidity has been extensively studied and Anosov systems with one-dimensional stable and unstable distributions were shown to be locally rigid [dlL87, dlLM88, dlL92, P90]. Local rigidity problem in higher dimensions is much less understood. Examples where the periodic condition is not sufficient were constructed by R. de la Llave [dlL92, dlL02]. However, the one-dimensional results were extended in two direc- tions. In the case when g is conformal on the full stable and unstable distributions, local rigidity was established for some classes ofsystems [dlL02, KS03, dlL04, KS09]. Inadifferent direction, localrigiditywasproved in[G08]foranirreducible Anosov toral automorphism L : Td → Td with real eigenvalues of distinct moduli, as well as for some nonlinear systems with similar structure. We recall that L is said to be irreducible if it has no rational invariant subspaces, or equivalently if its characteristic polynomial is irreducible over Q. It follows that all eigenvalues of L are simple. An important feature of this case is that Rd splits into a direct sum of one-dimensional L-invariantsubspaces. This splittinggivesrisetothecorresponding linear foliations on Td which are expanded or contracted by L at different rates. Such a splitting persist for C1-small perturbations of L and provides a framework for studying regularity of the conjugacy. Examples in [G08] show that irreducibility of L is a necessary assumption for local rigidity except when L is conformal on the stable and unstable distributions. The main result of this paper is the following theorem which establishes local rigidity for a broad class of irreducible toral automorphisms. We give a concise proof that uses techniques from [G08, dlL02, KS09] along with some new results on conformality of cocycles from [KS10]. Theorem 1.1. Let L : Td → Td be an irreducible Anosov automorphism such that no three of its eigenvalues have the same modulus. Let f be a C1-small perturbation of L such that the derivative D fn is conjugate to Ln whenever p = fnp. Then f is p C1+Hölder conjugate to L. We note that irreducibility of L implies that it is diagonalizable over C. Hence assuming that D fn is conjugate to Ln is equivalent to assuming that D fn is p p also diagonalizable over C and has the same eigenvalues as Ln. The only extra assumption in the theorem ensures that the dimensions of the subspaces in the splitting by rates of expansion/contraction are not higher than two. It allows L to have pairs of complex conjugate eigenvalues as well as pairs λ,−λ. We prove this theorem in Section 2. LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 3 In Section 3, we show that toral automorphisms satisfying the assumptions of the theoremaregenericinthefollowingsense. ConsiderthesetofmatricesinSL(d,Z)of norm at most T. Then the proportion of matrices corresponding to automorphisms that do not satisfy our assumptions goes to zero as T → ∞. Moreover, it can be estimated by cT−δ for some δ > 0. Example A.3 in the Appendix yields an Anosov toral automorphism conformal on a three-dimensional invariant subspace and a perturbation with conjugate periodic data whose derivative is not uniformly quasi-conformal on the corresponding three- dimensional invariant distribution. This, in particular, precludes smoothness of the conjugacy. The automorphism is reducible, so the example does not prove that the extra assumptionisindeed necessary forour theorem. However, itclearlyshows that currentmethodscannotbepushedfurthertogivetheresultwithoutthisassumption. 