The Hoelder Property for the Spectrum of Translation Flows in Genus Two
aa r X i v : . [ m a t h . D S ] J a n THE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATIONFLOWS IN GENUS TWO
ALEXANDER I. BUFETOV AND BORIS SOLOMYAK
Abstract.
The paper is devoted to generic translation flows corresponding to Abelian differen-tials with one zero of order two on flat surfaces of genus two. These flows are weakly mixing by theAvila-Forni theorem. Our main result gives first quantitative estimates on their spectrum, estab-lishing the H¨older property for the spectral measures of Lipschitz functions. The proof proceedsvia uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markovcompacta and arguments of Diophantine nature in the spirit of Salem, Erd˝os and Kahane.
To the memory of William Austin Veech (1938–2016) Introduction
Spectrum of translation flows.
Let M be a compact orientable surface. To a holomorphicone-form ω on M one can assign the corresponding vertical flow h + t on M , i.e., the flow atunit speed along the leaves of the foliation ℜ ( ω ) = 0. The vertical flow preserves the measure m = i ( ω ∧ ω ) / , the area form induced by ω . By a theorem of Katok [24], the flow h + t is nevermixing. The moduli space of abelian differentials carries a natural volume measure, called theMasur-Veech measure [25], [32]. For almost every Abelian differential with respect to the Masur-Veech measure, Masur [25] and Veech [32] independently and simultaneously proved that theflow h + t is uniquely ergodic. Under additional assumptions on the combinatorics of the abeliandifferentials, weak mixing for almost all translation flows has been established by Veech in [33]and in full generality by Avila and Forni [3]. The spectrum of translation flows is therefore almostsurely continuous and always has a singular component. No quantitative results have, however,previously been obtained about the spectral measure. Sinai [personal communication] posed thefollowing Problem.
Find the local asymptotics for the spectral measure of translation flows.
Formulation of the main result.
The aim of this paper is to give first quantitative esti-mates on the spectrum of translation flows. Let H (2) be the moduli space of abelian differentialson a surface of genus 2 with one zero of order two. The natural smooth Masur-Veech measure onthe stratum H (2) is denoted by µ . Our main result is that for almost all abelian differentials in ALEXANDER I. BUFETOV AND BORIS SOLOMYAK H (2), the spectral measures of Lipschitz functions with respect to the corresponding translationflows have the H¨older property. Recall that for a square-integrable test function f , the spectralmeasure σ f is defined by b σ f ( − t ) = h f ◦ h + t , f i , t ∈ R , see Section 3.3 for details. A point mass for the spectral measure corresponds to an eigenvalue,so H¨older estimates for spectral measures quantify weak mixing for our systems. Theorem 1.1.
There exists γ > such that for µ -almost every abelian differential ( M, ω ) ∈ H (2) the following holds. For any B > there exist constants C = C ( ω , B ) and r = r ( ω , B ) suchthat for any Lipschitz function f on M , for all x ∈ [ B − , B ] and r ∈ (0 , r ) we have σ f ([ x − r, x + r ]) ≤ C k f k L · r γ . The proof uses the symbolic formalism of [9], namely, the representation of translation flowsby flows along orbits of the asymptotic equivalence relation of a Markov compactum. The H¨olderproperty is then reformulated as a consequence of a statement on Diophantine approximationinvolving the incidence matrices of the graphs forming the Markov compactum that codes thetranslation flow. These incidence matrices are, in turn, realizations of a renormalization cocycle,isomorphic, under our symbolic coding, to the Kontsevich-Zorich cocycle over the Teichm¨uller flowon the moduli space of abelian differentials. By the Oseledets multiplicative ergodic theorem, thecocycle admits a decomposition into Oseledets subspaces corresponding to the distinct Lyapunovexponents. Our argument in this paper requires that the Kontsevich-Zorich cocycle admit exactlytwo positive Lyapunov exponents and not have zero Lyapunov exponents; this is the reason whyour main result only covers the stratum H (2).We stress that our proof only works for the Masur-Veech smooth measure. This can informallybe explained as follows. A translation flow can be represented as a suspension flow over aninterval exchange transformation with a piecewise constant roof function. Take a measure on astratum, invariant under the action of the Teichm¨uller flow. If one fixes an interval exchangetransformation, then the invariant measure yields a conditional measure on the polyhedron ofadmissible tuples of heights of the rectangles. Our argument in its present form requires that thismeasure be absolutely continuous with respect to the natural Lebesgue measure; this propertyonly holds for the Masur-Veech smooth measure.The H¨older exponent γ > x = 0 (assuming the testfunction f has zero mean), see Remark 3.2. Local estimates for spectral measures are obtainedvia growth estimates of twisted Birkhoff integrals. HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 3
The paper is organized as follows. In Section 2, we introduce the necessary setup of Markov com-pacta and Bratteli-Vershik (BV) transformations, as well the alternative, closely related frame-work, based on sequences of substitutions, or S -adic systems (see e.g. [36, 37, 38, 30, 9, 7] forfuther background). We will be working in this “symbolic” framework for most of the paper,only returning to translation flows in the last Section 11. Estimates of twisted Birkhoff integralsin terms of matrix product are considered in Section 3, which builds on our paper [12]. Thedifference is that [12] was concerned with a single substitution, or equivalently, with a station-ary Bratteli-Vershik diagram. In Section 4 we state the main theorem for random BV-systems,satisfying a certain list of conditions, among which the key condition is a uniform large devia-tions estimate. That theorem is proved in Sections 5-10. It should be emphasized that thereare substantial technical difficulties in the transition from the stationary framework of [12] tothe non-stationary setting of this paper. In the case of a single substitution matrix we couldrely on estimates of the Vandermonde matrix, its determinant and its inverse. In this paper, weneed similar estimates for the cocycle matrices. The Oseledets Theorem controls norms of thesematrices and angles between different subspaces only up to subexponential errors. It is preciselyin order to control these errors that we need the assumption that only two Lyapunov exponentsbe positive. In Section 10 we use a variant of the “Erd˝os-Kahane argument” that had originatedin the theory of Bernoulli convolution measures, see [14, 23]. Finally, in Section 11 we concludethe proof by explaining how the symbolic coding of translation flows gives rise to suspension flowsover BV-maps and by checking all the conditions required. The key probabilistic condition isderived from a (slight generalization of) large deviation estimate for the Teichm¨uller flow in [8].1.3.
General remarks.
For comparison, consider the work on weak mixing by Avila-Forni [3]and the recent Avila-Delecroix [2]. Let us parametrize all possible eigenvalues by the line in H ( M ; R ) through the imaginary part of the 1-form ω (we borrow here some of the wording fromthe introduction of [2], and also refer to that paper for the explanation of the terms used in thisparagraph). The Veech criterion [33] says that an eigenvalue is parametrized by an element of“weak-stable lamination” associated to an acceleration of the Kontsevich-Zorich cocycle actingmodulo H ( M, Z ). In [3], a probabilistic method is applied to exclude non-trivial intersections ofan arbitrary fixed line in H ( M ; R ) with the weak stable foliation to prove weak mixing for almostevery interval exchange, and a simpler, “linear exclusion” is used to exclude such intersectionsand prove weak mixing for almost every translation flow. (On the other hand, [2] uses additionalstructure to prove weak-mixing of the translation flow in almost every direction for a given non-arithmetic Veech surface.) In order to prove H¨older continuity, roughly speaking, we need thatthere is a positive density of iterates that fall outside of a fixed neighborhood of H ( M, Z )along the orbit of the acceleration of the Kontsevich-Zorich cocycle, uniformly for a fixed line in H ( M ; R ). This requires much more delicate estimates, see Sections 9 and 10, hence more limitedscope of our results. ALEXANDER I. BUFETOV AND BORIS SOLOMYAK
Our inspiration came from the work of Salem, Erd˝os, and Kahane on Diophantine approxima-tion and Bernoulli convolutions. A well-known problem, open since the 1930’s, asks whether(1.1) { λ > ∃ α > n →∞ k αλ n k R / Z = 0 } is the set of PV-numbers, that is, λ an algebraic integer all of whose conjugates are less than onein absolute value. Salem [27] showed that the set in (1.1) is countable. In short, if(1.2) αλ n = K n + ε n , with K n being the nearest integer, is such that | ε n + j | for j ≥
0, is sufficiently small, then all K n + j , j ≥ , are uniquely determined by K n , K n +1 , and λ may be recovered as lim j →∞ K n + j K n + j − ,hence there are only countable many possibilities for λ . This may be compared with the “linearexclusion” from [3] (in the sense that in both cases the question of (non)convergence is addressed;the method is actually different).The Bernoulli convolution measure ν λ can be defined by the formula for its Fourier transform: b ν λ ( t ) = ∞ Y n =0 cos(2 πλ − n t ) , t ∈ R . Erd˝os [13] observed that lim t →∞ | b ν λ ( t ) | 6 = 0 for PV-numbers λ , which implies that the correspond-ing Bernoulli convolution measure ν λ is singular. Salem (see [28]) proved that lim t →∞ | b ν λ ( t ) | = 0in all other cases, but it is an open problem whether such ν λ is necessarily absolutely continuous(see [21, 29, 31] for the recent progress in this direction). However, in another early paper, Erd˝os[14] proved that for almost all λ , b ν λ ( t ) = O ( | t | − C ( λ ) ), as | t | → ∞ , with C ( λ ) → ∞ as λ → λ sufficiently close to one(Kahane [23] later showed that in the argument of Erd˝os one can replace “almost all” by “alloutside of a set of any fixed positive Hausdorff dimension). The argument of Erd˝os and Kahanebuilds on the Salem idea: if all ε n in (1.2), outside a set of n ’s of small positive density are suffi-ciently small in absolute value, then “most” of the K n ’s are uniquely determined by the previousones, and we get a good covering of the “bad set” to yield the dimension estimate. A variation onthis idea, which came to be called the “Erd˝os-Kahane argument”, was used in [12], and a muchmore delicate version of it is developed in this work. Acknowledgements.
We are deeply grateful to Giovanni Forni for our useful discussions ofthe preliminary version of the paper, and to the referee for constructive criticism and helpfulsuggestions. The research of A. Bufetov on this project has received funding from the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020 research and innovation pro-gramme under grant agreement No 647133 (ICHAOS). It was also supported by the A*MIDEXproject (no. ANR-11-IDEX-0001-02) funded by the programme “Investissements d’Avenir” of theGovernment of the French Republic, managed by the French National Research Agency (ANR),
HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 5 by the Grant MD 5991.2016.1 of the President of the Russian Federation and by the Russian Aca-demic Excellence Project ‘5-100’. B. Solomyak has been supported by NSF grants DMS-0968879,DMS-1361424, and the ISF grant 396/15.2.
Preliminaries
Markov compacta and Bratteli-Vershik transformations.
