WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS
ARTUR AVILA AND GIOVANNI FORNI
Abstract. We show that a typical interval exchange transformation is either weak mixing or an irrational rotation. We also conclude that a typical translation flow on a surface of genus g ≥ 2 (with prescribed singularity types) is weak mixing.
1. Introduction Let d ≥ 2andletπ be an irreducible permutation of {1, ..., d},thatis,π{1, ..., k} = {1, ..., k}, d 1 ≤ k Date: May 14, 2004. 1 2 ARTUR AVILA AND GIOVANNI FORNI 1.1. Translation flows. Let M be a compact orientable surface of genus g ≥ 1, endowed with a flat metric, with finitely many singularities of conical type. Then the angles at the singularities are 2πκ1 ≤ ... ≤ 2πκr ,whereκ1, ..., κr are positive real numbers satisfying (κi − 1) = 2g − 2. A translation flow F on such surface is the flow given by a parallel vector field of unit length. The existence of a translation flow is equivalent to the condition κi ∈{1, 2, ...},1≤ i ≤ r. We will assume this condition from now on. Another, equivalent, point of view is to consider pairs (M,q) of a compact Riemann surface M and a (non-zero) abelian differential q.AflatmetriconM is given by |q| and a unit parallel vector field is specified by the condition q · v = 1, and one can go the other way around. Thus, specification of parameters g and κ =(κ1, ..., κr) determine a finite dimensional moduli space Q(g,κ)oftranslation flows (considered up to isometry), endowed with a natural complex structure and a Lebesgue measure class. We are interested in “typical” translation flows (with respect to Lebesgue in this sense). In genus 1 there are no singularities and translation flows are irrational flows on T2: typically, they are uniquely ergodic. In genus g ≥ 2, this is still the case as was shown by Masur [M] and Veech [V1]. As for interval exchange transformations, the question of weak mixing of translation flows is more delicate than unique ergodicity. In genus 1, translation flows are never weak mixing, but it has been expected that weak mixing holds typically in genus g ≥ 2. We will show that this is indeed the case: Theorem B. Almost every translation flow on a flat surface of genus g ≥ 2 and prescribed singu- larity type κ is weak mixing. Translation flows and i.e.t.’s are intimately related: the former can be viewed as suspension flows (of a particular type) over the latter. We point out that differently from the case of weak mixing for i.e.t., where weak mixing had been proved for infinitely many cases, there had been very little progress previously on weak mixing of translation flows, except for topological weak mixing which has been established in [L]. 1.2. Parameter exclusion. To prove our results, we will need to proceed a parameter exclusion to get rid of undesirable dynamics. With this in mind, instead or working in the direction of understanding the dynamics in the phase space (regularity of eigenfunctions1, etc.), we concentrate on the analysis of the parameter space. We analyze the parameter space of suspension flows over i.e.t.’s via a renormalization operator (the case of i.e.t.’s corresponds to constant roof function). This renormalization operator acts non- linearly on i.e.t.’s and linearly on roof functions, so it has the structure of a cocycle (the Zorich cocycle) over the renormalization operator over i.e.t.’s. One can work out a criteria for weak mixing (due to Veech) in terms of the dynamics of the renormalization operator. An important ingredient in our analysis is the result of [F] on the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle over the Teichm¨uller flow. This result is equivalent to hyperbolicity of the Zorich cocycle. (Actually we only need a weaker result, that the Kontsevich-Zorich cocycle has two positive Lyapunov exponents in the case of surfaces of genus at least 2.) In a first version of this work, this had been used to obtain the result on translation flows on sur- faces. The method was to do a “linear” parameter exclusion to show that “bad” roof functions form a small set (basically, each positive Lyapunov exponent for the Zorich cocycle gives one obstruction for the eigenvalue equation, which has only one free parameter). The situation for i.e.t.’s is much more complicate, because we do not have freedom to change the roof function. We need to do a “non-linear” exclusion process, based on a statistical argument. 