2. Proof of Theorem 1.1 2.1. Notation and outline of the proof. We denote by Es,L and Eu,L the stable and unstable distributions of L. Since f is C1 close to L, f is also Anosov and we denote its stable and unstable distributions by Es,f and Eu,f. They are tangent to the stable and unstable foliations Ws,f and Wu,f respectively (see, e.g. [KH95]). The leaves of these foliations are C∞ smooth, but in general the distributions Es,f and Eu,f are only Hölder continuous transversally to the corresponding foliations. Let 1 < ρ < ρ < ··· < ρ be the distinct moduli of the unstable eigenvalues of 1 2 l L, and let Eu,L = EL ⊕EL ⊕···⊕EL 1 2 l be the corresponding splitting of the unstable distribution. By the assumption, the distributions EL, k = 1,...,l, are either one- or two- k dimensional. As f is C1-close to L, the unstable distribution Eu,f splits into a direct sum of l invariant Hölder continuous distributions close to the corresponding distributions for L: Eu,f = Ef ⊕Ef ⊕···⊕Ef 1 2 l (see, e.g. [Pes04, Section 3.3]). We also consider the distributions Ef = Ef ⊕Ef ⊕...⊕Ef. (i,j) i i+1 j For any 1 < k ≤ l, the distribution Ef is a fast part of the unstable distribution (k,l) and thus it integrates to a Hölder foliation Wf with C∞ smooth leaves (see, (k,l) e.g. [Pes04, Section 3.3]). Moreover, the leaves Wf (x) depend C∞ smoothly on x (k,l) along the unstable leaves Wu,f (see, e.g. [KS07, Proposition 3.9]). Notation. We say that an object is C1+ if it is C1 and its differential is Hölder continuous with some positive exponent. We say that a homeomorphism h is C1+ LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 4 along a foliationF if the restrictions of h to the leaves of F is C1+ and the derivative Dh| is Hölder continuous on the manifold. F For any 1 ≤ k < l the distribution Ef is a slow part of the unstable distribu- (1,k) tion. It also integrates to an f-invariant foliation Wf with C1+ smooth leaves. (1,k) One way to see this is to view L as a partially hyperbolic automorphism with the splitting Es,L ⊕EL ⊕El . It follows from the structural stability of partially (1,k) (k+1,l) hyperbolic systems [HPS77, Theorem 7.1] that for a C1-small perturbation f the “central” foliation survives; that is, Ef integrates to a foliation Wf . For an (1,k) (1,k) alternative simple and short proof that uses specifics of our setup and also gives unique integrability, (as opposed to existence of some foliation tangent to Ef ) (1,k) see [G08, Lemma 6.1]. Thus within the unstable distribution Eu,f there are flags of weak and strong distributions Ef = Ef ⊂ Ef ⊂ ... ⊂ Ef = Eu,f, 1 (1,1) (1,2) (1,l) Ef = Ef ⊂ Ef ⊂ ... ⊂ Ef = Eu,f. l (l,l) (l−1,l) (1,l) Sincebothflagsareuniquely integrableandtheleaves ofthecorresponding foliations areatleast C1+, forany1 ≤ k ≤ l thedistributionEf = Ef ∩Ef alsointegrates k (1,k) (k,l) uniquely to a Hölder foliation Vf = Wf ∩Wf k (1,k) (k,l) with C1+ smooth leaves. Similarly, the distributions Ef = Ef ∩Ef , 1 ≤ i ≤ (i,j) (1,j) (i,l) j ≤ l, integrate to Hölder foliations Wf = Wf ∩Wf . (i,j) (1,j) (i,l) We use similar notation for the automorphism L: EL = EL ⊕ ... ⊕ EL, and (i,j) i j WL and VL are the linear foliations tangent to EL and EL respectively. (i,j) i (i,j) i Since L is Anosov and f is C1 close to L, there exists a bi-Ho¨lder continuous homeomorphism h : Td → Td close to the identity in C0 topology such that h◦L = f ◦h. The conjugacy h takes the flag of weak foliations