The reader is referred to[36, 37, 38, 30, 9] for further background.Let G be the set of all oriented graphs on m vertices such that there is an edge starting atevery vertex and an edge ending at every vertex (we allow loops and multiple edges). For an edge e we denote by I ( e ) and F ( e ) the initial and final vertices of e correspondingly. Assume we aregiven a sequence { Γ n } n ∈ Z of graphs belonging to G . To this sequence we associate the Markovcompactum of paths in our sequence of graphs:(2.1) X = { e = ( e n ) n ∈ Z : e n ∈ E (Γ n ) , F ( e n +1 ) = I ( e n ) } . We will also need the one-sided Markov compactum X + (respectively X − ), defined in the sameway with elements ( e n ) n ≥ (respectively ( e n ) n ≤ ). A one-sided sequence of graphs in G can alsobe considered as a Bratteli diagram of (finite) rank m . We then view its vertices as being arrangedin levels n ≥
0, so that the graph Γ n connects the vertices of level n to vertices of level n − A n ( X ) = A (Γ n ) be theincidence matrix of the graph Γ n given by the formula A ij (Γ) = { e ∈ E (Γ) : I ( e ) = i, F ( e ) = j } . On the Markov compactum X we define the “future tail” and “past tail” equivalence relations,in which two infinite paths are equivalent iff they agree from some point on (into the future orinto the past).There is a standard construction of telescoping (= aggregation): for any sequence 1 = n The sequence Γ n (after appropriate telescoping) contains infinitelymany occurrences of a single graph Γ with a strictly positive incidence matrix, both in the pastand in the future. In this case, as is well-known since the work of Furstenberg (see e.g. (16.13) in [20]), the Markovcompactum X + is uniquely ergodic , which means that there is a unique invariant probabilitymeasure for the “future tail” equivalence relation. We need to develop this point in more detail.In fact, in this case we have (see [9, 1.9.5]) that there exist strictly positive vectors ~ z ( ℓ ) , ~ u ( ℓ ) , for ℓ ∈ Z , such that ~ z ( ℓ ) = A tℓ ~ z ( ℓ +1) , ℓ ∈ Z ; ALEXANDER I. BUFETOV AND BORIS SOLOMYAK (2.2) \ n ∈ N A tℓ +1 · · · A tℓ + n R m + = R + ~ z ( ℓ ) , ℓ ∈ Z ; ~ u ( ℓ ) = A ℓ ~ u ( ℓ − , ℓ ∈ Z ; \ n ∈ N A ℓ − · · · A ℓ − n R m + = R + ~ u ( ℓ ) , ℓ ∈ Z . The vectors are normalized by | ~ z (0) | = 1 , h ~ z (0) , ~ u (0) i = 1 . As already mentioned, the Markov compactum X + is then uniquely ergodic, with the uniquetail-invariant probability measure ν + given by(2.3) ν + ( X + j ) = z (0) j , ν + ([ e . . . e ℓ ]) = z ( ℓ ) I ( e ℓ ) , where X + j is the set of one-sided paths e e . . . ∈ X + such that F ( e ) = j and [ e . . . e ℓ ] is thecylinder set corresponding to the finite path. The advantage of working with 2-sided Brattelidiagrams, which is one of the key ideas of [9], is that one can similarly define the “dual” measure ν − on the set of “negative paths” X − , invariant under the “past tail” equivalence relation. Then ν = ν + × ν − is a probability measure on X .Now suppose that a linear ordering (Vershik’s ordering) is defined on the set { e ∈ E (Γ ℓ ) : I ( e ) = i } for all i ≤ m and ℓ ∈ Z . This induces a partial lexicographic ordering o on the Markovcompactum X ; more precisely, two paths are comparable if they agree for n ≥ t for some t ∈ Z .(Also two paths in X − are comparable if they end at the same vertex.) Restricting this orderingto the 1-sided compactum X + , we obtain the adic , or Bratteli-Vershik (BV) transformation T ,defined as the immediate successor of a path e in the ordering o . (See also the work of S. Ito [22],where a similar construction had appeared earlier.) Let Max( o ) be the set of paths in X + suchthat its every edge is maximal. It is easy to see that card(Max( o )) ≤ m . Similarly define the setof minimal paths Min( o ), and let e X + be the set of paths, which are not tail equivalent to any ofthe paths in Min( o ) ∪ Max( o ). Then the Z -action { T n } n ∈ Z is well-defined on e X + . Since we areexcluding a countable set, the action is defined almost everywhere with respect to any non-atomicmeasure; certainly, ν + -a.e. in the uniquely ergodic case. We similarly define e X as the set of bi-infinite paths in X which are not forward tail-equivalent to any of the maximal or minimal paths.Note that invariant measures for the future tail equivalence relation are precisely the invariantmeasures for the BV map, so we get unique ergodicity of the system ( X + , T ) under our standingassumption. An easy, but important, fact is that the construction of telescoping/aggregationpreserves the Vershik ordering, and the corresponding BV-systems are isomorphic.We shall also consider suspension flows over BV-systems, with a piecewise-constant roof func-tion depending only on the vertex at the level 0. More precisely, let X + be a one-sided Markovcompactum with a Vershik ordering and BV-transformation T . For a strictly positive vector HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 7 ~s = ( s , . . . , s m ) define the roof function φ ~s to be equal to s j on the cylinder set X + j . Theresulting space will be denoted X ~s : X ~s := m G j =1 X + j × [0 , s j ] / ∼ , with ( e , φ ~s ( e )) ∼ ( Te , , on which we consider the usual suspension flow { h t } t ∈ R . It preserves the measure induced by the T -invariant measure ν + on X + and the Lebesgue measure on R . We will need the following Lemma 2.1. [9, Section 2.5] Given a Vershik ordering on a uniquely ergodic Markov compactum X with the unique invariant measure ν , there is a symbolic flow ( X, h + t ) t ∈ R , defined on e X , so ν -a.e. on X , which is measurably conjugate to the suspension flow ( X ~s , h t ) t ∈ R over ( X + , T ) , with s j = u (1) j , j ≤ m . Moreover , the conjugating map F : X → X ~s is given by F ( e ) = ( P + e , t ) , where t = ν − ( { a ∈ X − : I ( a ) = F ( e ) , a P − e } ) , where P + , P − are the truncations from X to X + , X − respectively and is the Vershik order. Themap F is well-defined on e X and its inverse is well-defined over e X + . Weakly Lipschitz functions. Following [10, 9], we consider the space of weakly Lipschitzfunctions on a uniquely ergodic Markov compactum X with the probability measure ν + invariantfor the “forward tail” equivalence relation and a Vershik ordering o . Recall that e X denotes theset of paths in x that are not (forward) tail equivalent to any of the maximal or minimal pathsin the Vershik ordering. We say that f is weakly Lipschitz and write f ∈ Lip + w ( X ) if f is definedand bounded on e X , and there exists C > e , e ′ ∈ e X , satisfying e k = e ′ k for −∞ < k ≤ n , with n ∈ N , we have(2.4) | f ( e ) − f ( e ′ ) | ≤ Cν + ([ e . . . e n ]) . The norm in Lip + w ( X ) is defined by(2.5) k f k L := k f k sup + e C, where e C is the infimum of the constants in (2.4). Note that a weakly Lipschitz function is mappedinto a weakly Lipschitz function when telescoping of the diagram is performed, and this does notincrease the norm k · k L .We analogously define the space Lip + w ( X ~s ) of weakly Lipschitz functions on the space X ~s of thesuspension flow over ( X + , T ) with the roof function determined by the vector ~s ∈ R m + . Namely, f ∈ Lip + w ( X ~s ) if it is defined and bounded on e X + and there exists C > e , e ′ ∈ e X + ,satisfying e k = e ′ k for k ≤ n , with n ∈ N , and all t ∈ [0 , s F ( e ) ] we have(2.6) | f ( e , t ) − f ( e ′ , t ) | ≤ Cν + ([ e . . . e n ]) . The norm k f k L is defined as in (2.5). ALEXANDER I. BUFETOV AND BORIS SOLOMYAK Note that weakly Lipschitz functions are not Lipschitz in the “transverse” direction, corre-sponding to the “past” in the 2-sided Markov compactum and to the vertical direction in thespace of the suspension flow. Note also that if f ∈ Lip + w ( X ), then f ◦ F − ∈ Lip + w ( X ~s ), with thesame norm, where F is defined in Lemma 2.1.2.3. Substitutions. Along with the Markov compactum and BV-transformation, it is convenientto use the language of substitutions (see e.g. [17] for further background). Consider the alphabet A = { , . . . , m } , which is identified with the vertex set of all the graphs Γ n . A substitution isa map ζ : A → A + , extended to A + and A N by concatenation. Given a Vershik ordering o ona 1-sided Bratteli diagram { Γ j } j ≥ , the substitution ζ j takes every b ∈ A into the word in A corresponding to all the vertices to which there is a Γ j -edge starting at b , in the order determinedby o . Formally, the length of the word ζ j ( b ) is | ζ j ( b ) | = m X a =1 A b,a (Γ j ) , and the substitution itself is given by(2.7) ζ j ( b ) = u b,j . . . u b,j | ζ j ( b ) | , b ∈ A , j ≥ , where ( b, u b,ji ) ∈ E (Γ j ), listed in the linear order prescribed by o . Substitutions, extended to A + ,can be composed in the usual way as transformations A + → A + . We will use the notation ζ [ n ,n ] := ζ n ◦ · · · ◦ ζ n , n ≥ n , and ζ [ n ] := ζ [1 ,n ] , n ≥ . Given a substitution ζ , its substitution matrix is defined by S ζ ( a, b ) := a in ζ ( b ) . Observe that S ζ ◦ ζ = S ζ S ζ . We will denote S n = S ζ n . Notice that the transpose S tn is exactlythe incidence matrix A n = A (Γ n ) by the definition of ζ n :(2.8) S tζ n = A (Γ n ) . We will use the notation S [ n ,n ] := S ζ [ n ,n and S [ n ] := S ζ [ n ] . Next, we associate to any e ∈ X + , its “horizontal sequence” in the alphabet A , defined by h ( e ) := x = ( x n ) n , where x n = x n ( e ) = b whenever F ( T n ( e ) ) = b, n ∈ Z , that is, we keep track of the vertex at level zero, applying the BV-transformation. The horizontalsequence is just the symbolic dynamics of T with respect to the 0-level cylinder partition. We geta full 2-sided infinite sequence h ( e ) = x = ( x n ) n ∈ Z whenever the (2-sided) orbit of e under T does HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 9 not hit a maximal or a minimal path. (By our assumptions, no orbit of T can hit both.) Thus h : e X + → A Z is well-defined. We obtain, by definition, the following commutative diagram: e X + T −−−−→ e X + h y y h A Z T −−−−→ A Z where T is the left shift on A Z . Of course, the map h is far from being surjective. In order tounderstand its image, it is useful to have a more explicit algorithm for h ( e ). Suppose that thepath e ∈ X + goes through the vertices b , b , . . . , that is, b n = F ( e n +1 ). Recalling the definitionof the substitutions ζ n we can write(2.9) ζ n ( b n ) = u n − b n − v n − , n ≥ , where u n − and v n − are words, possibly empty. Note that there may be more than one occurrenceof b n − in ζ n ( b n ), but we choose the representation (2.9) according to the edge e n . Consider thefollowing sequence of words U n , n ≥ 0, defined inductively. We start with U = u . b v , where the dot . separates negative and positive coordinates. Then U n +1 is obtained from U n inductively, by appending ζ n +1 ( u n +1 ) from the left and ζ n +1 ( v n +1 ) from the right. If we disregardthe location of the dot, we simply have U n = ζ ◦ · · · ◦ ζ n +1 ( b n +1 ) = ζ [ n +1] ( b n +1 ) , n ≥ . When we take the location of the dot into account, typically, the words U n will “grow” to infinity,both left and right, to a limiting 2-sided sequence, which is exactly h ( e ):(2.10) h ( e ) = . . . ζ ( u ) ζ ( u ) u . b v ζ ( v ) ζ ( v ) . . . The other alternative is that it grows to