1 In this respect, we should remark that Yoccoz has pointed out to us the existence of “strange” eigenfunctions for certain values of the parameter. WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS 3 While for linear exclusion it is enough to use ergodicity of the renormalization operator over i.e.t.’s, the statistical argument for the non-linear exclusion uses heavily its mixing properties. 1.3. Outline. We start this paper with the basic background on cocycles. We then prove our key technical result, an abstract parameter exclusion scheme for “sufficiently random integral cocycles”. We then discuss known results on the renormalization dynamics and show how the problem of weak mixing reduces to the abstract parameter exclusion theorem. The same argument also covers the case of translation flows. In the appendix we present our original linear exclusion argument, which is much simpler than the non-linear exclusion argument but is enough to deal with translation flows. Acknowledgements: A.A. would like to thank Jean-Christophe Yoccoz for several very pro- ductive discussions and Jean-Paul Thouvenot for proposing the problem and for his continuous encouragement. 2. Background 2.1. Strongly expanding maps. Let (∆,µ) be a probability space. We say that a measurable transformation T :∆→ ∆ which preserves µ is weakly expanding if there exists a partition (modulo 0) of ∆ by sets of positive µ-measure ∆(l), l ∈ Z such that T (l) = T :∆(l) → ∆ is invertible and (l) T∗ µ is equivalent to µ. l Let Ω be the set of finite words with integer entries. If l ∈ Ω, l =(l1, ..., ln), let ∆ = {x ∈ ∆,Tk−1(x) ∈ ∆(li), 1 ≤ k ≤ n}.LetT l = T n|∆l.Thenµ(∆l) > 0. Let 1 (2.1) µl = T lµ. µ(∆l) ∗ Let M = {µl,l∈ Ω}.WesaythatT is strongly expanding if there exists a constant K>0such that dν (2.2) K−1 ≤ ≤ K, ν ∈M. dµ This has the following consequence. If Y ⊂ ∆ is such that µ(Y ) > 0then l T∗ν(Y ) (2.3) K−2µ(Y ) ≤ ≤ K2µ(Y ),ν∈M,l∈ Ω. µ(∆l) d−1 d−1 d 2.2. Projective transformations. We let P+ ⊂ P be the projectivization of R+. We will call it the standard simplex.Aprojective contraction is a projective transformation taking the standard simplex into itself. Thus a projective contraction is the projectivization of some matrix B ∈ GL(p, R) with non-negative entries. The image of the standard simplex by a projective contraction is called a simplex. We need the following easy fact. d−1 (l) Lemma 2.1. Let ∆ be a simplex compactly contained in P+ and ∆ ⊂ ∆, l ∈ Z be a partition of ∆ (modulo sets of Lebesgue measure 0) in sets of positive Lebesgue measure. Let T :∆→ ∆ be a map such that T (l) :∆(l) → ∆ is invertible and its inverse is the restriction of a projective contraction. Then T preserves a probability measure µ which is absolutely continuous with respect to Lebesgue measure and has a density which is continuous and positive in ∆. Moreover, T is strongly expanding with respect to µ. Proof. Let d([x], [y]) be the projective distance between [x]and[y] xiyj (2.4) d([x], [y]) = sup ln . 1≤i,j≤p xjyi 4 ARTUR AVILA AND GIOVANNI FORNI Let N be the class of absolutely continuous probability measures on ∆ whose densities have loga- rithms which are 1-Lipschitz with respect to the projective distance. Since ∆ has finite projective diameter, it suffices to show that there exists µ ∈N invariant under T and such that µl ∈N for all N .NoticethatN is compact in the weak* topology and convex. l −1 l l Notice that (T ) be the projectivization of some matrix B =(bij ) in GL(2, R)withnon- negative entries. Then one sees that x (2.5) | det D(T l)−1(x)| = , Bl · x so that 2 1/2 l −1 | det D(T l) (y)| Bl · x y i j bij xj y x y ≤ i ≤ d([x],[y]) (2.6) l −1 = l = 2 sup e . | det D(T ) (x)| B · y x l x 1≤i≤p yix i j bij yj Thus l 1 l (2.7) Leb = T∗Leb ∈N, Leb(∆l) and −1 1 n 1 (2.8) ν = T kLeb = Leb(∆l)Lebl ∈N. n n ∗ n k=0 l∈Ω,|l|≤n l Take µ as any limit of νn.Thenµ is invariant under T , belongs to N and any µ is a limit of −1 0 l l0l l0l l l ∈N (2.9) νn = Leb(∆ ) Leb(∆ )Leb , l0∈Ω,|l0|≤n l0∈Ω,|l0|≤n which implies that µl ∈N. 2.3. Cocycles. A cocycle is a pair (T,A)whereT :∆→ ∆andA :∆→ GL(p, R), viewed as a p n n linear skew-product (x, w) → (T (x),A(x) · w)on∆× R .Noticethat(T,A) =(T ,An), where n−1 (2.10) An(x)=A(T (x)) ···A(x),n≥ 0. If (∆,µ) is a probability space, µ is an invariant ergodic measure for T (in particular T is measurable) and (2.11) ln A(x)dµ(x) < ∞, ∆ we say that (T,A)isameasurable cocycle. Define s p (2.12) E (x)={w ∈ R , lim An(x) · w =0}, cs p 1/n (2.13) E (x)={w ∈ R , lim sup An(x) · w ≤ 1. Then Es(x) ⊂ Ecs(x) are subspaces of Rp (called the stable and central stable spaces respectively), and we have A(x) · Ecs(x)=Ecs(T (x)), A(x) · Es(x)=Es(T (x)). If (T,A) is a measurable cocycle, dim Es and dim Ecs are constant almost everywhere. WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS 5 1/n If (T,A) is a measurable cocycle, Oseledets Theorem implies that lim An(x) · w exists for p almost every x ∈ ∆andforeveryw ∈ R , and thus there are p Lyapunov exponents θ1 ≥···≥θp, characterized by (2.14) p 1/n θ p 1/n θ #{i, θi = θ} =dim{w ∈ R , lim An(x) · w ≤ e }−dim{w ∈ R , lim An(x) · w (2.17) ln A(x)0dµ(x) < ∞ ∆ we will call (T,A)auniform cocycle. Lemma 2.2. Let (T,A) be uniform and let 1 (2.18) ω(κ)= sup sup ln AN (x)0dµ(x). µ(U)≤κ N>0 U N Then (2.19) lim ω(κ)=0. κ→0 Proof. Let 1 (2.20) ωN (κ)= sup ln AN (x)0dµ(x). µ(U)≤κ U N Since ln AN 0 is integrable for every N>0, we have (2.21) lim ωN (κ)=0. κ→0 1 N−1 k 1 ≤ Let s = ∆ ln A(x) 0dµ(x). Let SN (x)= N k=0 ln A(T (x)) 0,sothatN ln AN (x) 0 SN (x). We have 1 (2.22) ωN (κ)= sup ln AN (x)0dµ(x) ≤ sup SN (x)dµ(x) ( )≤ N ( )≤ µ U κ U µ U κ U ≤ sup s + |Sn(x) − s|dµ(x) ≤ κs + SN − sL1(µ). µ(U)≤κ U By Birkhoff’s Theorem, SN − sL1(µ) → 0, and by (2.20) the result follows. 2.3.1. Locally constant cocycles. We say that (T,A)isalocally constant cocycle if T :∆→ ∆is (l) (l) strongly expanding and A|∆ is a constant A . In this case, we denote for l ∈ Ω, l =(l1, ..., ln), (2.23) Al = A(ln) ···A(l1). 2 s It is also possible to show that dim E (x)=#{i, θi < 0}. 6 ARTUR AVILA AND GIOVANNI FORNI 2.3.2. Weak stable manifold, integral cocycles. We define the weak stable manifold by s p (2.24) W (x)={w ∈ R , An(x) · wRp/Zp → 0}. Obviously W s(x) is a union of translates of Es(x). We say that a cocycle (T,A)isintegral if A(x) ∈ GL(p, Z), x ∈ ∆. An integral cocycle can be viewed as a skew product on ∆ × Rp/Zp.Inthiscase,W s(x) has a natural interpretation as the stable manifold of the zero section in ∆ × Rp/Zp.IfA is actually bounded, then it is easy to see s s s that W (x)=∪c∈Zp E (x)+c. In general W (x) may be the union of uncountably many translates of Es(x). 3. Exclusion of the weak-stable manifold Let Θ ⊂ Pp−1 be a compact set. We say that Θ is adapted to (T,A)ifA(l) · Θ ⊂ Θ for all l and if for almost every x ∈ ∆wehave (3.1) A(x) · w≥w, (3.2) An(x) · w→∞ whenever w ∈ Rp \{0} projectivizes to an element of Θ. Let J = J (Θ) be the set of lines in Rp, parallel to some element of Θ and not passing through 0. The main result in this section is the following. Theorem 3.1. Let (T,A) be a locally constant integral uniform cocycle, and let Θ be adapted to (T,A). Assume that for every line J ∈J = J (Θ), J ∩ Ecs(x)=∅ for almost every x ∈ ∆.Thenif L is a line contained in Rp parallel to some element of Θ, L ∩ W s(x) ⊂ Zp for almost every x ∈ ∆. The proof will take the rest of this section. For J ⊂J,weletJ be the distance between J and 0. Lemma 3.2. There exists 0 > 0, such that An(x) · J (3.3) lim sup µ x, ln <0n → 0. →∞ n J∈J J Proof. Let C(x, J) be largest such that λn/2 (3.4) An(x) · J≥C(x, J)e J,n≥ 0. By Oseledets Theorem, C(x, J) ∈ [0, 1] is positive for every J and almost every x, and depends continuously on J for almost every x. Thus for every J ∈J, δ>0, there exists Cδ(J) > 0 such that µ{C(x, J) s ∈ · p p ≤ For δ<1/10, let Wδ,n(x)bethesetofallw Bδ(0) such that Ak(x) w R /Z <δfor all k n, s ∩ s and let Wδ (x)= Wδ,n(x). ∈J ∈ ∩ s ∅ Lemma 3.3. There exists δ>0 such that for all J and for almost every x ∆, J Wδ (x)= . l p Proof. Let φδ(l,J) be the number of connected components of A (J ∩ Bδ(0)) ∩ Bδ (Z \{0}), and let → φδ(l)=supJ∈J φδ(l,J). Notice that δ φδ(l) is non-decreasing and for l fixed, for δ sufficiently small φδ(l) = 0. Notice that l (3.5) φδ(l) ≤A 0. ∈ l · − Given J with J <δand l Ω, let Jl,1, ..., Jl,φδ(l,J) be all the lines of the form A J c where l p l A (J ∩ Bδ(0)) ∩ Bδ(c) = ∅ with c ∈ Z \{0}.LetJl,0 = A · J. WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS 7 Notice that (3.6) Jl,k <δ, k≥ 1. l Let w ∈ Jl,k satisfy w = J.Thenw−w <δfor some w ∈ A ·(J∩Bδ(0))−c.SinceJ is parallel to some element of Θ, it is expanded by Al (see (3.1)), so (Al)−1 · (w + c) − (Al)−1 · (w + c) <δ, and we get (Al)−1 · (w + c) < 2δ.Since(Al)−1 · c ∈ Zp \{0}, l l −1 l −1 (3.7) A 0w≥(A ) · (w + c) − (A ) · c≥1 − 2δ, and we get l −1 (3.8) Jl,k≥(1 − 2δ)A 0 ,k≥ 1. On the other hand, it is clear that l l −1 (3.9) A 0J≥Jl,0≥A 0 J. Given measurable sets X, Y ⊂ ∆ such that µ(Y ) > 0, we let ν(X ∩ Y ) (3.10) P (X|Y )= ,ν∈M, ν ν(Y ) (3.11) P (X|Y )= supPν (X|Y ). ν∈M For N>0, let ΩN be the set of all words of length N,andΩN be the set of all words of length multiple of N. N l Fix 0 <η<1/10, select a finite set Z ⊂ Ω such that µ(∪l∈Z ∆ ) > 1 − η.Ifη<η0(N), we have l l 1 (3.12) ln A 0µ(∆ ) < . 