for L into the corresponding weak flag for f: Lemma 2.1. For any 1 ≤ k ≤ l, h(WL ) = Wf . (1,k) (1,k) The proof is the same as that of Lemma 6.3 in [G08]. We give the argument for the reader’s convenience. Proof. Let h˜, f˜ and L˜ be the lifts of h, f and L to Rd. Similarly we use the tilde sign to denote lifts of various foliations. LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 5 Since h˜(W˜ u,L) = W˜ u,f we have that h˜(W˜ L ) ⊂ W˜ u,f. Let y ∈ W˜ u,L(x), then (1,k) y ∈ W˜ L (x) if and only if dist(Lñ(x),Lñ(y)) ≤ (ρ +ǫ)ndist(x,y) for all n > 0, (1,k) k where dist is the standard metric on Rd. Since ˜h is C0 close to Id we further obtain that y ∈ W˜ L (x) if and only if for all n > 0 (1,k) dist(fñ(h˜(x)),fñ(h˜(y))) = dist(h˜(Lñ(x)),˜h(Lñ(y))) ≤ (ρ +ǫ)ndist(x,y)+c. k The latter condition in turn is equivalent to h˜(y) ∈ W˜ f (h˜(x)). (cid:3) (1,k) We note that Lemma 2.1 holds for any sufficiently C1-small perturbation of an Anosov automorphism of Td. The coarse strategy of the proof of Theorem 1.1 is showing inductively that h is C1+ along WL for any k and thus along WL = Wu(L). By the same argument, (1,k) (1,l) h is C1+ along Ws(L) and hence h is C1+ by Journé Lemma: Lemma 2.2 (Journé [J88]). Let M be a manifold and Fs, Fu be continuous j j j transverse foliations on M with uniformly smooth leaves, j = 1,2. Suppose that j h : M → M is a homeomorphism that maps Fs into Fs and Fu into Fu. More- 1 2 1 2 1 2 over, assume that the restrictions of h to the leaves of these foliations are uniformly Cr+ν, r ∈ N, 0 < ν < 1. Then h is Cr+ν. The main steps of the proof of the Theorem are the following statements: • h(VL) = Vf; i i • h is a C1+ diffeomorphism along VL. i Their proofs are interdependent and organized into an inductive process given by Propositions 2.3 and 2.4. Proposition 2.3. If h(VL) = Vf, then h is a C1+ diffeomorphism along VL. i i i The proof of this proposition is given in Section 2.2 below. Since VL = WL , 1 (1,1) Lemma 2.1 implies that h(VL) = Vf, and then Proposition 2.3 yields that h is C1+ 1 1 along VL. This provides the base of the induction. The inductive step is given by 1 the following proposition. Proposition 2.4. Suppose that h(VL) = Vf, 1 ≤ i ≤ k − 1, and h is a C1+ i i diffeomorphism along WL . Then h(VL) = Vf and h is a C1+ diffeomorphism (1,k−1) k k along WL . (1,k) The proof of this proposition is given in Section 2.3 (and also uses an inductive argument). In the proof we only need to establish that h(VL) = Vf. Then Propo- k k sition 2.3 implies the smoothness of h along VL, and the smoothness along WL k (1,k) follows from the Journé Lemma 2.2. LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 6 2.2. Proof of Proposition 2.3. In this subsection we write VL d=ef VL, Vf d=ef Vf, EL d=ef EL = TVL, Ef d=ef Ef = TVf. i i i i The proof is an adaptation of arguments of de la Llave [dlL02]. First we show that h is Lipschitz along VL as a limit of smooth maps with uniformly bounded derivatives. Then we prove that the measurable derivative of h along VL is actually Hölder continuous. Both steps use Livˇsic Theorem for commutative and noncom- mutative cocycles and rely on conformality of L and f along VL and Vf respectively. Conformality of f along Vf is crucial and to establish it we use a result from [KS10]. First we construct a map h close to h and satisfying the following conditions: 0 (1) h (VL) = Vf, moreover, h (VL(x)) = Vf(h(x)) for all x in Td; 0 0 (2) sup d (h (x),h(x)) < +∞, where d is the distance along the leaves; x∈Td Vf 0 Vf (3) h is C1+ diffeomorphism along the leaves of VL. 0 Let V¯L be the linear integral foliation of Es,L ⊕EL ⊕...