infinity only on one side, which happens if and only if e is tail equivalent to either minimal or a maximal path. Denote Y := clos( h ( e X + )) , where “clos” denotes the closure in the product topology. Now the following is clear. Lemma 2.2. The space Y ⊂ A Z is exactly the set of 2-sided sequences x with the property thatany subword of x appears as a subword of ζ [ n ] ( b ) for some b ∈ A and n ≥ . Remark. Dynamical systems ( Y , T ) have been studied under the name of S -adic systems . Theywere originally introduced by S. Ferenczi [15], with the additional assumption that there arefinitely many different substitutions in the sequence { ζ j } ; however, more recently this restrictionhas been removed, see e.g. [7]. Let h ( n ) i be the number of finite paths e , . . . , e n such that I ( e n ) = i . (This is the height ofthe Rokhlin tower for the BV-map.) We get a sequence of real vectors ~h ( n ) which satisfy theequations:(2.11) ~h ( n +1) = A n +1 ~h ( n ) = S tn +1 ~h ( n ) , h ( n ) i = | ζ [ n ] ( i ) | , n ≥ . Let s be the left shift transformation on the 1-sided Markov compactum: s ( e , e , e , . . . ) =( e , e , . . . ). We thus obtain a sequence of Markov compacta X ( ℓ )+ for ℓ ≥ 0, with X (0)+ := X + ,so that s : X ( ℓ )+ → X ( ℓ +1)+ for all ℓ . The Vershik ordering and BV-transformation are naturallyinduced on the whole family. We can then consider the horizontal sequence map h : e X ( ℓ )+ → A Z . Its image, denoted by Y ( ℓ ) , is described similarly to Y = Y (0) , as the set of all sequences in A Z whose every subword occurs as a block in ζ [ ℓ +1 ,n ] ( b ) for some n ≥ ℓ + 1 and b ∈ A .A substitution ζ acts on A Z as follows: ζ ( . . . a − a . a . . . ) = . . . ζ ( a − ) ζ ( a ) . ζ ( a ) . . . Definitions imply that we have a sequence of surjective maps ζ ℓ : Y ( ℓ ) → Y ( ℓ − , ℓ ≥ . It followsfrom definitions (and the explicit formulas for h ) that x = T k − ζ ( x ′ ) , where k is the number (rank) of the edge e in the Vershik ordering. Of course, similar formulasrelate h ( e ) and h ( se ) for e ∈ X ( ℓ )+ . (Recall that T denotes the left shift on A Z .)3. Estimating the growth of exponential sums and matrix products We use the following convention for the Fourier transform of functions and measures: given ψ ∈ L ( R ) we set b ψ ( t ) = R R e − πiωt ψ ( ω ) dω, and for a probability measure ν on R we let b ν ( t ) = R R e − πiωt dν ( ω ) . Spectral measures and twisted Birkhoff integrals. Since our goal is to obtain estimatesof spectral measures, we recall how they are defined for flows. Given a measure-preserving flow( Y, h t , µ ) t ∈ R and a test function f ∈ L ( Y, µ ), there is a finite positive Borel measure σ f on R such that b σ f ( − τ ) = Z ∞−∞ e πiωτ dσ f ( ω ) = h f ◦ h τ , f i for τ ∈ R . In order to obtain local bounds on the spectral measure, we can use growth estimates of the“twisted Birkhoff integral”(3.1) S ( y ) R ( f, ω ) := Z R e − πiωτ f ◦ h τ ( y ) dτ. The following lemma is standard; a proof may be found in [12, Lemma 4.3]. HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 11 Lemma 3.1. Suppose that for some fixed ω ∈ R , R > , and α ∈ (0 , we have (3.2) (cid:13)(cid:13)(cid:13) S ( y ) R ( f, ω ) (cid:13)(cid:13)(cid:13) L ( Y,µ ) ≤ C R α for all R ≥ R . Then (3.3) σ f ([ ω − r, ω + r ]) ≤ π − α C r − α ) for all r ≤ (2 R ) − . Remark 3.2. 1. Since ( Y, µ ) is a probability space, the L -estimate (3.2) obviously follows froma uniform estimate. We only need L estimates for the proof of our main result. Nonetheless, weexpect that the uniform estimates of this paper would have further applications.2. Estimates of twisted Birkhoff sums have been used for a number of different dynamicalsystems recently; in particular, see the work of Forni and Ulcigrai [19] on the Lebesgue spectrumfor smooth time changes of the horocycle flow.3. For ω = 0 the expression (3.1) reduces to the usual Birkhoff integral, for which sharpestimates and asymptotics are known (under the assumption that the test function f has zeromean) in a number of cases. It should be possible to obtain precise asymptotics of the spectralmeasure at zero for almost every translation flow in the context of Theorem 1.1, even for a generalstratum in genus g > 1. We expect that it is governed by the second Lyapunov exponent, basedon the results of [10], analogously to [12, Theorem 6.2]4. It is easy to see that (cid:13)(cid:13)(cid:13) S ( y ) R ( f, ω ) (cid:13)(cid:13)(cid:13) L ( Y,µ ) = O ( R ), and o ( R ) indicates the absence of an eigen-value at ω . The exponent α = 1 / α < / σ f -a.e. ω , the infimum of α , forwhich (3.2) holds, is at least 1 / 2. Indeed, (3.3) implies that d ( σ f , ω ) = lim inf r → σ f ( B ( ω,r ))log r ≥ − α ), and it is well-known thatdim H ( σ f ) = sup { s : d ( σ f , ω ) ≥ s for σ f -a.e. ω } ≤ , see e.g. [16, Prop. 10.2].3.2. Exponential sums corresponding to suspension flows. Let X + be a one-sided Markovcompactum with a Vershik ordering and BV-transformation T . For a strictly positive vector ~s = ( s , . . . , s m ) we define the roof function φ ~s to be equal to s a on the cylinder set X + a , as inSection 2, and obtain the suspension flow ( X ~s , h t ).Recall that for e ∈ X + (minus a countable exceptional set) we defined its horizontal sequence h ( e ) = ( x n ) n ∈ Z , in such a way that the BV-transformation intertwines the left shift. Similarly,we can associate to ( e , t ) ∈ X ~s a tiling of the line R : a symbol a corresponds to a closed linesegment of length s a (labeled by a ), and these line segments are “strung together” according tothe symbolic sequence h ( e ). The tile corresponding to x should contain the origin at the distance t from the left endpoint. This defines a map e h from X ~s to a “tiling space,” which intertwines theflow h τ and the left shift by τ .Our goal is to obtain growth estimates for twisted Birkhoff integrals (3.1). We start with testfunctions depending only on the cylinder set X a and the height t . More precisely, given somefunctions ψ a ∈ C ([0 , s a ]), a ∈ A , let f = X a ∈A c a f a , with f a ( e , t ) = 11 X a ψ a ( t ) , where X a = X a × [0 , s a ] . For a word v in the alphabet A denote by ~ℓ ( v ) ∈ Z m its “population vector” whose j -th entry isthe number of j ’s in v , for j ≤ m . We will need the “tiling length” of v defined by(3.4) | v | ~s := h ~ℓ ( v ) , ~s i . The following property will be used frequently: for an arbitrary substitution ξ , ~s > 0, and word U ∈ A + we have(3.5) | U | S tξ ~s = h ~ℓ ( U ) , S tξ ~s i = h S ξ ~ℓ ( U ) , ~s i = h ~ℓ ( ξ ( U )) , ~s i = | ξ ( U ) | ~s . For v = v . . . v N − ∈ A + (3.6) Φ ~sa ( v, ω ) = N − X j =0 δ v j ,a exp( − πiω | v . . . v j − | ~s ) , where the term for j = 0 is equal to one by convention. Then a straightforward calculation shows(3.7) S ( e , R ( f a , ω ) = b ψ a ( ω ) · Φ ~sa ( x [0 , N − , ω ) for R = | x [0 , N − | ~s , where ( x n ) n ∈ Z = h ( e ). Moreover, the horizontal sequence can be represented as a concatenationof long blocks of the form ζ [ n ] ( b ) , b ∈ A (not necessarily starting at the 0-th position); therefore,estimates of twisted Birkhoff sums for an arbitrary sequence may be reduced to those for ζ [ n ] ( b ),and for the latter the renormalization naturally leads to matrix products. This is what we donext.3.3. Setting up matrix products. Observe that for any two words u, v and the concatenatedword uv we have(3.8) Φ ~sa ( uv, ω ) = Φ ~sa ( u, ω ) + e − πiω | u | ~s · Φ ~sa ( v, ω ) . Recalling (2.7), we can write ζ [ n ] ( b ) = ζ [ n − ( ζ n ( b )) = ζ [ n − ( u b,n ) . . . ζ [ n − ( u b,n | ζ n ( b ) | ) , n ≥ , where we use the convention ζ [0] := Id . Hence (3.8) implies for all a, b ∈ A :Φ ~sa ( ζ [ n ] ( b ) , ω ) = | ζ n ( b ) | X j =1 exp h − πiω (cid:16) | ζ [ n − ( u b,n . . . u b,nj − ) | ~s (cid:17)i Φ ~sa ( ζ [ n − ( u b,nj ) , ω ) , n ≥ , HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 13 (for j = 1, the exponential reduces to exp(0) = 1 by definition). For n ≥ m × m matrix-function Π n ( ω ) defined by(3.9) Π ~sn ( ω ) = Φ ~s ( ζ [ n ] (1) , ω ) . . . Φ ~sm ( ζ [ n ] (1) , ω ) . . . . . . . . . . . . . . . . . . . . . Φ ~s ( ζ [ n ] ( m ) , ω ) . . . Φ ~sm ( ζ [ n ] ( m ) , ω ) . . It follows that(3.10) Π ~sn ( ω ) = M ~sn ( ω ) Π ~sn − ( ω ) , n ≥ , where M ~sn ( ω ) is an m × m matrix-function, whose matrix elements are trigonometric polynomialsgiven by(3.11) ( M ~sn ( ω ))( b, c ) = X j ≤| ζ n ( b ) | : u b,nj = c exp h − πiω (cid:16) | ζ [ n − ( u b,n . . . u b,nj − ) | ~s (cid:17)i , n ≥ . Note that M ~sn (0) = S tn , the transpose of the n -th substitution matrix, for all n ≥ 1. Since Π ~s ( ω ) = I (the identity matrix), it follows from (3.10) that(3.12) Π ~sn ( ω ) = M ~sn ( ω ) M ~sn − ( ω ) · · · M ~s ( ω ) . There is another way to express the matrices M ~sn ( ω ) which will be useful below. It follows from(3.6), (3.11), and (3.5) that M ~sn ( ω )( b, c ) = (Φ c ) S tζ [ n − ~s ( ζ n ( b ) , ω ) . This motivates the following definition for two arbitrary substitutions ξ , ξ :(3.13) M ~sξ ,ξ ( ω )( b, c ) = (Φ c ) S tξ ~s ( ξ ( b ) , ω ) , so that(3.14) M ~sn = M ~sζ [ n − ,ζ n . A straightforward verification yields the following identity for arbitrary substitutions ξ , ξ , ξ :(3.15) M ~sξ ,ξ ξ = M ~sξ ξ ,ξ · M ~sξ ,ξ . Estimating matrix products.Definition 3.3. A word v is called a good return word for the substitution ζ if v starts withsome symbol c (which can be any element of A ) and vc occurs in the word ζ ( b ) for every b ∈ A .Denote by GR ( ζ ) the set of good return words for ζ . One of our main assumptions will be that a fixed substitution ζ , with a strictly positive matrix S ζ =: Q and nonempty set of good return words, appears infinitely often in the sequence ζ j . Thus,it is convenient to perform “telescoping” of the Bratteli-Vershik diagram and assume from thestart that every substitution ζ j has the form(3.16) ζ j = ζξ j ζ, where ξ j is an arbitrary substitution.Denote by ~ k x k R / Z be the distance from x ∈ R to the nearestinteger. For a strictly positive square m × n matrix A letcol( A ) = max i,j,k A ij A kj . It is well-known and easy to check that if Q is strictly positive m × m matrix and A is anynon-negative m × n matrix, then we have(3.17) col( QA ) ≤ col( Q ) . Proposition 3.4. Let X + be a one-sided Markov compactum with a Vershik ordering, and let ζ j be the corresponding sequence of substitutions, given by (2.7). Suppose that there exists asubstitution ζ with a nonempty set of good return words, such that Q = S ζ is strictly positive and ζ j = ζξ j ζ for some substitution ξ j for all j ≥ . Then there exists c ∈ (0 , , depending only onthe substitution ζ , such that for all a, b ∈ A and N ∈ N , ω ∈ [0 , , and ~s > ~ , (3.18) | Φ ~sa ( ζ [ N ] ( b ) , ω ) | ≤ k S [ N ] k Y n ≤ N − (cid:18) − c · max v ∈ GR ( ζ ) k ω | ζ [ n ] ( v ) | ~s k R / Z (cid:19) . In fact, we can take (3.19) c = (cid:0) m · max i,j Q ij · col( Q t ) (cid:1) − . Proof. This is similar to the proof of [12, Proposition 3.2], but there are a number of new technicaldetails. Let ~e a denote the unit basis vector in R m corresponding to a ∈ A . In view of (3.9), itsuffices to estimate Π ~sN ( ω ) ~e a . We will use the following notation: • for vectors ~x, ~y ∈ R m , the inequality ~x ≤ ~y means