2 l∈ΩN \Z Claim 3.4. If N>N0 and η<η0(N), for every Y ⊂ ∆ with µ(Y ) > 0, we have 1 Jl ,0 l1 l −N (3.13) inf ln Pν (∆ | ∆ ∩ T (Y )) ≥ 2. ν∈M J l1∈Z l∈Z Proof. By the previous lemma, for every κ>0, if N>N0(κ) then for every J ∈J there exists Z = Z(J) ⊂ Z such that Al · J (3.14) ln ≥ 0N, J (3.15) µ ∆l <κ. l∈Z\Z Let J 1 0 1 (3.16) (I)= ln l , P (∆l | ∆l ∩ T −N (Y )) J ν 1∈ l∈Z l Z l l −N 4 l l ≥ 0NPν ( ∆ | ∆ ∩ T (Y )) ≥ 0N(1 − K Pµ( ∆ | ∆ )) ∈ ∈ ∈ \ ∈ l Z l Z l Z Z l Z 4 κ ≥ 0N 1 − K , 1 − η 8 ARTUR AVILA AND GIOVANNI FORNI J 1 0 1 (3.17) (II)= ln l , P (∆l | ∆l ∩ T −N (Y )) J ν 1∈ \ l∈Z l Z Z l1 l1 l −N ≥− ln A 0Pν (∆ | ∆ ∩ T (Y )) l1∈Z\Z l∈Z ≥− l1 4 l1 | l ≥− 4 1 l ln A 0K Pµ(∆ ∆ ) K − ln A (x) dµ 1 η ∪ ∆l l1∈Z\Z l∈Z l∈Z\Z 1 ≥−K4 ω(κ)N 1 − η (where ω(κ) is as in Lemma 2.2), so that for κ small, N>N0(κ)andη sufficiently small we have J 1 0 1 0 (3.18) ln l , P (∆l | ∆l ∩ T −N (Y )) ≥ (I)+(II) ≥ N. J ν 2 l1∈Z l∈Z Claim 3.5. If N>N0 and η<η0(N), for every 0 <ρ<ρ0(Z), for every Y ⊂ ∆ with µ(Y ) > 0, we have −ρ l1 l −N −ρ 1 | ∩ ≤ − (3.19) sup Jl ,0 0 Pν (∆ ∆ T (Y )) (1 ρ) J 0 . ν∈M l1∈Z l∈Z Proof. Let −ρ J 1 0 1 (3.20) Φ(ν, Y, ρ)= l , P (∆l | ∆l ∩ T −N (Y )) J−ρ ν l1∈Z l∈Z Then Φ(ν, Y, 0) = 1 and −ρ d J 1 0 J 1 0 1 (3.21) Φ(ν, Y, ρ)= − ln l , l , P (∆l | ∆l ∩ T −N (Y )), dρ J J−ρ ν l1∈Z l∈Z so by the previous claim there exists ρ0 = ρ0(Z) such that d (3.22) Φ(ν, Y, ρ) < −1, 0 ≤ ρ ≤ ρ0, dρ which gives the result. At this moment we fix N, η, Z,andρ so that (3.12) and (3.19) hold and let δ<δ0(Z, ρ)sosmall that we have l l ρ l l (3.23) (ρ ln A 0 +ln(1+A 0(2δ) ))µ(∆ ) − ρµ( ∆ )=α<0, l∈ΩN \Z l∈Z (this is possible by (3.12) and (3.19)) and (3.24) φδ(l)=0,l∈ Z (this is possible since Z is finite). m { ∈ ∩ s ∅} m → ∈J Let Γδ (J)= x ∆,J Wδ,mN (x) = . We must show that Γδ (J) 0 for every J . Let ψ :ΩN → Z be such that ψ(l)=0ifl ∈ Z and ψ(l) = ψ(l ) whenever l = l and l ∈/ Z.We (1) ( ) (1) ( ) ( ) let Ψ : ΩN → ΩbegivenbyΨ(l ...l m )=ψ(l )...ψ(l m ), where the l i are in ΩN .Welet d l ∆ = ∪l∈Ψ−1(d)∆ . WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS 9 For d ∈ Ω, let C(d) ≥ 0 be minimal so that (3.25) P (Γm(J)|∆ d) ≤ C(d)J−ρ,J∈J. ≤ m ∅ It follows that C(d) 1 for all d (since Γδ (J)= , J >δ). Claim 3.6. If d =(d1, ..., dm), we have li ρ li ρ (3.26) C(d) ≤ (1 − ρ) A 0(1 + A 0(2δ) ). i di=0 di=0,ψ(l )=di Proof. Let d =(d1, ..., dm+1), d =(d2, ..., dm+1). There are two possibilities: (1) If d1 = 0, we have by (3.19) and (3.24) m+1 d m d l1 d −ρ | ≤ 1 | | ≤ − (3.27) P (Γ (J) ∆ ) sup P (Γ (Jl ,0) ∆ )Pν (∆ ∆ ) (1 ρ)C(d ) J , ν∈M l1∈Z 1 1 m+1 | d (2) If d1 =0,letl be given by ψ(l )=d1. Then either J >δ(and P (Γδ (J) ∆ ) = 0) or by (3.5), (3.8) and (3.9) (3.28) 1 φ δ (l ) m+1 d m d −ρ 1 −ρ | ≤ 1 | ≤ 1 1 P (Γ (J) ∆ ) P (Γ (Jl ,k) ∆ ) C(d )( Jl ,0 + φδ(l )sup Jl ,k ) k≥1 k=0 ρ l1 1+ρ 1 2 A 1 1 1+ ≤ C(d )J−ρ Al ρ + 0 ≤ C(d )J−ρ(Al ρ +(2δ)ρAl ρ). 0 J−ρ 0 0 The result follows. Let l −ρ, x ∈∪l∈Z ∆ , (3.29) γ(x)= l l ρ l ρ ln A 0 +ln(1+A 0(2δ) ),x∈∪l∈ΩN \Z ∆ . We have chosen δ so that (see (3.23) (3.30) γ(x)dµ(x)=α<0. ∆ d Let Cm(x)=C(d)wherex ∈ ∆ , |d| = m. Then by (3.26) m −1 kN (3.31) ln Cm(x) ≤ γ(T (x)). k=0 so that, by Birkhoff’s Theorem, Cm(x) → 0 for almost every x. By dominated convergence (since Cm(x) ≤ 1), we have (3.32) lim Cm(x)dµ(x)=0. m→∞ ∆ Notice that m d m d −ρ (3.33) µ(Γ (J)) ≤ µ(∆ )Pµ(Γ (J)|∆ ) ≤ Cm(x)J dµ(x), ∆ d∈Ωb,|d|=m so lim µ(Γm(J)) = 0. 