⊕EL ⊕EL ⊕...⊕EL. 1 i−1 i+1 l We define the map h by intersecting local leaves: 0 h (x) = Vf,loc(h(x))∩V¯L,loc(x). 0 The map is well-defined and satisfies (2) since h is close to the identity. Condition (1) holds since h(VL(x)) = Vf(h(x)) by the assumption, and (3) is satisfied since for any x the leaf Vf(h(x)) is C1+ and C1 close to VL(x). It follows easily as in [dlL02, Theorem 2.1] that h = lim h , where h = f−n ◦h ◦Ln. n n 0 n→∞ Indeed, let us endow the space of maps satisfying (1) and (2) with the metric d(k ,k ) = sup d (k (x),k (x)). Then, since f−1 contracts the leaves of Vf, it 1 2 x Vf 1 2 follows that the map k 7→ f−1 ◦k ◦L is a contraction with the fixed point h. Now we prove that h is Lipschitz along Vf. For this it suffices to show that the derivatives of the maps h along VL are uniformly bounded. We estimate n kD h k ≤ kDf−n| k·kD h k·kLn| k VL(x) n Ef(h0(Lnx)) VL(Lnx) 0 EL(x) = k Dfn|Ef(f−n(h0(Lnx))) −1k·kDVL(Lnx)h0k·kLn|ELk ≤ k(cid:0)Dfn|Ef(hn(x)) −1k·(cid:1)kLn|ELk·supkDVL(z)h0k. z (cid:0) (cid:1) Since D h is continuous on Td, the supremum on the right is finite. Now we show VL 0 that the product k(Dfn| )−1k·kLn| k is uniformly bounded in y and n. Ef(y) EL We concentrate on the case when Vf is two-dimensional. The one-dimensional case is similar except for conformality of L along VL and of f along Vf is trivial. Since L is irreducible it is diagonalizable over C. Therefore, as the eigenvalues of L| have the same modulus, L| is conformal with respect to some norm on EL. EL EL We can assume that our background norm k·k is chosen so that L| is conformal. EL LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 7 By the assumption of the theorem, D fn is conjugate to Ln whenever fnp = p. It p follows that D fn| is also diagonalizable over C and has eigenvalues of the same p Ef(p) modulus. To obtain conformality of Df| , we apply the following result to vector Ef bundle E = Ef and cocycle F = Df| . Ef [KS10,Theorem1.3]LetE be a Höldercontinuouslinearbundlewith two-dimensional fibers over a compact Riemannian manifold M. Let F : E → E be a Hölder continuous linear cocycle over a transitive Anosov diffeomorphism f : M → M. If for each periodic point p ∈ M, the return map Fn : E → E is diagonalizable over C and p p p its eigenvalues are equal in modulus, then F is conformal with respect to a Hölder continuous Riemannian metric on E. Wedenoteby k·kf thenorminduced bythemetriconEf(x)given bythetheorem. x The conformality of Df| with respect to this norm means that Ef kDf(v)kf = c(x)·kvkf for any x ∈ Td and v ∈ Ef(x). f(x) x Clearly, c(x) = kDf| kf, the norm of Df : (Ef(x),k·kf) → (Ef(f(x)),k·kf ). Ef(x) x f(x) We set a(x) = kL| k = kL| k and b(x) = c(h(x)) = kDf| kf. EL(x) EL Ef(h(x)) The function a(x) is constant in our context, however we will keep the variable for consistency with b(x). Since L is conformal on EL, a(x) satisfies a (x) d=ef a(x)a(Lx)···a(Ln−1x) = kLn| k. n EL The function b(x) is Hölder continuous, and using the relation fm◦h = h◦Lm and the conformality of Df| we obtain Ef b (x) d=ef b(x)b(Lx)···b(Ln−1x) = n = kDf|Ef(h(x))kf ·kDf|Ef(h(Lx))kf ···kDf|Ef(h(Ln−1x))kf = kDfn|Ef(h(x))kf. We claim that the functions a and b are cohomologous, i.e. the exists a continuous function φ : Td → R such that + a(x)/b(x) = φ(Lx)/φ(x). This follows from the Livˇsic Theorem [Liv71], [KH95, Theorem 19.2.1] once we show that a (p) = b (p) for any periodic point p = Lnp. We note that b (p) = n n n kDfn| kf is the modulus of the eigenvalues of Dfn| since this linear Ef(h(p)) Ef(h(p)) map is conformal with respect to norm k·kf . A similar statement holds for a (p) h(p) n and Ln| . The coincidence of the periodic data for f and L implies that indeed EL a (p) = b (p)andhence thefunctions aandbarecohomologous. Using conformality n n we obtain that kLn| k·k(Dfn| )−1kf = kLn| k· kDfn| kf −1 = EL Ef(h(x)) EL Ef(h(x)) = a (x)/b (x) = φ(Lnx)/φ(x) n n (cid:0) (cid:1) LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 8 is uniformly bounded since φ is continuous on Td. Since the norm k·kf is equivalent to k·k we obtain that k(Dfn| )−1k·kLn| k is uniformly bounded in y and n. We Ef(y) EL conclude that kD h k is uniformly bounded in x and n, and hence h is Lipschitz VL(x) n along Vf. A similar argument shows that k(D h )−1k is uniformly bounded and hence VL(x) n h is bi-Lipschitz along Vf. In particular, D h exists and is invertible almost ev- VL erywhere. Differentiating f◦h = h◦L along VL on a set of full Lebesgue measure we obtain Df| ◦D h = D h◦L| , Ef(h(x)) VL(x) VL(Lx) EL(x) i.e., the cocycles Df| and L| are cohomologous with transfer function Ef(h(x)) EL(x) D h. The bundle Ef is trivial since it is close to the trivial bundle EL. Therefore, VL(x) Df| and L| can be viewed as Hölder continuous GL(2,R)-valued cocycles Ef(h(x)) EL(x) over the automorphism L. Moreover, the existence of conformal metrics implies that they are cohomologous to cocycles with values in the conformal subgroup. We remark that in general measurable transfer functions are not necessarily continuous [PW01, Section9]. However, forconformalcocyclesthemeasurable transferfunction coincidesalmost everywhere withaHöldercontinuousone. Thisfollowsfrom[Sch99, Theorem 6.1] or from [PP97, Theorem 1] after reducing cocycles to orthogonal ones by factoring out the norms. See also [S10] for stronger results on GL(2,R)-valued cocycles. We conclude that D h is Hölder continuous, and hence h is a C1+ VL(x) diffeomorphism along VL. (cid:3) 2.3. Proof of Proposition 2.4. The proof is based on the following proposition. Proposition 2.5. Assume that h(WL ) = Wf , h(VL) = Vf and h is a C1+ (i,k) (i,k) i i diffeomorphism along VL. Then h(WL ) = WL . i (i+1,k) (i+1,k) We apply Proposition 2.5 inductively with i = 1,...,k − 1. At every step the assumption of the proposition is fulfilled due to the assumptions in the Proposi- tion 2.4 and the conclusion of Proposition 2.5 at the previous step. We obtain the conclusion of Proposition 2.4 at the final step when WL = WL = VL. (i+1,k) (k,k) k It remains to prove Proposition 2.5. We will use the following simplified notation: (WL,VL,UL) = (WL ,VL,WL ), (i,k) i (i+1,k) (Wf,Vf,Uf) = (Wf ,Vf,Wf ). (i,k) i (i+1,k) We note that VL and UL are slow and fast sub-foliations of WL respectively. Sim- ilarly, Vf and Uf are slow and fast sub-foliations in Wf. We also note that Uf = Wf∩Wf . The foliation Wf is a fast part of the unstable foliation and hence (i+1,l) (i+1,l) is C∞ inside the unstable leaves, see for example [KS07, Proposition 3.9]. There- fore, the foliation Uf is C1+ inside the leaves of Wf and the holonomies between the leaves of Vf along Uf are uniformly C1+. LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 9 Let F = h−1(Uf). Then F is a continuous foliation with continuous leaves that subfoliates WL. We need