componentwise inequality, and similarlyfor real-valued matrices; • the operation of taking absolute values of all entries for a vector ~x and a matrix A will bedenoted ~x |·| and A |·| .It is clear that for any, generally speaking, rectangular matrices A, B such that the product AB is well-defined, we have(3.20) ( AB ) |·| ≤ A |·| B |·| . HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 15 We fix ω and ~s and omit them from notation, so that M n ≡ M ~sn ( ω ) and Π n ≡ Π ~sn ( ω ). Observethat (3.20) and (3.10) imply for all n ∈ N :(3.21) ( Π n ~e a ) |·| = ( M n · · · M ~e a ) |·| ≤ M |·| n · · · M |·| ~e a ≤ M |·| n · · · M |·| ~ . In view of (3.14), (3.15), and (3.16), we have M n = M ζ [ n − ,ζ n = M ζ [ n − ,ζξ n ζ = M ζ [ n − ζξ n ,ζ · M ζ [ n − ,ζξ n = M ζ [ n − ζξ n ,ζ · M ζ [ n − ζ,ξ n · M ζ [ n − ,ζ . (3.22)By the definition of a good return word, for any v ∈ GR ( ζ ) and b ∈ A , we can write(3.23) ζ ( b ) = p ( b ) vcq ( b ) , where p ( b ) and q ( b ) are words, possibly empty, and v starts with c . Let ξ be any substitution on A . First we are going to estimate the absolute value of M ξ,ζ ( b, c ) = (Φ c ) S tξ ~s ( ζ ( b ) , ω )from above. This is a trigonometric polynomial with S tζ ( b, c ) exponential terms and all coefficientsequal to one. By (3.6) and (3.23), this polynomial includes the termsexp (cid:0) − πiω | p ( b ) | S tξ ~s (cid:1) + exp (cid:0) − πiω | p ( b ) v | S tξ ~s (cid:1) = exp (cid:0) − πiω | ξ ( p ( b ) (cid:1) | ~s ) + exp (cid:0) − πiω | ξ ( p ( b ) ) ξ ( v ) | ~s (cid:1) = exp (cid:0) − πiω | ξ ( p ( b ) ) | ~s (cid:1) · (cid:0) e − πiω | ξ ( v ) | ~s (cid:1) . It follows that | M ξ,ζ ( b, c ) | ≤ S tζ ( b, c ) − (cid:12)(cid:12) e − πiω | ξ ( v ) | ~s (cid:12)(cid:12) , and from the inequality | e πiτ | ≤ − (1 / k τ k R / Z , τ ∈ R , we obtain(3.24) | M ξ,ζ ( b, c ) | ≤ S tζ ( b, c ) − (cid:13)(cid:13) ω | ξ ( v ) | ~s (cid:13)(cid:13) R / Z . Now, for an arbitrary ~x = [ x , . . . , x m ] t > ~ b ∈ A , using (3.24) we can estimate( M |·| ξ,ζ ~x ) b = m X j =1 | M ξ,ζ ( b, j ) | · x j ≤ m X j =1 S tζ ( b, j ) x j − (cid:13)(cid:13) ω | ξ ( v ) | ~s (cid:13)(cid:13) R / Z · x c ≤ (cid:16) − c ψ ( ~x ) (cid:13)(cid:13) ω | ξ ( v ) | ~s (cid:13)(cid:13) R / Z (cid:17) · m X j =1 S tζ ( b, j ) x j = (cid:16) − c ψ ( ~x ) (cid:13)(cid:13) ω | ξ ( v ) | ~s (cid:13)(cid:13) R / Z (cid:17) · ( S tζ ~x ) b , (3.25)where c = 12 m max i,j S tζ ( i, j ) = 12 m max i,j Q ij and ψ ( ~x ) = min j x j max j x j . Thus,(3.26) M |·| ξ,ζ ~x ≤ (cid:16) − c ψ ( ~x ) (cid:13)(cid:13) ω | ξ ( v ) | ~s (cid:13)(cid:13) R / Z (cid:17) · S tζ ~x, and v ∈ GR ( ζ ) is arbitrary. We will apply the last inequality with ξ = ζ [ n − and ~x = ~x n = ( S [ n − ) t ~ ∈ Q t R m + , recalling that S n − = S ζ n − = Q S ξ n − Q . Since the matrix Q is strictly positive, we have ψ ( ~x ) = (col( ~x )) − ≥ (col( Q t )) − for any ~x ∈ Q t R m + by (3.17). Note also that for any substitutions ξ , ξ we have M |·| ξ ,ξ ≤ S tξ by the definition (3.13). Therefore, taking (3.22) and (3.26) into account, we obtain M |·| n ( S [ n − ) t ~ ≤ M |·| ζ [ n − ζξ n ,ζ · M |·| ζ [ n − ζ,ξ n · M |·| ζ [ n − ,ζ ( S [ n − ) t ~ ≤ S tζ S tξ n (cid:18) − c max v ∈ GR ( ζ ) (cid:13)(cid:13) ω | ζ [ n − ( v ) | ~s (cid:13)(cid:13) R / Z (cid:19) S tζ ( S [ n − ) t ~ (cid:18) − c max v ∈ GR ( ζ ) (cid:13)(cid:13) ω | ζ [ n − ( v ) | ~s (cid:13)(cid:13) R / Z (cid:19) ( S [ n ] ) t ~ . where c is given by (3.19). Iterating this inequality yields( Π n ~e a ) |·| ≤ Y n ≤ N − (cid:18) − c max v ∈ GR ( ζ ) (cid:13)(cid:13) ω | ζ [ n ] ( v ) | (cid:13)(cid:13) R / Z (cid:19) · ( S [ N ] ) t ~ . Finally, note that the maximal component of ( S [ N ] ) t ~ S [ N ] , whichis k S [ N ] k , and the proposition is proved completely. We emphasize that we used in an essentialway that every ζ n = ζξ n ζ both starts and end with ζ . (cid:3) HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 17 Cylindrical functions of higher order. Suppose now that f on X is a “cylindrical functionof level ℓ ”, that is, its value depends only on the first ℓ edges of the path e and on the height t .It is then convenient to represent h τ as a suspension flow with a different height function, basedon the decomposition (disjoint in measure) X ~s = [ a ∈A X ( ℓ ) a , where X ( ℓ ) a = { ( e , t ) ∈ X ~s : e ∈ X + , x ℓ = F ( e ℓ +1 ) = a } . The BV-transformation T “changes” a vertex a at the ℓ -th level after h ( ℓ ) a = | ζ [ ℓ ] ( a ) | iterates.Thus, after we enter the cylinder X ( ℓ ) a , the flow h τ stays in it for the time equal to(3.27) s ( ℓ ) a := | ζ [ ℓ ] ( a ) | ~s = [( S [ ℓ ] ) t ~s ] a . More precisely, if F ( e ℓ +1 ) = a and ( e , t ) ∈ X ~s , then(3.28) ∃ t ′ ∈ h , s ( ℓ ) a i such that ( e , t ) = h t ′ ( e ′ . . . e ′ ℓ e ℓ +1 e ℓ +2 . . . , , where e ′ . . . e ′ ℓ is the minimal path in the Bratteli diagram from the vertex a on level ℓ to the level0. Observe that the horizontal sequence of a path e ′ . . . e ′ ℓ e ℓ +1 e ℓ +2 . . . , with F ( e ℓ +1 ) = a , beginswith ζ [ ℓ ] ( a ) and can be written as ζ [ ℓ ] ( x ( ℓ ) ) for some x ( ℓ ) ∈ A N . (In fact, x ( ℓ ) = h ( σ ℓ e ) ∈ Y ( ℓ ) , seeSection 2). To summarize this discussion, for any real-valued continuous cylindrical function f oflevel ℓ on X + there exist c a ∈ R and ψ ( ℓ ) a ∈ C ([0 , s ( ℓ ) a ]) , a ∈ A , such that(3.29) f = X a ∈A c a f ( ℓ ) a , where f ( ℓ ) a ( e , t ) = 11 X ( ℓ ) a ψ ( ℓ ) a ( t ′ ) , with t ′ from (3.28) . Now we can also write down a generalization of (3.7). Denote ~s ( ℓ ) = ( s ( ℓ ) a ) a ∈A and assume e ′ = e ′ . . . e ′ ℓ e ℓ +1 e ℓ +2 . . . , with F ( e ℓ +1 ) = a . Then(3.30) S ( e ′ , R ( f ( ℓ ) a , ω ) = b ψ ( ℓ ) a ( ω ) · Φ ~s ( ℓ ) a (cid:0) x ( ℓ ) [0 , N − , ω (cid:1) for R = (cid:12)(cid:12) x ( ℓ ) [0 , N − (cid:12)(cid:12) ~s ( ℓ ) , where ( x n ) n ≥ = ζ [ ℓ ] ( x ( ℓ ) n ) n ≥ and ( x n ) n ∈ Z = h ( e ). In the next corollary we extend Proposition 3.4to cylindrical functions of level ℓ . Corollary 3.5. Under the assumptions of Proposition 3.4, for any ℓ ≥ , a, b ∈ A , n ≥ ℓ + 1 , ~s > ~ , and ω ∈ R , we have (3.31) (cid:12)(cid:12) Φ ~s ( ℓ ) a ( ζ [ ℓ +1 ,n ] ( b ) , ω ) (cid:12)(cid:12) ≤ k S [ ℓ +1 ,n ] k · Y ℓ +1 ≤ k ≤ n − (cid:18) − c · max v ∈ GR ( ζ ) k ω | ζ [ k ] ( v ) | ~s k R / Z (cid:19) , where c is given by (3.19).Proof. It is immediate from Proposition 3.4 by shifting the indices that (cid:12)(cid:12) Φ ~s ( ℓ ) a ( ζ [ ℓ +1 ,n ] ( b ) , ω ) (cid:12)(cid:12) ≤ k S [ ℓ +1 ,n ] k · Y ℓ +1 ≤ k ≤ n − (cid:18) − c · max v ∈ GR ( ζ ) k ω | ζ [ ℓ +1 ,k ] ( v ) | ~s ( ℓ ) k R / Z (cid:19) . It remains to note that | ζ [ ℓ +1 ,k ] ( v ) | ~s ( ℓ ) = h ~ℓ ( ζ [ ℓ +1 ,k ] ( v )) , ~s ( ℓ ) i = h S [ ℓ +1 ,k ] ~ℓ ( v ) , ( S ℓ ) t ~s i = h S ℓ S [ ℓ +1 ,k ] ~ℓ ( v ) , ~s i = h S [ k ] ~ℓ ( v ) , ~s i = h ~ℓ ( ζ [ k ] ( v )) , ~s i = | ζ [ k ] ( v ) | ~s . (cid:3) Next we need to pass from the exponential sum corresponding to the word ζ [ ℓ +1 ,n ] ( b ) to theone corresponding to a general word in the space Y ( ℓ ) . To this end, we will use the well-knownprefix-suffix decomposition. Lemma 3.6. Let x ( ℓ ) ∈ Y ( ℓ ) and N ≥ . Then (3.32) x ( ℓ ) [0 , N − 1] = ζ [ ℓ +1] ( u ℓ +1 ) ζ [ ℓ +1 ,ℓ +2] ( u ℓ +2 ) . . . ζ [ ℓ +1 ,n ] ( u ℓ + n ) ζ [ ℓ +1 ,n ] ( v ℓ + n ) . . . ζ [ ℓ +1] ( v ℓ +1 ) , where u j , v j , j = ℓ + 1 , . . . , ℓ + n , are respectively proper suffixes and prefixes of the words ζ j +1 ( b ) , b ∈ A . The words u j , v j may be empty, except that at least one of u ℓ + n , v ℓ + n is nonempty.Moreover, (3.33) min b ∈A | ζ [ ℓ +1 ,n ] ( b ) | ≤ N ≤ b ∈A | ζ [ ℓ +1 ,n +1] ( b ) | . Proof. This is immediate from the description of Y ( ℓ ) at the end of Section 2. (cid:3) Proposition 3.7. Under the assumptions of Proposition 3.4, for any ℓ ≥ , a ∈ A , N ∈ N , ~s > ~ , x ( ℓ ) ∈ Y ( ℓ ) , and ω ∈ R , we have, (3.34) (cid:12)(cid:12) Φ ~s ( ℓ ) a (cid:0) x ( ℓ ) [0 , N − , ω (cid:1)(cid:12)(cid:12) ≤ n X j = ℓ k S [ ℓ +1 ,j ] k ·k S j +1 k · Y ℓ +1 ≤ k ≤ j − (cid:0) − c · max v ∈ GR ( ζ ) k ω | ζ [ k ] ( v ) | ~s k R / Z (cid:1) , where c is given by (3.19) and n ∈ N is such that (3.33) holds. Here we let S [ ℓ +1 ,ℓ ] =: I .Proof. We use Lemma 3.6, and apply Corollary 3.5 to each term. The factor 2 k S j +1 k in (3.34)appears, because | u j | , | v j | ≤ max b | ζ j ( b ) | = k S j +1 k in (3.32). (cid:3) Random BV-transformations: statement of the theorem and plan of the proof Here we consider dynamical systems generated by a random sequence of Markov compacta. Inorder to state our results, we need some preparation; specifically, the Oseledets Theorem.Recall that G denotes the set of all oriented graphs on m vertices such that there is an edgestarting at every vertex and an edge ending at every vertex (we allow loops and multiple edges). HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 19 We also assume that each graph is equipped with a Vershik ordering. Let Ω be the space ofsequences of graphs: Ω = { a = . . . a − n . . . a .a . . . a n . . . , a i ∈ G , i ∈ Z } . For a ∈ Ω we denote by X ( a ) the Markov compactum corresponding to a according to the ruleΓ n = a n , n ∈ Z , and let σ the left shift on Ω. We also consider the corresponding one-sidedcompactum X + ( a ). For a word q = q . . . q k ∈ G k we can “concatenate” the graphs to obtainthe “aggregated” graph Γ q , also belonging to G . By the definition of incidence matrix, we have A ( q ) := A (Γ q ) = A ( q k ) · . . . · A ( q ) . Since the graphs are equipped with the Vershik ordering, we also have a corresponding sequenceof substitutions, so that ζ ( q ) = ζ ( q ) . . . ζ ( q k ). We will also need a “2-sided cylinder set”:[ q . q ] = { a ∈ Ω : a − k +1 . . . a = a . . . a k = q } . Following [11], we say that the word q = q . . . q k is “simple” if for all 2 ≤ i ≤ k we have q i . . . q k = q . . . q k − i +1 . If the word q is simple, two occurrences of q in the sequence a cannotoverlap. Let P be an ergodic σ -invariant probability measure on Ω satisfying the following Conditions:(C1) There exists a word q ∈ G k such that all the entries of the matrix A ( q ) are positive and (4.1) P ([ q . q ]) > . (C2) The matrices A ( a n ) are almost surely invertible with respect to P . (C3) The functions a log(1 + k A ± ( a ) k ) are integrable. (Here and below k A k denotes the Euclidean operator norm of the matrix.)Observe that (C3) , together with the Birkhoff ergodic theorem, immediately gives(4.2) lim n →∞ n − log(1 + k A ( a n ) k ) = 0 for P -a.e. a ∈ Ω . We obtain a measurable cocycle A : Ω → GL ( m, R ), defined by A ( a ) = A ( a ), called the renor-malization cocycle . Denote(4.3) A ( n, a ) = A ( σ n − a ) · . . . · A ( a ) , n > Id, n = 0; A − ( σ − n a ) · . . . · A − ( σ − a ) , n < , so that A ( n, a ) = A ( a n ) · · · A ( a ) , n ≥ . As in Section 2, we consider the sequence of substitutions ζ ( a k ) , a ∈ Ω , k ∈ Z , and theirsubstitution matrices S ζ k ( a ) = A t ( a k ). (Recall that all graphs a k are equipped with a Vershikordering.) Thus A ( n, a ) = S t ( a n . . . a ) = S tζ ( a ... a n ) n ≥ . By the Oseledets Theorem [26] (for a detailed survey, see Barreira-Pesin [6]), there exist Lyapunovexponents θ > θ > . . . > θ r and, for P -a.e. a ∈ Ω, a direct-sum decomposition(4.4) R m = E a ⊕ · · · ⊕ E r a that depends measurably