10 ARTUR AVILA AND GIOVANNI FORNI Proof of Theorem 3.1. Assume that there exists a positive measure set X such that for every x ∈ X, s p there exists w(x) ∈ (L ∩ W (x)) \ Z .Thus,foreveryδ>0 for every x ∈ X,thereexistsn0(x) > 0 ≥ ∈ Zp · − ∈ s n such that for every n n0(x), there exists cn(x) such that An(x) w(x) cn(x) Wδ (T (x)). If An(x) · L − cn(x) passes through 0 for all n ≥ n0, we get a contradiction as follows. Since n0 −1 · − · − n0 −1 · An−n0 (T (x)) (An(x) w(x) cn(x)) = An0 (x) w(x) An−n0 (T (x)) cn(x)andAn(x) n0 −1 · − → expands L (see (3.2)) we get An−n0 (T (x)) (An(x) w(x) cn(x)) 0. But this implies that · An0 (x) w(x)=cn0 (x), contradiction. Thus for every x ∈ X there exists n(x) ≥ n0(x) such that An(x) · L − cn(x) does not pass through 0, that is, An(x) · L − cn(x) ∈J. By restricting to a subset of X of positive measure, we may assume that n(x), An(x)(x)andcn(x)(x) do not depend on x.ThenAn(x)(x) · L − cn(x)(x) ∈J intersects s n(x) n(x) Wδ (x)forasetT (X) such that µ(T (X)) > 0. This contradicts the previous lemma. 4. Renormalization schemes Fix some number d ≥ 2. Let S be the space of irreducible permutations on {1, ..., d},thatis π{1, ..., k} = {1, ..., k} for 1 ≤ k ≤ d. An i.e.t. f = f(λ, π)ond intervals is specified by a pair d (λ, π) ∈ R+ × S as described in the introduction. 4.1. Rauzy induction. Let (λ, π) be such that λd = λπ−1(d). Then the first return map (under f(λ, π)) to the interval d (4.1) 0, λi − min{λπ−1(d),λd} i=1 can be seen again as an i.e.t. f(λ,π)ond intervals as follows: (1) If λd <λπ−1(d),let λ , 1 ≤ i<π−1(d), i −1 −1 − λπ (d) λd,i= π (d), (4.2) λi = λ ,i= π−1(d)+1, d −1 λi−1,π(d)+1 (2) If λd >λπ−1(d),let ≤ λi, 1 i d d We define a map QR : R+ ×S → R+ ×S by taking QR(λ, π)=(λ ,π ). Notice that QR is defined almost everywhere (in the complement of finitely many hyperplanes). The Rauzy class of π ∈ S is the set R(π)ofall˜π that can be obtained from π by a finite number of elementary operations. It is a basic fact that the Rauzy classes partition S. d−1 d−1 d Let P+ ⊂ P be the projectivization of R+.SinceQR comutes with dilatations (4.6) QR(αλ, π)=(αλ ,π ), d−1 d−1 QR projectivizes to a map RR : P+ × S → P+ × S. 4.2. Zorich induction. Zorich modified Rauzy induction as follows. Given (λ, π), let n = n(λ, π) n+1 ≤ ≤ i i−1 be such that QR (λ, π) is defined and, for 1 i n, QR(λ, π) is obtained from QR (λ, π)by n+1 n elementary operations of the same type, while QR (λ, π) is obtained from QR(λ, π)byanelementary operation of the other type. Then he sets n(λ,π) (4.7) QZ (λ, π)=QR (λ, π). d d Then QZ : R+ × S → R+ × S is defined almost everywhere (in the complement of countably many hyperplanes). We can again consider the projectivization of QZ , denoted by RZ . d−1 Theorem 4.1 (Zorich [Z]). Let π ∈ S.ThenRZ |P+ × R(π) admits a unique ergodic absolutely continuous probability measure µR(π). Its density is positive and analytic. 4.3. Cocycles. Let (λ,π) be obtained from (λ, π) by Rauzy or Zorich induction. Let f = f(λ, π). ∈ Let rj(x), x I(λ ,π ) be the number of intersections of x, f(x), ... before the first return of x to I = I(λ ,π ). In particular, r(x)= j rj (x) is the first return time of x to I .Noticethatrj(x)is Rd a constant rij on each Ii = Ii(λ ,π ). We let B be the linear operator of given in matricial form R Z by rij . We distinguish Rauzy and Zorich induction by denoting B or B . It is easy to see that BR,BZ ∈ GL(2, Z), and Z R n(λ,π)−1 ··· R (4.8) B (λ, π)=B (QR (λ, π)) B (λ, π). Notice that Q(λ, π)=(λ,π) implies λ = B∗λ (B∗ denotes the adjoint of B). Thus (4.9) λ, w =0 ifandonlyif λ,B· w =0. Obviously we can projectivize B. Theorem 4.2 (Zorich [Z]). Let π ∈ S. We have Z (4.10) ln B 0dµR(π) < ∞ 4.4. An invariant subspace. Let π ∈ S. Define a permutation σ on {0, ..., d} by −1 − π (1) 1,i=0, (4.11) σ(i)= d, i = π−1(d), π−1(π(i)+1),i=0 ,π−1(d). Let S(j), 0 ≤ j ≤ d be the orbit of j under σ. This defines a partition of {0, ..., d}.LetΣ=Σ(π)= {S(j), 0 ≤ j ≤ d}. Define a vector bS ∈ Rd, S ∈ Σby S − − ≤ ≤ (4.12) bi = χS(i 1) χS(i), 1 i d, S where χS denotes the characteristic function of S.LetN = N(π) be the space generated by the b , S ∈ ΣandletH = H(π) be the orthogonal complement to N.LetΥ=Υ(π)={bS,S∈ Σ}. 