to show that F = UL. Since VL = h−1(Vf), F is topologically transverse to VL, that is, any leaf of F and and any leaf of VL in the same leaf of WL intersect at exactly one point. First we prove an auxiliary statement that gives some insight into relative structure of F and VL. For any point a ∈ Td and any b ∈ F(a) we denote by H : VL(a) → VL(b) the a,b holonomy along the foliation F, i.e. for every x ∈ VL(a) we define H (x) to be the a,b unique point of intersection F(x)∩VL(b). Lemma 2.6. For any point a ∈ Td and any b ∈ F(a) the holonomy map H is a a,b restriction to VL(a) of a parallel translation inside WL. Proof. For any point c ∈ Td and any d ∈ Uf(c) we denote by H˜ : Vf(c) → Vf(d) c,d the holonomy along the foliation Uf, which is C1+ as we noted above. Since F = h−1(Uf) and h(VL) = Vf we have H = h−1 ◦H˜ ◦h. a,b h(a),h(b) Since h is a C1+ diffeomorphism along VL we conclude that H is C1+. a,b UL(a) Uf(h(a)) b Ha,b(x) h−1 H˜h(a),h(b)(x) VL(b) h(b) Vf(h(b)) F(a) h a x VL(a) h(a) h(x) Vf(h(a)) L−n Figure 1. To show that H is the restriction of a parallel translation, we prove that the a,b differential DH = Id. We apply L−n and denote a = L−n(a), b = L−n(b). Since a,b n n F = h−1(Uf) and f preserves the foliation Uf, L preserves F and we can write H = Ln ◦H ◦L−n. a,b an,bn LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 10 Differentiating and denoting D H = Id+∆ we obtain an an,bn n D H = Ln| ◦D H ◦L−n| = Id+Ln| ◦∆ ◦L−n| . a a,b VL an an,bn VL VL n VL Since L is conformal on VL with respect to some inner product, kLn| k·kL−n| k ≤ C and hence kLn| ◦∆ ◦L−n| k ≤ Ck∆ k for all n. VL VL VL n VL n Itremainstoshowthatk∆ k → 0. Thisfollowseasilybydifferentiatingtheequation n H = h−1 ◦H˜ ◦h. an,bn h(an),h(bn) Indeed, we obtain D H = (D h)−1 ◦D H˜ ◦D h. an an,bn bn h(an) h(an),h(bn) an Since dist(a ,b ) → 0 as n → ∞ we obtain that D H˜ → Id. Also, by n n h(an) h(an),h(bn) choosing a subsequence if necessary, we can assume that lima = limb = c ∈ Td. n n ThusD h → D h,D h → D handhence∆ = D H −Id → 0asn → ∞. (cid:3) an c bn c n an an,bn Now let a be a fixed point of L and let B be the unit ball in UL(a) centered at a. If B ⊂ F(a), then UL(a) = F(a) by invariance under L. Since L is irreducible and UL is invariant, the leaf UL(a) is dense in Td. It follows that the set of points x such that UL(x) = F(x) is dense in Td and hence UL = F. Therefore, it suffices to show that B ⊂ F(a). We argue by contradiction. Assume that there is z ∈ B such that z ∈/ F(a). 1 1 Let x = VL(z )∩F(a). 1 1 Since VL has dense leaves we can choose a sequence {b ,n ≥ 1} ⊂ VL(a) so that n b → x as n → ∞. Let n 1 y = H (b ), n a,x1 n where H is the holonomy map along F from VL(a) to VL(x ). Continuity of F a,x1 1 implies that the sequence y converges to a point x ∈ F(a). Moreover, Lemma 2.6 n 2 implies that {x ,x } is a parallel translation of {a,x }. 1 2 1 We continue this procedure inductively to construct the sequence {x ,n ≥ 1} ⊂ n F(a). Let z = VL(x )∩UL(a). n n Then according to the construction (2.1) d (z ,a) = n·d (z ,a) and d (x ,z ) = n·d (x ,z ). UL n UL 1 VL n n VL 1 1 For every n we take N(n) to be the smallest integer such that L−N(n)(z ) ∈ B. n Since L−1 contracts UL exponentially faster than VL, equation (2.1) implies that d (L−N(n)(x ),L−N(n)(z )) → ∞ as n → ∞. VL n n This contradicts an obvious bound due to compactness of B: maxd (z,VL(z)∩F(a)) < ∞. VL z∈B

Local rigidity for Anosov automorphisms

Similar Local rigidity for Anosov automorphisms