on a ∈ Ω and satisfies the following:(i) for P -a.e. a ∈ Ω, any n ∈ Z , and any i = 1 , . . . , r we have A ( n, a ) E i a = E iσ n a ;(ii) for any v ∈ E i a , v = 0, we havelim | n |→∞ log k A ( n, a ) v k n = θ i . (iii) lim | n |→∞ n log ∠ (cid:16)L i ∈ I E iσ n a , L j ∈ J E jσ n a (cid:17) = 0 whenever I ∩ J = ∅ .Let P i a be the projection to E i a arising from (4.4). Denote by σ f the spectral measure for thesystem ( X ~s , h t ) with the test function f (assuming the system is uniquely ergodic). Now we canstate our theorem. Theorem 4.1. Let (Ω , P , σ ) be an invertible ergodic measure-preserving system satisfying condi-tions (C1)-(C3) above. Consider the cocycle A ( n, a ) defined by (4.3). Assume that (a) the Lyapunov spectrum satisfies θ > θ > > θ > . . . , and the two top exponents are simple (i.e. dim( E a ) = dim( E a ) = 1 for P -a.e. a ); (b) there exists a simple word q ∈ G k for some k ∈ N , such that all the entries of the matrix A ( q ) are strictly positive and P ([ q . q ]) > ; (c) there exist “good return words” { u j } mj =1 for ζ = ζ ( q ) (see Definition 3.3), such that { ~ℓ ( u j ) } mj =1 is a basis for R m ; (d) Let ℓ q ( a ) be the “negative” waiting time until the first appearance of q . q , i.e. ℓ q ( a ) = min { n ≥ σ − n a ∈ [ q . q ] } . Let P ( a | a + ) be the conditional distribution on the set of a ’s conditioned on the future a + = a a . . . We assume that there exist ε > and < C < ∞ such that (4.5) Z [ q . q ] (cid:13)(cid:13)(cid:13) A ( ℓ q ( a ) , σ − ℓ q ( a ) a ) (cid:13)(cid:13)(cid:13) ε d P ( a | a + ) ≤ C for all a + starting with q . HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 21 Then there exists γ > such that for P -a.e. a ∈ Ω the following holds:Let ( X + , T ) be the Bratteli-Vershik system corresponding to a + , which is uniquely ergodic. Let ( X ~s , h t ) be the suspension flow over ( X + , T ) under the piecewise-constant roof function determinedby ~s . Then for all β > and B > there exists r = r ( a , β, B ) > , such that for Lebesgue-a.e. ~s , with k ~s k = 1 and min j =1 , | P j a ( ~s ) | ≥ β , for any f ∈ Lip + w ( X ~s ) , (4.6) σ f ( B ( ω, r )) ≤ C ( a , k f k L ) · r γ for all ω ∈ [ B − , B ] and < r < r , with the constant depending only on a and k f k L . Remarks. 1. It is clear that condition (C1) follows from assumption (b), but we chose to state (C1) explicitly, since this is the condition which appears in the literature and implies uniqueergodicity. The unique ergodicity of the system ( X + , T ) for P -a.e. a under the given assumptionsis well-known and goes back to the work of Furstenberg [20] (see the beginning of Section 2).2. The assumption that q is a simple word ensures that occurrences of q do not overlap. Thenwe have(4.7) A ( ℓ q ( a ) , σ − ℓ q ( a ) a ) = A ( q ) A ( p ) A ( q ) , for some p ∈ G (possibly trivial). For our application, it will be easy to make sure that q issimple, as we show in Section 11, unlike in the paper [11], where additional efforts were neededto achieve the desired aims.The scheme of the proof is as follows: first we reduce the theorem to the case where all thesymbols have the form a n = qp n q . This is done by considering the first return map to the cylinderset [ q . q ]. Then we apply Proposition 3.7, with the goal to use Lemma 3.1. In order to achieve thedesired estimate, roughly speaking, we need to show that for P -a.e. sequence of substitutions, forLebesgue a.e. ~s , the distance from ω | ζ [ n ] ( v n ) | ~s to the nearest integer (for some choice of a goodreturn word v n which depends on n ) is bounded away from zero for a positive frequency of n ’s(uniformly in ω bounded away from zero and infinity). The proof splits into two parts, separatingthe two “almost every”. The first part is probabilistic, showing that certain assumptions on thesequence of substitutions ζ ( p n ) hold P -almost surely. In the second part we fix a typical sequence ζ ( p n ) and obtain estimates for a.e. ~s . This is done using the “Erd˝os-Kahane argument.”5. Reduction In this short section, we show that Theorem 4.1 reduces to the case when(5.1) a n = qp n q for all n ∈ Z , where q is a fixed graph with fixed Vershik ordering, such that its incidence matrix is strictlypositive, and p n is arbitrary. In the next theorem we use the same notation as in Theorem 4.1. Theorem 5.1. Let (Ω q , P , σ ) be an invertible ergodic measure-preserving system of the form (5.1)satisfying conditions (C1)-(C3) from Section 4. Consider the cocycle A ( n, a ) defined by (4.3).Assume that (a ′ ) the Lyapunov spectrum satisfies θ > θ > > θ > . . . , and the two top exponents are simple; (b ′ ) the substitution ζ = ζ ( q ) is such that its substitution matrix S ζ = Q has strictly positiveentries; (c ′ ) there exist “good return words” { u j } mj =1 for ζ = ζ ( q ) (see Definition 3.3), such that { ~ℓ ( u j ) } mj =1 is a basis for R m ; (d ′ ) there exist ε > and < C < ∞ such that (5.2) Z Ω q k A ( a ) k ε d P ( a | a + ) ≤ C for all a + . Then there exists γ > such that for P -a.e. a ∈ Ω q the following holds:Let ( X + , T ) be the Bratteli-Vershik system corresponding to a + , which is uniquely ergodic. Let ( X ~s , h t ) be the suspension flow over ( X + , T ) under the piecewise-constant roof function determinedby ~s . Then for all β > and B > there exists r = r ( a , β, B ) > , such that for Lebesgue-a.e. ~s , with k ~s k = 1 and min j =1 , | P j a ( ~s ) | ≥ β , for any f ∈ Lip + w ( X ~s ) , (5.3) σ f ( B ( ω, r )) ≤ C ( a , k f k L ) · r γ for all ω ∈ [ B − , B ] and < r < r , with the constant depending only on a and k f k L . Remark. If we assume that P is “quasi-Bernoulli”, i.e. it satisfies the “bounded distortionproperty” of [5], then (5.2) can be replaced by the “unconditional” estimate R Ω q k A ( a ) k ε d P ( a ) ≤ C . However, we prefer the current formulation. Proof of Theorem 4.1 assuming Theorem 5.1. Given an ergodic system (Ω , P , σ ) from the state-ment of Theorem 4.1, we consider the induced system on the cylinder set Ω q := [ q . q ]. Thensymbolically we can represent elements of Ω q as sequences satisfying (5.1). Denote by P q theinduced (conditional) measure on Ω q . Since P ([ q . q ]) > 0, standard results in Ergodic Theoryimply that the resulting induced system (Ω q , P q , σ ) is also ergodic and the associated cocyclehas the same properties of the Lyapunov spectrum (with the values of the Lyapunov exponentsmultiplied by 1 / P ([ q . q ])); that is, (a ′ ) holds.The properties (b ′ ) and (c ′ ) follow from (b) and (c)automatically. Finally, note that (5.2) is identical to (4.5). On the level of Bratteli-Vershik dia-grams this corresponds to the “aggregation-telescoping procedure”, which results in a naturallyisomorphic Bratteli-Vershik system. Observe also that a weakly-Lipschitz function on Ω inducesa weakly-Lipschitz function on Ω q without increase of the norm k f k L , see Section 2.1. Thus,Theorem 5.1 applies, and the reduction is complete. (cid:3) HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 23 The next five sections are devoted to the proof of Theorem 5.1. In the last Section 11 we returnto the setting of Theorem 4.1 and deduce Theorem 1.1 from it.6. Exponential tails For a ∈ Ω q we consider the sequence of substitutions ζ ( a n ), n ≥ 1. In view of (5.1), we have ζ ( a n ) = ζ ( q ) ζ ( p n ) ζ ( q ) . Recall that A ( p n ) = S tζ ( p n ) . Denote(6.1) W n = W n ( a ) := log k A ( a n ) k = log k Q t A ( p n ) Q t k . Since all the matrices in the product have non-negative integer entries and a.s. invertible, and thefirst and last one are equal to Q with strictly positive entries, we have W n > , n ≥ , for P -a.e. a . This will always be assumed below, without loss of generality. Proposition 6.1. Under the assumptions of Theorem 5.1, there exists a positive constant L suchthat for P -a.e. a , the following holds: for any δ > , for all N sufficiently large ( N ≥ N ( a , δ ) ), (6.2) max (X n ∈ Ψ W n : Ψ ⊂ { , . . . , N } , | Ψ | ≤ δN ) ≤ L · log(1 /δ ) · δN. We will prove the proposition at the end of the section, but first point out the following. Remark 6.2. It follows from (6.2) that for any e δ > , for all n sufficiently large, (6.3) W n ≤ e δn. Indeed, in (6.2) we just need to take δ > such that L · log(1 /δ ) · δ < e δ , and then Ψ = { n } ,which clearly satisfies the condition | Ψ | ≤ δN for N sufficiently large.As the referee pointed out, this also follows directly from the Birkhoff Ergodic Theorem, since n ( W + · · · + W n ) → R Ω q log k A ( a ) k d P < ∞ for a.e. a ∈ Ω q . Lemma 6.3. We have for all N and n , and for any (deterministic!) increasing sequence ≤ j < j < . . . < j n : (6.4) P " n X i =1 W j i ≥ Kn ≤ exp( − εKn/ for K ≥ Cε , where ε > and C > are the constants from (4.5). This is a standard large deviation result, but we provide a proof for completeness. Thanks toChris Hoffman who showed us the argument. Proof. Let X i = W j i − K . Then P [ P ni =1 W j i ≥ Kn ] = P (cid:2)P ni =1 X i ≥ (cid:3) . Observe that the valuesof W j , . . . W j n are determined by the “future” of σ a , that is, by ( σ a ) + = a a . . . , hence by (4.5)we have E (cid:2) e εW j | W j , . . . , W j n (cid:3) < C. Therefore,(6.5) E (cid:2) e εX | X , . . . , X n (cid:3) < Ce − εK ≤ e − εK/ , provided K ≥ ε − log C . Let S ℓ = P ni = n − ℓ +1 X i . Now, E (cid:2) e εS n (cid:3) = X b E (cid:2) e εS n | e εS n − = b (cid:3) · P (cid:2) e εS n − = b (cid:3) = X b b · E (cid:2) e εX n | e εS n − = b (cid:3) · P (cid:2) e εS n − = b (cid:3) ≤ e − εK/ X b b · P (cid:2) e εS n − = b (cid:3) = e − εK/ E (cid:2) e εS n − (cid:3) , taking (6.5) into account. Iterating the last inequality yields E (cid:2) e εS n (cid:3) ≤ e − εKn/ , and since P [ S n ≥ ≤ E (cid:2) e εS n (cid:3) , the estimate (6.4) is proved. (cid:3) Proof of Proposition 6.1. Consider the event W ( N, δ, K ) = max Ψ ⊂{ ,...,N }| Ψ |≤ δN X n ∈ Ψ W n ≥ K ( δN ) Then we have for K ≥ C/ε , P ( W ( N, δ, K )) ≤ X Ψ ⊂{ ,...,N }| Ψ |≤ δN P "X n ∈ Ψ W n ≥ K ( δN ) ≤ X i ≤ δN (cid:18) Ni (cid:19) e − εK ( δN ) / , in view of Lemma 6.3. By Stirling, there exists C ′ > X i ≤ δN (cid:18) Ni (cid:19) ≤ exp (cid:2) C ′ δ log(1 /δ ) N (cid:3) for δ < e − and all N > . Therefore, X i ≤ δN (cid:18) Ni (cid:19) ≤ exp[ − εK ( δN ) / 4] for K = 4 C ′ ε log(1 /δ ) , HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 25 whence, by Borel-Cantelli, the event W ( N, δ, L log(1 /δ )) does not occur for all N sufficientlylarge, with L = ε − max(4 C ′ , C ) , which means that condition (6.2) holds. (cid:3) Estimating twisted Birkhoff integrals In this section we continue to work with a P -generic 2-sided sequence a ∈ Ω q . Under theassumptions of Theorem 5.1, for P -a.e. a , the sequence of substitutions ζ ( a n ), n ∈ Z , satisfiesseveral conditions. First of all, we can assume that the point a is generic for the OseledetsTheorem; that is, assertions (i)-(iii) from Section 4 hold. We further assume that the conclusionsof Proposition 6.1 hold. Recall that ζ ( a n ) = ζ ( q ) ξ n ζ ( q ), where Q = S ζ is a strictly positivematrix. Below we denote by O Q (1) a generic constant which depends only on Q = S ( ζ ) andwhich may be different from line to line. Proposition 7.1. Suppose that the conditions of Theorem 5.1 are satisfied. Then for