12 ARTUR AVILA AND GIOVANNI FORNI An important fact from [V3] is that if Q(λ,π)=(λ, π)then (4.13) B(λ, π)∗ · Υ(π)=Υ(π), which implies (4.14) B(λ, π)∗ · N(π)=N(π), (4.15) B(λ, π) · H(π)=H(π), and in particular the dimensions of N(π), H(π) only depend on the Rauzy class of π. Veech showed that the dimension of H(π) is a non-zero even number 2g, g = g(π). It is also easy to see that H ∩ Zd is a co-compact lattice in H, and in particular (4.16) dist(H, Zd \ H) > 0. Z d−1 4.5. Lyapunov exponents. Let π ∈ S. We can consider B |H as a integral cocycle over RZ |P+ × 3 R(π). This cocycle is uniform (with respect to the measure µR(π)) by Theorems 4.1 and 4.2. We will call this cocycle the Zorich cocycle. Let θ1 ≥ ... ≥ θ2g be the Lyapunov exponents of the Zorich cocycle. In [Z], Zorich showed that θi = −θ2g−i and that θ1 >θ2 (this result corresponds to non-uniform hyperbolicity of the Teichm¨uller flow [V2]). He conjectured that θ1 > ... > θ2g. Theorem 4.3 (Forni, [F]). The Zorich cocycle is hyperbolic. Thus (4.17) θ1 >θ2 ≥ ... ≥ θg > 0 >θg+1 ≥ ... ≥ θ2g−1 >θ2g. Actually [F] proved hyperbolicity of a related cocycle GKZ (the Kontsevich-Zorich cocycle), which is a continuous time version of the Zorich cocycle, whose 2g Lyapunov exponents are given by θ˜i = θi/θ1. Thus the hyperbolicity of the Kontsevich-Zorich cocycle is equivalent to the hyperbolicity of the Zorich cocycle. 5. Exclusion of the central stable manifold for the Zorich cocycle Theorem 5.1. Let π ∈ S with g>1 and let L ⊂ H be a line not passing through 0.LetEcs denote d−1 the (g-dimensional) central stable space of the Zorich cocycle. Then for almost every [λ] ∈ P+ , L ∩ Ecs([λ],π)=∅. Our original proof of this theorem was based on the behavior of the Kontsevich-Zorich cocycle over the Teichm¨uller flow, and the key mechanism was that the stable space of two measured foliations in general position are transverse. Yoccoz pointed out to us that it was possible to argue directly with the Zorich cocycle, and later we found out that such an argument was already present in [NR]. The claim of [NR] is slightly different than what we need, but the modification is straightforward, so we will give only a short sketch of proof. Proof. Following Nogueira-Rudolph [NR], define π ∈ S to be standard if π(1) = n and π(n)=1. They proved that every Rauzy class contains at least one standard permutation [NR], Lemma 3.2. Clearly it suffices to consider the case where π is standard. Notice that cs R cs (5.1) E (RR([λ],π)) = B · E ([λ],π) for almost every ([λ],π). It is easy to see (using Perron-Frobenius together with (4.9)) that Ecs([λ],π) is orthogonal to λ. 3 Strictly speaking, to get in the setting of §2.3wemustfirstfix,foreach˜π ∈ R(π), an isomorphism H(˜π) → R2g that takes H ∩ Zd to Z2g . WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS 13 Define vectors v(i) ∈ Rd by 1,π(j) <π(i),j>i, ( ) (5.2) v i = −1,π(j) >π(i),j cs (5.6) (B(i) · L) ∩ E ([λ],π) = ∅, 1 ≤ i ≤ d. d cs Write L = {r+ts, t ∈ R} with r, s ∈ H, s ∈ R+ linearly independent. Then from L∩E ([λ],π) = ∅, we get λ, r (5.7) r − s ∈ Ecs([λ],π), λ, s and analogously we get (5.8) (i) − (d) − (d) (i) − (d) − (d) − λ, r + ri(v v ) rdv (i) − (d) − (d) ∈ cs (r + ri(v v ) rdv ) (i) (d) (d) (s + si(v v ) sdv ) E ([λ],π) λ, s + si(v − v ) − sdv for 1 ≤ i 6. Weak mixing for interval exchange tranformations Weak mixing for the interval exchange transformation f means no non-constant measurable so- lutions φ : I → C to (6.1) φ(f(x)) = e2πitφ(f(x)), for some t ∈ R. This can be rephrased in two condition: (1) f is ergodic, (2) For every t ∈ R \ Z, there are no non-zero measurable solutions φ : I → C to (6.2) φ(f(x)) = e2πitφ(f(x)). By [M], [V1], the first condition is not really an obstruction: f(λ, π) is ergodic for almost every λ. Our criteria to deal with the second condition is the following. d−1 Theorem 6.1. Let π ∈ S. There exists an open set U ⊂ P+ with the following property. Assume d−1 that ([λ],π) ∈ P ×{π} visits U infinitely many times under iteration by RR. If there is a non-zero measurable solution φ : I → C to (6.3) φ(f(x)) = e2πithi φ(f(x)), with t ∈ R, h ∈ Rd,then R (6.4) lim B (λ, π) · thRd Zd =0. n n / n→∞,RR([λ])∈U Notice that (6.3) reduces to (6.2) when h =(1, ..., 1) and can thus be used to rule out eigenvalues for i.e.t.’s. The more general form (6.3) will be used in the case of translation flows. We thank Jean-Christophe Yoccoz for pointing out to us that this result is actually a known result of Veech (our original proof does not differ from Veech’s), see [V3] §7. We will call it Veech’s criteria for weak mixing. It has the following consequences: Theorem 6.2 (Katok-Stepin, [KS]). If g =1then either π is a rotation or (λ, π) is weak mixing for almost every λ. (Of course the proof of Katok-Stepin’s result precedes Veech’s criteria.) Theorem 6.3 (Veech, [V3]). If (1, ..., 1) ∈/ H(π) then (λ, π) is