P -a.e. a ∈ Ω , for any η ∈ (0 , , there exists ℓ η = ℓ η ( a ) ∈ N , such that for all ℓ ≥ ℓ η and any boundedcylindrical function f ( ℓ ) of level ℓ , for any ( e , t ) ∈ X ~s , with e ∈ X + ( a ) , and ω ∈ R , (7.1) | S ( e ,t ) R ( f ( ℓ ) , ω ) | ≤ O Q (1) · k f ( ℓ ) k ∞ (cid:16) R / + R η Y ℓ +1 ≤ k< log R θ (cid:16) − c · max v ∈ GR ( ζ ) (cid:13)(cid:13) ω | ζ [ k ] ( v ) | ~s (cid:13)(cid:13) R / Z (cid:17)(cid:17) , for all R ≥ e θ ℓ . Remark 7.2. By (3.5) we have k ω | ζ [ n ] ( v ) | ~s k R / Z = kh ~ℓ ( v ) , ω ( S [ n ] ) t ~s ik R / Z = kh ~ℓ ( v ) , A ( n, a )( ω~s ) ik R / Z . In fact, our assumption (namely, condition (c ′ ) in Theorem 5.1) is that the substitution ζ possesses m good return words v , . . . , v m such that their population vectors ~ℓ ( v ) , . . . ~ℓ ( v m ) form a basisof R m . Observe that h ~ℓ ( v j ) , ~x i , for j = 1 , . . . , m , are the coordinates of a vector ~x ∈ R m withrespect to the basis dual to { ~ℓ ( v ) , . . . ~ℓ ( v m ) } . Let Γ be the free Abelian group generated by ~ℓ ( v ) , . . . ~ℓ ( v m ). Then Γ < Z m is a full rank lattice. Let b Γ be the dual lattice. Observe that C − ζ k ~x k R m / b Γ ≤ max j ≤ m kh ~ℓ ( v j ) , ~x ik R / Z ≤ C ζ k ~x k R m / b Γ , with C ζ > ~ℓ ( v ) , . . . ~ℓ ( v m ). Thus, estimating the product in (7.1) is equivalentto estimating(7.2) Y ℓ +1 ≤ k< log R θ (cid:16) − e c · (cid:13)(cid:13) A ( k, a )( ω~s ) (cid:13)(cid:13) R m / b Γ (cid:17) , making it similar to the expression appearing in the Veech criterion alluded to in Section 1.3.However, although the form of (7.2) may be more appealing, for technical reasons we prefer towork with the expression in (7.1). Proof of Proposition 7.1. Without loss of generality we can assume that f ( ℓ ) = f ( ℓ ) a for some a ∈ A , as in (3.29) and find e ′ and t ′ as in (3.28). Since ( e , t ) = h t ′ ( e ′ , | S ( e ,t ) R ( f ( ℓ ) a , ω ) | = (cid:12)(cid:12)(cid:12)Z R e − πiωτ f ( ℓ ) a ◦ h τ + t ′ ( e ′ , dτ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z R + t ′ t ′ e − πiωτ f ( ℓ ) a ◦ h τ ( e ′ , dτ (cid:12)(cid:12)(cid:12) . Recall that ~s ( ℓ ) = ( S [ ℓ ] ) t ~s , and we let s ( ℓ )max and s ( ℓ )max be the maximal and minimal components ofthe vector ~s ( ℓ ) , respectively. Note that | t ′ | ≤ s ( ℓ )max , so we obtain(7.3) (cid:12)(cid:12)(cid:12) S ( e ,t ) R ( f ( ℓ ) a , ω ) − S ( e ′ , R ( f ( ℓ ) a , ω ) (cid:12)(cid:12)(cid:12) ≤ k f ( ℓ ) k ∞ s ( ℓ )max . Next, consider x ( ℓ ) ∈ Y ( ℓ ) as in (3.30) and take the maximal N such that R ′ := | x ( ℓ ) [0 , N − | ~s ( ℓ ) ≤ R . Then | R − R ′ | ≤ s ( ℓ )max , hence(7.4) (cid:12)(cid:12)(cid:12) S ( e ′ , R ( f ( ℓ ) a , ω ) − S ( e ′ , R ′ ( f ( ℓ ) a , ω ) (cid:12)(cid:12)(cid:12) ≤ k f ( ℓ ) k ∞ s ( ℓ )max , and for S ( e ′ , R ′ ( f ( ℓ ) a , ω ) the formula in (3.30) applies (with R replaced by R ′ ). Thus, the combinederror in the above estimates (7.3), (7.4) is bounded by 3 k f ( ℓ ) k ∞ · s ( ℓ )max . By Oseledets Theorem,we can make sure that ℓ η is such that(7.5) (cid:12)(cid:12)(cid:12) ℓ − log k S [ ℓ ] k − θ (cid:12)(cid:12)(cid:12) ≤ θ η/ , for all ℓ ≥ ℓ η . Then s ( ℓ )max ≤ k S [ ℓ ] k ≤ e θ ℓ (1+ η/ ≤ e θ ℓ < R / , for ℓ ≥ ℓ η and R ≥ e θ ℓ . Taking O Q (1) ≥ 3, we thus guarantee that the first term in the right-hand side of (7.1), equal to O Q (1) · k f ( ℓ ) k ∞ R / , dominates the combined error. Thus it sufficesto consider the case of (3.30). Since R ′ ≤ R and | R − R ′ | ≤ s ( ℓ )max < R / , and ( R − R / ) ≥ R / for R ≥ 9, the proposition will follow from the following lemma. (cid:3) Lemma 7.3. Suppose that the assumptions of Proposition 7.1 are satisfied. Then for P -a.e. a ∈ Ω , for any η ∈ (0 , , there exists ℓ η = ℓ η ( a ) ∈ N , such that for all ℓ ≥ ℓ η and any boundedcylindrical function f ( ℓ ) of level ℓ , for any e ′ ∈ X + ( a ) such that h ( e ′ ) = ζ [ ℓ ] ( x [ ℓ ] ) , with x ( ℓ ) ∈ Y ( ℓ ) and ω ∈ R , (7.6) | S ( e ′ , R ( f ( ℓ ) , ω ) | ≤ O Q (1) · k f ( ℓ ) k ∞ R η Y ℓ +1 ≤ k< log R θ (cid:16) − c · max v ∈ GR ( ζ ) (cid:13)(cid:13) ω | ζ [ k ] ( v ) | ~s (cid:13)(cid:13) R / Z (cid:17) , HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 27 whenever R = (cid:12)(cid:12) x ( ℓ ) [0 , N − (cid:12)(cid:12) ~s ( ℓ ) ≥ e θ ℓ . Proof. Again we assume without loss of generality that f ( ℓ ) = f ( ℓ ) a , where f ( ℓ ) a ( e , t ) = 11 X ( ℓ ) a ψ ( ℓ ) a ( t ′ ),with t ′ ∈ [0 , s ( ℓ ) a ] and ψ ( ℓ ) a ∈ C ([0 , s ( ℓ ) ]), see Section 3.5 for details. Recall the formula (3.30) whichapplies here:(7.7) S ( e ′ , R ( f ( ℓ ) a , ω ) = b ψ ( ℓ ) a ( ω ) · Φ ~s ( ℓ ) a (cid:0) x ( ℓ ) [0 , N − , ω (cid:1) . First observe that(7.8) | b ψ ( ℓ ) a ( ω ) | ≤ k ψ ( ℓ ) a k ≤ k ψ ( ℓ ) a k ∞ s ( ℓ ) a ≤ k f ( ℓ ) a k ∞ s ( ℓ )max . Next we apply (3.34), which we copy here for convenience:(7.9) (cid:12)(cid:12) Φ ~s ( ℓ ) a (cid:0) x ( ℓ ) [0 , N − , ω (cid:1)(cid:12)(cid:12) ≤ n X j = ℓ k S [ ℓ +1 ,j ] k ·k S j +1 k · Y ℓ +1 ≤ k ≤ j − (cid:16) − c · max v ∈ GR ( ζ ) (cid:13)(cid:13) ω | ζ [ k ] ( v ) | ~s (cid:13)(cid:13) R / Z (cid:17) , where(7.10) min b ∈A | ζ [ ℓ +1 ,n ] ( b ) | ≤ N ≤ b ∈A | ζ [ ℓ +1 ,n +1] ( b ) | . By (4.2), we can assume that(7.11) k S j +1 k ≤ e jθ ( η/ for all j ≥ ℓ η . Further, note that(7.12) k S [ ℓ +1 ,j +1] k = k S [ ℓ +1 ,j ] Q S ξ j +1 Q k ≥ k S [ ℓ +1 ,j ] k , since Q has strictly positive entries, and hence all entries of Q S ξ j +1 Q are not less than m ≥ c ≤ / 4, we obtain that the sum in (7.9) is boundedabove by O Q (1) times the last, n -th term, yielding(7.13) (cid:12)(cid:12) Φ ~s ( ℓ ) a (cid:0) x ( ℓ ) [0 , N − , ω (cid:1)(cid:12)(cid:12) < O Q (1) · k S [ ℓ +1 ,n ] k · e nθ ( η/ · Y ℓ +1 ≤ k ≤ n − (cid:16) − c · max v ∈ GR ( ζ ) (cid:13)(cid:13) ω | ζ [ k ] ( v ) | ~s (cid:13)(cid:13) R / Z (cid:17) . This, together with (7.7), (7.8), is already very close to the desired (7.6), but we need to relate N, n , and R . First observe that(7.14) R = (cid:12)(cid:12) x ( ℓ ) [0 , N − (cid:12)(cid:12) ~s ( ℓ ) ∈ [ N s ( ℓ )min , N s ( ℓ )max ] . Note that(7.15) s ( ℓ )max ≤ col( Q t ) · s ( ℓ )min , since s ( ℓ )max and s ( ℓ )max are respectively the maximal and minimal components of ~s ( ℓ ) = ( S [ ℓ ] ) t ~s = Q t ( S [ ℓ − ) t ~s (recall that S ℓ = Q by assumption). Further, by (7.10) and (3.17), N ≥ (col( Q t )) − · k S [ ℓ +1 ,n ] k , since | ζ [ ℓ +1 ,n ] ( b ) | is a column sum of S [ ℓ +1 ,n ] , a matrix which starts and ends with Q . Therefore,(7.16) R ≥ N s ( ℓ )min ≥ (col( Q t )) − · k S [ ℓ +1 ,n ] k · s ( ℓ )max . Comparing (7.7), (7.8), and (7.13), we see that in order to conclude the proof of (7.6), it remainsto show, first, that(7.17) R η ≥ e nθ ( η/ , and second, that(7.18) n ≥ (log R ) / (3 θ ) . Since s ( ℓ )max ≥ (col( Q t )) − k S [ ℓ ] k , we obtain from (7.16) and (7.5) that R ≥ (col( Q t )) − · k S [ ℓ +1 ,n ] k · | S [ ℓ ] k ≥ (col( Q t )) − · k S [ n ] k ≥ (col( Q t )) − · e θ n (1 − η/ , confirming (7.17) once ℓ , and hence n is sufficiently large. On the other hand, by the upper boundin (7.10) and (7.14), R ≤ N s ( ℓ )max ≤ k S [ ℓ ] k · k S [ ℓ +1 ,n ] k . Since all matrices involved begin and end with Q , it is easy to see that k S [ ℓ ] k · k S [ ℓ +1 ,n ] k ≤ col( Q t ) · k S [ ℓ ] · S [ ℓ +1 ,n ] k = col( Q t ) · k S [ n ] k , and then (7.5) yields that R ≤ O Q (1) · e θ n (1+ η/ , which certainly guarantees (7.18), once ℓ , and hence n , is sufficiently large. Now the lemma andthe proposition are proved completely. (cid:3) Linear algebra and the choice of good return words In this section, as well as the next one, we continue to work with a P -generic 2-sided sequence a ∈ Ω q . In view of the assumption (a ′ ) of Theorem 5.1, we can fix unit basis vectors ~e ( n ) j , j = 1 , E jσ n a , j = 1 , n ≥ 0, such that(8.1) A ( n, a ) ~e (0) j = A ( n, j ) ~e ( n ) j for some A ( n, j ) > . By (ii) in Oseledets Theorem, we have n log A ( n, j ) → θ j , j = 1 , { ~e ( n ) j } j =1 of the unstablesubspace. All the matrices A ( n, a ) are non-negative. Thus, k A ( n, a ) ~x k ≤ k A ( n, a ) ~x |·| k , HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 29 hence ~e ( n )1 ∈ R m + (the positive cone) for all n ≥ 0, and so(8.2) ~e ( n )1 ∈ A ( a n ) R m + = Q t A ( p n ) Q t R m + ⊂ Q t R m + . On the other hand, under our assumptions the image of the positive cone A ( n, a ) R m + shrinksto a single direction exponentially fast. (The fact that the cone shrinks to a single direction isequivalent to unique ergodicity, see (2.2) and Veech [32, 33].) It follows that the basis vector ~e ( n )2 does not lie in R m + for all n , otherwise we would get a contradiction with (iii) in OseledetsTheorem. Combined with (8.2), this implies that the angle between ~e ( n )1 and ~e ( n )2 is bounded awayfrom zero by a constant depending only on Q .We will next need an elementary fact from linear algebra. Lemma 8.1. Let B = { ~x j } j ≤ m be a basis of R m , and let { ~ξ , . . . , ~ξ r } ⊂ R m be a linearly indepen-dent set, with r ≤ m . Then there exists a subset { x i } i ∈ I ⊂ B of cardinality r such that | D I | := (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) h ~x i , ~ξ j i (cid:17) i ∈ I,j ≤ r (cid:12)(cid:12)(cid:12)(cid:12) ≥ C B k ~ξ ∧ · · · ∧ ~ξ r k , where C B depends only on the basis B .Proof. Let T be the linear isomorphism which takes the standard basis { ~ e j } j ≤ m of R m into B .Then D I = det (cid:16) h ~x i , ~ξ j i (cid:17) i ∈ I,j ≤ r = det (cid:16) hT ~ e i , ~ξ j i (cid:17) i ∈ I,j ≤ r = det (cid:16) h ~ e i , T ∗ ~ξ j i (cid:17) i ∈ I,j ≤ r The latter determinant is the order- r minor of the matrix whose columns are T ∗ ~ξ j , j = 1 , . . . , r ,corresponding to the rows indexed by I . Thus, X I = r | D I | = kT ∗ ~ξ ∧ · · · ∧ T ∗ ~ξ r k ≥ (cid:13)(cid:13) ( V r T ∗ ) − (cid:13)(cid:13) − k ~ξ ∧ · · · ∧ ~ξ r k . We are using here that T ∗ is invertible, hence its exterior power is invertible. Thus,max {| D I | : I = r } ≥ (cid:13)(cid:13) ( V r T ∗ ) − (cid:13)(cid:13) − / (cid:18) mr (cid:19) − / k ~ξ ∧ · · · ∧ ~ξ r k , and the proof is complete. (cid:3) We now return to our theorem, in which r = 2. Let { u j } j ≤ m be the good return words from theAssumption (c ′ ) of Theorem 5.1. We will choose a sequence of words v n ∈ { u j } j ≤ m , dependingon our generic a ∈ Ω. For n ≥ Θ n := A ( n, h ~ℓ ( v n ) , ~e ( n )1 i A ( n, h ~ℓ ( v n ) , ~e ( n )2 i A ( n + 1 , h ~ℓ ( v n +1 ) , ~e ( n +1)1 i A ( n + 1 , h ~ℓ ( v n +1 ) , ~e ( n +1)2 i ! . Below C ζ denotes a constant depending only on the substitution ζ = ζ ( q ). Lemma 8.2. For P -a.e. a ∈ Ω we can choose the words v n ∈ { u j } j ≤ m , so that for all n ≥ , (8.4) k Θ − n k ∞ ≤ C ζ · max { A ( n + j, i ); j = 0 , i = 1 , } A ( n, A ( n + 1 , and (8.5) k Θ n +1 Θ − n k ∞ ≤ C ζ · max j =0 , , A ( n + j, A ( n, · max j =0 , , A ( n + j, A ( n, . Proof. We are going to choose v n inductively. Pick v arbitrarily, and suppose v , . . . , v n havebeen chosen. For i ≤ m consider∆ i := det A ( n, h ~ℓ ( v n ) , ~e ( n )1 i A ( n, h ~ℓ ( v n ) , ~e ( n )2 i A ( n + 1 , h ~ℓ ( u i ) , ~e ( n +1)1 i A ( n + 1 , h ~ℓ ( u i ) , ~e ( n +1)2 i ! . Observe that(8.6) det h ~ℓ ( u i ) , ~e ( n +1)1 i ∆ i h ~ℓ ( u j ) , ~e ( n +1)1 i ∆ j ! = A ( n, A ( n + 1 , h ~ℓ ( v n ) , ~e ( n )1 i D ij , where D ij := det h ~ℓ ( u i ) , ~e ( n +1)1 i h ~ℓ ( u i ) , ~e ( n +1)2 ih ~ℓ ( u j ) , ~e ( n +1)1 i h ~ℓ ( u j ) , ~e ( n +1)2 i ! . Note that ~ξ := ~e ( n +1)1 ∈ Q t R m + and ~ξ := ~e ( n +1)2 R m + by the comments above. Thus, the anglebetween ~ξ and ~ξ is bounded away from zero, uniformly in n . Hence we can apply Lemma 8.1to these vectors and find i = j such that | D ij | ≥ c > , independent of n . Note that for all i ≤ m ,(8.7) 0 < c ≤ |h ~ℓ ( u i ) , ~e ( n )1 i| ≤ C := max i ≤ m k ~ℓ ( u i ) k < ∞ for some positive constant c = c ( Q ) independent of n , since ~ℓ ( u i ) , i ≤ m, are positive integervectors. It follows from (8.6) and (8.7) thatmax i | ∆ i | ≥ c c C A ( n, A ( n + 1 , . We choose v n +1 ∈ { u i } i ≤ m to maximize | ∆ i | . Denote ∆ ( n ) = det( Θ n ). As a result, we will havefor all n ≥ | ∆ ( n ) | ≥ c c C A ( n, A ( n + 1 , , which implies (8.4). Also, a direct calculation, combined with (8.7) and (8.8), yields (8.5). (cid:3) HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 31 Corollary 8.3. For P -a.e. a ∈ Ω q we can choose the words v n ∈ { u j } j ≤ m , so that for any δ > there exists n ∈ N such that for all n ≥ n , (8.9) k Θ − n k ∞ ≤ C ζ exp[ − ( θ − δ ) n ] and (8.10) k Θ n +1 Θ − n k ∞ ≤ C ζ exp[2( W n + W n +1 )] , where C ζ is the constant from Lemma 8.2 and W n are defined in (6.1).Proof. This is a combination of the last lemma and Oseledets Theorem. First we prove (8.9). ByOseledets Theorem, for P -a.e. a ∈ Ω q , for all n sufficiently large,exp[( θ i − δ / n ] ≤ A ( n, i ) ≤ exp[( θ i + δ / n ] , i = 1 , . We will use (8.4), where clearly the maximum in the numerator is (eventually) attained for i = 1,to obtain for n sufficiently large: k Θ − n k ∞ ≤ C ζ exp[( θ + δ / n + 1 − ( θ − δ / n − ( θ − δ / n + 1]= C ζ exp[( θ − θ + δ / − ( θ − δ / n ] . For n sufficiently large, θ − θ + δ / ≤ ( δ / n, and (8.9) follows. Next, let us verify (8.10). Equation (8.1) implies that A ( n + 1 , j ) ~e ( n +1) j = A ( a n ) A ( n, j ) ~e ( n ) j , j = 1 , , hence A ( n + 1 , j ) /A ( n, j ) ≤ e W n by (6.1). Now (8.10) follows from (8.5). (cid:3) Beginning of the proof of Theorem 5.1 Now we proceed with the proof of Theorem 5.1. As before, we fix a P -generic point a ∈ Ω q .Under the assumptions of Theorem 5.1,(9.1) ~s = X j =1 a j ~e (0) j + P st a ~s, where a j = P j a ( ~s ) and P st a is the projection to the stable subspace E a ⊕ · · · ⊕ E r a in (4.4). Recallthat ~s ∈ ∆ m := { ~s ∈ R m + : k ~s k = 1 } . Our goal is to prove that for all β > B > 1, for a.e. ~s , with min j =1 , | a j | ≥ β ,(9.2) σ f ( B ( ω, r )) ≤ C ( a , k f k L ) · r γ for ω ∈ [ B − , B ] and 0 < r ≤ r ( a , β, B ) . Fix β > B > ~s in the estimate is“hidden” in σ f , which is the spectral measure of the suspension flow corresponding to the rooffunction given by ~s . Fix the sequence of good return words v n from Lemma 8.2. For n ∈ N and ω ∈ [ B − , B ], let(9.3) ω | ζ [ n ] ( v n ) | ~s = K n + ε n , where K n ∈ N , | ε n | ≤ / , so that k ω | ζ [ n ] ( v n ) | ~s k R / Z = | ε n | . We should keep in mind that K n and ε n depend on ω and on ~s , although this is suppressed innotation to avoid clutter. Given β, ̺, δ > B > 1, define E N ( ̺, δ, β, B ) := n ~s ∈ ∆ m : min j =1 , | P j a ( ~s ) | ≥ β and ∃ ω ∈ [ B − , B ]such that card { n ≤ N : | ε n | ≥ ̺ } < δN o . and E ( ̺, δ, β, B ) := ∞ \ N =1 ∞ [ N = N E N ( ̺, δ, β, B ) . Proposition 9.1. There exist ̺ > such that for P -a.e. a ∈ Ω q we have ∀ ǫ > , ∃ δ > , ∀ β > , ∀ B > δ < δ = ⇒ dim H ( E ( ̺, δ, β, B )) < m − ǫ. In the remaining part of this section, we derive Theorem 5.1 from Proposition 9.1. Then in thenext section we use the “Erd˝os-Kahane argument” to prove the proposition. Proof of Theorem 5.1 assuming Proposition 9.1. In view of Lemma 3.1, it suffices to show(9.4) | S ( e ,t ) R ( f, ω ) | ≤ e C ( a , k f k L ) · R − γ/ , for R ≥ R ( a , β, B ) , for some γ ∈ (0 , e , t ) ∈ X ~s . We will specify γ at the end of the proof, see (9.10).Since f is weakly Lipschitz on X ~s (see Section 2.1), for almost every a we can approximate f ,for any ℓ ∈ N , by a function f ( ℓ ) , which is cylindrical of level ℓ , and has sup-norm bounded by k f k ∞ , so that k f − f ( ℓ ) k ∞ ≤ k f k L · ν + ([ e . . . e n ]) . We can do this simply taking f ( ℓ ) ( e , t ) := f ( e ( ℓ ) , t ), where e ( ℓ ) agrees with e down to level ℓ afterwhich it is extended to infinity in any fixed way. By (2.6), (2.3), and (2.2), we havelim n →∞ log ν + ([ e . . . e n ]) n = − θ , P -almost surely, hence for ℓ sufficiently large we have(9.5) k f − f ( ℓ ) k ∞ ≤ k f k L · e − θ ℓ . Recall that S ( e ,t ) R ( f, ω ) = R R e − πiωt f ◦ h τ ( e , t ) dτ . Let(9.6) ℓ := (cid:22) γ log Rθ (cid:23) . HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 33 Then (9.5) yields | S ( e ,t ) R ( f, ω ) − S ( e ,t ) R ( f ( ℓ ) , ω ) | ≤ R · k f k L · e − θ ℓ ≤ e θ / · k f k L · R − γ . Thus, it is enough to obtain (9.4), with f replaced by f ( ℓ ) . For the latter, we can apply Proposi-tion 7.1. Recall the inequality (7.1), using the sequence of good return words { v n } and η = γ/ | S ( e ,t ) R ( f ( ℓ ) , ω ) | ≤ O Q (1) · k f ( ℓ ) k ∞ (cid:16) R / + R γ/ Y ℓ +1 ≤ n< log R θ (cid:0) − c · k ω | ζ [ n ] ( v n ) | ~s k R / Z (cid:1)(cid:17) , for ℓ ≥ ℓ γ and all R ≥ e θ ℓ . We can ensure that ℓ ≥ ℓ γ by taking R sufficiently large, and R ≥ e θ ℓ will follow from (9.6) if γ ≤ / 16. Since our goal is (9.4), we can discard the R / termimmediately. Now choose ̺ > δ > H ( E ( ̺, δ, β, B )) 1. It is enough to verify(9.7) Y ℓ +1 ≤ n< log R θ (cid:0) − c · k ω | ζ [ n ] ( v n ) | ~s k R / Z (cid:1) ≤ O a , k f k L (1) · R − γ , ω ∈ [ B − , B ] , for R ≥ R ( a , β, B ) , for all vectors ~s ∈ ∆ m \ E ( ̺, δ, β, B ), for which min j =1 , | P j a ( ~s ) | ≥ β , thusobtaining an even stronger than ‘almost every ~s ’ statement.By definition, ~s 6∈ E ( ̺, δ, β, B ) means that there exists N = N ( a , β, B ) ∈ N such that ~s E N ( ̺, δ, β, B ) for all N ≥ N . Let(9.8) N = (cid:22) log R θ (cid:23) and R = e θ ( N +1) . Then R ≥ R implies N ≥ N , and the product in (9.7) is less than or equal to N − Y n = ℓ +1 (1 − c | ε n | ) , where we also use (9.3). By definition, ~s E N ( ̺, δ, β, B ) means that there are at least ⌈ δN ⌉ numbers n ∈ { , . . . , N } with | ε k n | ≥ ̺, hence the left-hand side of (9.7) is bounded above by(1 − c ̺ ) δN − − ℓ . Recalling (9.8) and (9.6), we see that(9.9) (1 − c ̺ ) δN − − ℓ ≤ O (1) · (1 − c ̺ ) ( δ log R ) / (8 θ ) ≤ O (1) · R − γ , provided that(9.10) γ ≤ min n δ , − δ log(1 − c ̺ )8 θ o , with the two conditions for γ needed for the left and right inequality in (9.9) correspondingly.Now the proof of (9.7) is complete, and it remains to verify Proposition 9.1 to conclude the proofof Theorem 5.1. (cid:3) The Erd˝os-Kahane method and the conclusion of the proof of Theorem 5.1 In this section, we prove Proposition 9.1. We need some preparation first. Fix β ∈ (0 , 1) andsuppose that ~s ∈ ∆ m is such that min j =1 , | P j a ( ~s ) | ≥ β . Further, let B > ω ∈ [ B − , B ].Recall (9.1) and (9.3). In view of (8.1), we have for n ≥ 1, denoting ξ n = h ~ℓ ( v n ) , ( S [ n ] ) t P st a ~s i : | ζ [ n ] ( v n ) | ~s = h ~ℓ ( ζ [ n ] ( v n )) , ~s i = h S [ n ] ~ℓ ( v n ) , ~s i = h ~ℓ ( v n ) , ( S [ n ] ) t ~e (0) j i = X j =1 a j A ( n, j ) h ~ℓ ( v n ) , ~e ( n ) j i + ξ n . (10.1)By the Assumption (a ′ ) of Theorem 5.1, for P -a.e. a ∈ Ω q , we have lim sup n →∞ n − log k ( S [ n ] ) t P st a k ≤ θ < 0, hence(10.2) lim sup n →∞ n − log | ξ n | ≤ θ < . Now let ~s ∈ ∆ m and ω ∈ [ B − , B ] be from the definition of E N ( ̺, δ, β, B ). Recall (9.3), anddenote ~a = a a ! , ~K n = K n K n +1 ! , ~ε n = ε n ε n +1 ! , ~ξ n = ξ n ξ n +1 ! . We need the matrices Θ n defined in (8.3): Θ n = A ( n, h ~ℓ ( v n ) , ~e ( n )1 i A ( n, h ~ℓ ( v n ) , ~e ( n )2 i A ( n + 1 , h ~ℓ ( v n +1 ) , ~e ( n +1)1 i A ( n + 1 , h ~ℓ ( v n +1 ) , ~e ( n +1)2 i ! . The equations (9.3) for n, n + 1, in view of (10.1), combine into ω Θ n ~a = ~K n + ~ε n − ω~ξ n , hence(10.3) ~a = ω − Θ − n ( ~K n + ~ε n − ω~ξ n ) . It follows that(10.4) a j = ω − [ Θ − n ( ~K n + ~ε n − ω~ξ n )] j , j = 1 , , where [ · ] j denotes j -th component of a vector. Observe that(10.5) 0 < β ≤ | a | , | a | ≤ C a , where the upper bound comes from the fact that k ~s k = 1 and the angles between Lyapunovsubspaces at a depend on a . Choose δ > θ − δ > θ + δ < 0. Note that k ~ε n k ∞ ≤ for all n , and for n ≥ n ( a ),(10.6) | ξ k n | ≤ e ( θ + δ ) k n ≤ e ( θ + δ ) n . HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 35 Since | ω | ≤ B , we have k ω~ξ n k ∞ ≤ / n ≥ n ( a ) + C Lyap · log B, where C Lyap = | θ + δ | − . Here and below we denote by n ( a ) a generic integer constant depending on a , and possibly alsoon the Lyapunov spectrum (below it will also depend on β ), and by C Lyap a constant whichdepends only on the Lyapunov spectrum. Similarly, C a ≥ a ,and possibly also on the Lyapunov spectrum. These constants may be different from line to line.Thus, k ~ε n − ω~ξ n k ∞ ≤ n , so (10.4), the lower bound for ω , and (8.9) yieldfor j = 1 , | a j − ω − ( Θ − n ~K n ) j | ≤ B k Θ − n k ∞ ≤ BC ζ e − ( θ − δ ) k n ≤ BC ζ e − ( θ − δ ) n , for n ≥ n ( a ) . (10.7)In view of (10.5) and (10.7), we obtain for j = 1 , < β/ ≤ | ω − ( Θ − n ~K n ) j | ≤ C a , for n ≥ n ( a , β ) + C Lyap · log B. From these bounds and (10.7), we obtain, again for n ≥ n ( a , β ) + C Lyap · log B : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a − [ Θ − n ~K n ] [ Θ − n ~K n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | a − ω − ( Θ − n ~K n ) || a | + | ω − ( Θ − n ~K n ) | · | a − ω − ( Θ − n ~K n ) || a | · | ω − ( Θ − n ~K n ) |≤ C ( a ) β − BC ζ exp[ − ( θ − δ ) n ] . (10.8)On the other hand, comparing (10.3) for n and n + 1 yields ~K n +1 + ~ε n +1 − ω~ξ n +1 = Θ n +1 Θ − n [ ~K n + ~ε n − ω~ξ n ] , hence, using | ω | ≤ B , we obtain k ~K n +1 − Θ n +1 Θ − n ~K n k ∞ ≤ k ~ε n +1 k ∞ + B k ~ξ n +1 k ∞ + k Θ n +1 Θ − n k ∞ ( k ~ε n k ∞ + B k ~ξ n k ∞ ) . This implies, in view of (8.10), for n ≥ n ( a ): (cid:12)(cid:12)(cid:12) K n +2 − [ Θ n +1 Θ − n ~K n ] (cid:12)(cid:12)(cid:12) ≤ (1 + C ζ exp[2( W n + W n +1 )] ×× (max {| ε n | , | ε n +1 | , | ε n +2 |} + B max {| ξ n | , | ξ n +1 | , | ξ n +2 |} ) . (10.9)Let(10.10) M n := 1 + C ζ exp[2( W n + W n +1 )] and ρ n = 14 M n . Lemma 10.1. For all n ≥ n ( a , β ) + C Lyap log B , we have the following, independent of ω ∈ [ B − , B ] and ~s ∈ ∆ m , satisfying | a j | = | P j a ( ~s ) | ≥ β > : (i) Given K n , K n +1 , there are at most M n + 1 possibilities for the integer K n +2 ; (ii) if max {| ε n | , | ε n +1 | , | ε n +2 |} < ρ n , then K n +2 is uniquely determined by K n , K n +1 . Proof. For part (i), we just use that k ~ε n k ∞ ≤ for all n and k ~ξ n k ∞ ≤ (2 B ) − for n ≥ n ( a ) + C Lyap log B , and that the number of integer points in an interval of length 2 M n is at most 2 M n +1.For part (ii), we claim that K n +2 belongs to a neighborhood of radius less than , centered at[ Θ n +1 Θ − n ~K n ] , under the given assumptions, for n sufficiently large. We have M n ρ n = 1 / 4, soit remains to make sure that max {| ξ n | , | ξ n +1 | , | ξ n +2 |} ≤ ρ n /B for n sufficiently large. Note that for any e δ > ρ n ≥ e − e δn , for n ≥ n ( a ) . Taking e δ < | θ + δ | and combining the last inequality with (10.6) implies