weak mixing for almost every λ. 6.1. ProofofTheoremA. By Theorems 6.2 and 6.3 it is enough to consider the case where g>1 and (1, ..., 1) ∈ H(π). By Veech’s criteria Theorem 6.1, Theorem A is a consequence of the following. d−1 Theorem 6.4. Let π ∈ S with g>1 and let h ∈ H \{0}.IfU ⊂ P+ is an open set then for d−1 d almost every [λ] ∈ P+ , and for every t ∈ R,eitherth ∈ Z or R · d d (6.5) lim sup Bn ([λ ],π ) th R /Z > 0. n n→∞,RR([λ])∈U Proof. Choosing n sufficiently large, we see that there exists a connected component ∆ of the domain n n of RZ which is compactly contained in U. Indeed, the connected component of the domain of RR d−1 containing [λ]shrinksto[λ]asn →∞, for almost every [λ] ∈ P+ (this is exactly the criteria for unique ergodicity used in [V1]). d If the result does not hold, a density point argument implies that there exists h ∈ H ∩ R+ and a positive measure set of [λ] ∈ ∆ such that R d (6.6) lim B ([λ ],π ) · thRd Zd =0, for some t ∈ R such that th∈ / Z . n n / n→∞,RR([λ])∈U WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS 15 Let T :∆→ ∆ be the first return map to ∆ under iteration of RZ .ThenT is ergodic, and by Lemma 2.1 it is also strongly expanding. A(x)=BZ (T r(x)−1(x)) ···BZ (x)|H where r(x)isthe d−1 return time of x ∈ ∆. Then the cocycle (T,A) is locally constant integral and uniform, and Θ = P+ is adapted to (T,A). The central stable space of (T,A) coincides with the central stable space of Z (RZ ,B |H) almost everywhere. Using Theorem 5.1, we see that all the hypothesis of Theorem 3.1 are satisfied. Thus for almost every [λ] ∈ ∆, the line L = {th, t ∈ R} intersects the weak stable manifold in a subset of H ∩ Zd. This implies (together with (4.16)) that (6.6) fails for almost every [λ] ∈ ∆, as required. 7. Translation flows 7.1. Special flows. Translation flows on flat surfaces are a particular case of special flows,thatis, suspension flows over an i.e.t. with a roof function which is constant on each Ii. To specify such a d roof function, it is enough to give an extra parameter h ∈ R+. The phase space of the special flow F = F (λ, h, π) (over the i.e.t. f = f(λ, π)) is the union of disjoint rectangles D = ∪iIi × [0,hi), and the flow F is completely determined by the conditions Fs(x, 0) = (x, s), x ∈ Ii, s Fhi (x, 0) = (f(x), 0). Weak mixing of the flow F means no non-constant measurable solutions φ : D → C to 2πits (7.1) φ(Fs(x)) = e φ(F (x)), for some t ∈ R. This translates to two conditions in terms of the i.e.t. f: (1) f is ergodic, (2) There are no non-zero measurable solutions φ : I → C to (6.3) for some t =0. d d Theorem 7.1. Let π ∈ S with g>1. For almost every (λ, h) ∈ R+ × (H ∩ R+), F = F (λ, h, π) is weak mixing. Proof. This is an immediate consequence of Veech’s criteria and Theorem 6.4. This theorem is all we will need for the discussion of translation flows. For completeness, we will first discuss the general case of special flows. ∈ Rd ×{ } n Lemma 7.2 (Veech, [V3]). Assume that (λ, π) + π is such that QR(λ, π) is defined for all n>0.If R · d d (7.2) lim inf Bn (λ, π) th R /Z =0 n→∞ with t ∈ R, h ∈ Rd then the orthogonal projection of th on N(π) belongs to Zd. This lemma, together with Veech’s criteria can be used to establish typical weak mixing for special flows with some specific combinatorics (such that dim H ≤ d − 2). But this lemma does not help at all with the case h ∈ H, which as we will see is the important case for translation flows. d d−1 Theorem 7.3. Let π ∈ S with g>1 and let h ∈ R \{0}.IfU ⊂ P+ is an open set then for d−1 d almost every [λ] ∈ P+ , and for every t ∈ R,eitherth ∈ Z or R · d d (7.3) lim sup Bnk ([λ ],π ) th R /Z > 0. n n→∞,RR([λ])∈U Proof. If h ∈ N, then this is just a consequence of the previous lemma. So we assume that h ∈ Rd\Zd. d Let w be the orthogonal projection of h on H. By Theorem 6.4, for almost every λ ∈ R+,there d−1 d exists a set of [λ] ∈ P+ of full measure such that if tw∈ / Z then R · d d (7.4) lim sup Bnk ([λ ],π ) tw R /Z > 0. n n→∞,RR([λ])∈U 16 ARTUR AVILA AND GIOVANNI FORNI By the previous lemma, if (7.3) does not hold for some t then th = c + w with c ∈ Zd.Butthis implies that R R · d d · d d (7.5) Bnk ([λ ],π ) tw R /Z = Bnk ([λ ],π ) th R /Z , and the result follows. d d Theorem 7.4. Let π ∈ S with g>1. For almost every (λ, h) ∈ R+ × R+, F = F (λ, h, π) is weak mixing. Proof. This is an immediate consequence of the previous theorem and Veech’s criteria. 7.2. ProofofTheoremB.Let Q(g,κ) be the moduli space of translation flows on surfaces of genus g with with singularity type κ.ThenQ(g,κ) is obtained as a quotient of a “strata” T (g,κ) of the Teichm¨uller flow by the modular group of surfaces of genus g. d + Veech showed that each connected component of T (g,κ) can be parametrized by R+×H (π)×{π} where π is some irreducible permutation with g(π)=g and H+(π) ⊂ H is a convex cone of non- + d empty interior specified by a finite number of linear inequalities, such that H ⊂ R+.Moreover, this parametrization (which preserves the Lebesgue measure class) is such that the special flow F = F (λ, h, π) is conjugate to the corresponding translation flow. Thus Theorem B follows from Theorem 7.1. Appendix A. Linear exclusion Theorem A.1. Let (T,A) be a measurable cocycle. For almost every x ∈ ∆,ifG ⊂ Rp is parallel to a subspace transverse to Ecs(x) then the Hausdorff dimension of G ∩ W s(x) is 0. Proof. Let 1 (A.1) β(δ) = lim sup ln(1 + (3δA(T n(x)))p). n→∞ n By Birkhoff’s Theorem, for almost every x ∈ ∆, limδ→0 β(δ)=0. p If n ≥ m ≥ 0letSδ,m,n(x)bethesetofw ∈ G such that Ak(x) · w ∈ Bδ(Z ), m ≤ k ≤ n.Thus s (A.2) G ∩ W (x)=∩δ>0 ∪m≥0 ∩n≥mSδ,m,n. If δ<1/2, the connected components of Sδ,m,n are convex open sets and have diameter bounded by −1 −λn (A.3) 2δC(x, G0) e , where G0 is the subspace parallel to G,andC(x, G0)andλ are given by Oseledets Theorem as in (2.15). For n ≥ 0, let ρn be the maximal over all m ≤ n and over all connected components U of Sm,n of the number of connected components of Sm,n+1 intersecting U.Then n p (A.4) ρn ≤ 1+(3δA(T (x))) . It follows that each connected component U of Sδ,m,m intersects at most n−1 (δ,n)n β(δ)n (A.5) ρk ≤ e e k=m connected components of Sδ,m,n, where limn→∞ (δ, n)=0.Thus∩n≥mSδ,m,n intersects U in a set β(δ) ∪ ∩ of upper box dimension at most λ . We conclude that m≥0 n≥m Sδ,m,n has Hausdorff dimension β(δ) ∩ s at most λ and G W (x) has Hausdorff dimension 0. This theorem, together with Veech’s criteria has the following consequence, which implies Theo- rem B. WEAK MIXING FOR INTERVAL EXCHANGE TRANSFORMATIONS AND TRANSLATION FLOWS 17 d d Theorem A.2. Let π ∈ S. Then for almost every λ ∈ R+,thesetofh ∈ H ∩R+ such that (λ, h, π) is not weak mixing has Hausdorff dimension at most g +1.4 Proof. It is enough to show that W s(x) has Hausdorff dimension at most g where W s(x)istheweak stable manifold of the cocycle (T,A) considered in Theorem 6.4: in this case, the set of h ∈ H \{0} such that the line through h intersects W s(x)insomew = 0 will have Hausdorff dimension at most s s s g + 1, and the result then follows by Veech’s criteria. But W (x)=(G0 ∩ W (x)) + E (x). Thus it s is enough to show that W (x) ∩ G0 has Hausdorff dimension 0. But this follows from the previous theorem and hyperbolicity of (T,A). References [F] Forni, Giovanni Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2) 155 (2002), no. 1, 1–103. [KS] Katok, A. B.; Stepin, A. M. Approximations in ergodic theory. Uspehi Mat. Nauk 22 1967 no. 5 (137), 81–106. [L] Lucien, Isabelle Melange faible topologique des flots sur les surfaces. Ergodic Theory Dynam. Systems 18 (1998), no. 4, 963–984. [M] Masur, Howard Interval exchange transformations and measured foliations. Ann. of Math. (2) 115 (1982), no. 1, 169–200. [NR] Nogueira, A.; Rudolph, D. Topological weak-mixing of interval exchange maps. Ergodic Theory Dynam. Systems 17 (1997), no. 5, 1183–1209. [V1] Veech, William A. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), no. 1, 201–242. [V2] Veech, William A. The Teichm¨uller geodesic flow. Ann. of Math. (2) 124 (1986), no. 3, 441–530. [V3] Veech, William A. The metric theory of interval exchange transformations. I. Generic spectral properties. Amer. J. Math. 106 (1984), no. 6, 1331–1359. [Z] Zorich, Anton Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 325–370. Laboratoire de Probabilites´ et Modeles` aleatoires,´ Universite´ Pierre et Marie Curie–Boite courrier 188, 75252–Paris Cedex 05, France E-mail address: [email protected] Topologie et Dynamique, Universite´ Paris-Sud, F-91405 Orsay (Cedex), France. E-mail address: [email protected] 4 In order to get a weaker, measure zero, statement (when g>1), it is enough to use as before that the Zorich cocycle has two positive Lyapunov exponents.