the desired claim. (cid:3) Proof of Proposition 9.1. Let e E N ( δ, β, B ) be defined by e E N ( δ, β, B ) := n ~s ∈ ∆ m : min j =1 , | P j a ( ~s ) | ≥ β and ∃ ω ∈ [ B − , B ] such thatcard { n ≤ N : max {| ε n | , | ε n +1 | , | ε n +2 |} ≥ ρ n } < δN o . First we claim that P -almost surely,(10.11) e E N ( δ, β, B ) ⊃ E N ( ̺, δ/ , β, B )for N ≥ N ( a ), where(10.12) ̺ = (1 / C ζ e K ) − , with K = 25 L log(1 /δ ) . Here C ζ is from Lemma 8.2 and L is from Proposition 6.1. Suppose ~s e E N ( δ, β, B ). Then forall ω ∈ [ B − , B ] there exists a subset Γ N ⊂ { , . . . , N } of cardinality ≥ δN/ | ε k n | ≥ ρ n for all k n ∈ Γ N . Observe that there are fewer than δN/ n ≤ N for which W n + W n +1 > K , for N ≥ N ( a ).Indeed, otherwise we can find Ψ ⊂ { , . . . , N } , with | Ψ | ≥ δN/ 12, such that W n > K/ n ∈ Ψ,hence X { W n : n ∈ Ψ } ≥ KδN/ , which contradicts (6.2) for K > L log(1 /δ ). In view of (10.10) and (10.12), it follows thatcard (cid:8) n ∈ Γ N : ρ n ≥ ̺ (cid:9) ≥ δN/ . Thus ~s E N ( ̺, δ/ , β, B ) which confirms (10.11).It follows that it is enough to estimate the dimension of e E := e E ( δ, β, B ) := ∞ \ N =1 ∞ [ N = N e E N ( δ, β, B ) . HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 37 Suppose ~s ∈ e E N := e E N ( δ, β, B ); choose ω from the definition of e E N , and find the correspondingsequence K n , ε n . In order to prove that dim H ( e E ) < m − ε , it is enough to show that the set of a /a corresponding to ~s ∈ e E has Hausdorff dimension smaller than ε . We have from (10.8) that a /a is covered by an interval of radius(10.13) C ( a , β ) · B exp[ − ( θ − δ ) N ] , for N ≥ N ( a , β ) + C Lyap log B, centered at [ Θ − n ~K n ] / [ Θ − n ~K n ] . Thus we need to estimate the number of sequences K n , n ≤ N ,which may arise. Let Ψ N be the set of n ∈ { , . . . , N } for which we have max {| ε n | , | ε n +1 | , | ε n +2 |} ≥ ρ n . By the definition of e E N we have | Ψ N | < δN . There are P i<δN (cid:0) Ni (cid:1) such sets. For a fixed Ψ N the number of possible sequences { K n } is at most B N := Y n ∈ Ψ N (2 M n + 1) , times the number of “beginnings” K , . . . , K n , by Lemma 10.1. The number of possible “begin-nings” is bounded, independent of N by a constant depending on β and B , in view of the a prioribounds on ω and ~s . By the definition of M n and (6.2), we have, for N sufficiently large: B N . exp C ′′ X n ∈ Ψ N ( W n + W n +1 ) ≤ exp h e L log(1 /δ )( δN ) i . Thus, by (10.13), the number of balls of radius O β,B (1) e − ( θ − δ ) N needed to cover e E is at most(10.14) O β,B (1) · X i<δN (cid:18) Ni (cid:19) exp h e L log(1 /δ )( δN ) i ≤ O β,B (1) · exp h ( e L + C ′ ) log(1 /δ )( δN ) i , using (6.6) in the last inequality. Since δ log(1 /δ ) → δ → 0, we can choose δ > δ < δ implies h ( e L + C ′ ) log(1 /δ )( δN ) i < ǫ ( θ − δ ) N, whence e E has Hausdorff dimension less than ǫ , as desired. The proof of Proposition 9.1, and henceof Theorem 5.1, is complete. (cid:3) Derivation of Theorem 1.1 from Theorem 4.1 Consider our surface M of genus 2. By the results of [9, Section 4] there is a correspondencebetween almost every translation flow and an element a ∈ Ω (space of 2-sided Markov compacta),such that the (uniquely ergodic) flow is measure-theoretically conjugate to the uniquely ergodicflow ( X ( a ) , h + t ) and hence to the suspension flow ( X ~s , h t ) over the Vershik map ( X + ( a ) , T ) withthe roof function corresponding to an appropriate vector ~s , see Lemma 2.1. By construction (see[9]), this correspondence intertwines the Teichm¨uller flow on the space of Abelian differentials anda measure-preserving system (Ω , P , σ ), as considered at the beginning of our Section 4. The key point here is that the Masur-Veech measure on the space of abelian differentials is taken, underthis correpondence, to a measure mutually absolute continuous with the product of the measure P + on Ω + and the Lebesgue measure on the 3-dimensional set of possible vectors ~s defining thesuspension.More precisely: as is well-known, the translation flow on the surface can be realized as asuspension flow over an interval exchange transformation (IET), see [40] for details. Veech [32]constructed, for any connected component of a stratum H , a measurable finite-to-one map fromthe space V ( R ) of zippered rectangles corresponding to the Rauzy class R , to H , which intertwinesthe Teichm¨uller flow on H and a renormalization flow P t that Veech defined on V ( R ). Observethat in our case the stratum H (2) is connected and corresponds to the Rauzy class of the IETwith permutation (4 , , , V ( R ) to H is not bijective (itcorresponds to passing from absolute to relative real cohomologies in the manifold M ), but in ourcase the kernel is trivial, since the manifold has only one singularity and therefore there are nosaddle connections. For background and complete details the reader is referred to [9] and [40].Section 4.3 of [9] constructs the symbolic coding of the flow P t on V ( R ), namely, a map(11.1) Ξ R : ( V ( R ) , e µ ) → (Ω , P ) , defined almost everywhere, where e µ is the pull-back of the Masur-Veech measure µ from H and(Ω , P ) is a space of Markov compacta. The first return map of the flow P t for an appropriatePoincar´e section is mapped by Ξ R to the shift map σ on (Ω , P ). This correspondence maps theRauzy-Veech cocycle over the Teichm¨uller flow into the renormalization cocycle for the Markovcompacta. Moreover, the map Ξ R induces a map defined for a.e. X ∈ V ( R ), from the correspond-ing Riemann surface M ( X ) to a Markov compactum X ( a ) ∈ Ω, intertwining their vertical andhorizontal flows.Now let f be a Lipschitz function on M with an abelian differential ω . Under the symboliccoding from [9], it is mapped into a weakly Lipschitz function on X ( a ) and then to a weaklyLipschitz function f ~s on X ~s with ~s given by Lemma 2.1. By definition, its norm k f ~s k L is dominatedby k f k L for all ~s . Once we check all the assumptions, we can apply Theorem 4.1 and obtain theH¨older property of the spectrum of the suspension flow for P -a.e. a ∈ Ω, for Lebesgue-a.e. ~s , whichin view of the mutual absolute continuity indicated above, is equivalent to the H¨older propertyfor the flow h + t , as desired. Note that the dependence of r on β (determined by a and ~s ) inTheorem 4.1 will be subsumed by the dependence of r on a in Theorem 1.1.In order to reduce Theorem 1.1 to Theorem 4.1, we must now check that the assumptions ofTheorem 4.1 hold for the left shift on the space of Markov compacta endowed with the push-forward of the Masur-Veech smooth measure under the isomorphism of [9]. It is clear thatcondition (C1) follows from (b), which we discuss below. Condition (C2) holds because therenormalization matrices in the Rauzy-Veech induction all have determinant ± 1. Condition (C3)holds by a theorem of Zorich [41]. The condition (a) on the Lyapunov spectrum from Theorem 4.1 HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 39 follows from results of Forni [18] in our case (later Avila and Viana [5] proved this for an arbitrarygenus ≥ λ, π ), where π is the permutation of m subintervals and λ is the vectorof their lengths. The well-known Rauzy induction (operations “a” and “b”) proceeds by inducingon a smaller interval, so that the first return map is again an exchange of m intervals. The Rauzygraph is a directed labeled graph, whose vertices are permutations of IET’s and the edges leadto permutations obtained by applying one of the operations. Moreover, the edges are labeledby the type of the operation (“a” or “b”). As is well-known, for almost every IET, there is acorresponding infinite path in the Rauzy graph, and the length of the interval on which we inducetends to zero. For any finite “block” of this path, we have a pair of intervals J ⊂ I and IET’son them, denoted T I and T J , such that both are exchanges of m intervals and T J is the firstreturn map of T I to J . Let I , . . . , I m be the subintervals of the exchange T I and J , . . . , J m thesubintervals of the exchange T J . Let r i be the return time for the interval J i into J under T I ,that is, r i = min { k > T kI J i ⊂ J } . Represent I as a Rokhlin tower over the subset J and itsinduced map T J , and write I = G i =1 ,...,m,k =0 ,...,r i − T k J i . By construction, each of the “floors” of our tower, that is, each of the subintervals T kI J i , is asubset of some, of course, unique, subinterval of the initial exchange, and we define an integer n ( i, k ) by the formula T kI J i ⊂ I n ( i,k ) . To the pair I, J we now assign a substitution ζ IJ on the alphabet { , . . . , m } by the formula(11.2) ζ IJ : i → n ( i, n ( i, . . . n ( i, r i − . This is the sequence of substitutions arising from the Bratteli-Vershik realization of an IET. Remark. Veech [35] proved that if the IET satisfies the Keane condition and is uniquely ergodic,then it is uniquely determined by the sequence of renormalization matrices arising from the Rauzy-Veech induction. This implies that the map (11.1) is injective on the set of zippered rectanglesof full measure (those which correspond to uniquely ergodic horizontal and vertical flows).Condition (c) is verified in the next lemma. Words obtained from finite paths in the Rauzygraph will be called admissible. Lemma 11.1. There exists an admissible word q , which is simple , whose associated matrix A ( q ) has strictly positive entries, and the corresponding substitution ζ , with Q = S ζ = A ( q ) t having theproperty that there exist good return words u , . . . , u m ∈ GR ( ζ ) , such that { ~ℓ ( u j ) : j ≤ m } isa basis of R m . We start with a preliminary claim. Lemma 11.2. There exists a letter c and an admissible simple word W such that η ( j ) starts with c , for all letters j ≤ m , where η = η ( W ) is the substitution associated to W .Proof. Indeed, start with an arbitrary loop V in the Rauzy graph such that the correspondingrenormalization matrix has all entries positive. Consider the interval exchange transformationwith periodic Rauzy-Veech expansion obtained by going along the loop repeatedly (it is knownfrom [32] that such an IET exists). As the number of passages through the loop grows, the lengthof the interval forming phase space of the new interval exchange (the result of the inductionprocess) goes to zero. In particular, after sufficiently many moves, this interval will be completelycontained in the first subinterval of the initial interval exchange — but this means, in view of(11.2) that n ( i, 0) = 1 for all i , and hence the resulting substitution ζ ( V n ) has the propertythat ζ ( V n )( j ) starts with c = 1 for all j . It remains to make sure that the admissible wordis simple. Observe that concatenating two loops V , V starting at the same vertex we obtain ζ ( V V ) = ζ ( V ) ζ ( V ). If ζ ( j ) starts with c for all j , then ξζ ( j ) starts with the first letter of ξ ( c )for all j . Thus, we can make sure that V = V n starts and ends with the same Rauzy operationsymbol — either a or b , by appending another loop at the end. We can then take V to be theloop of the other Rauzy operation symbol starting at the same vertex. As a result, we obtain theword in the alphabet { a, b } corresponding to W := V V has the form a e V ab k or b e V ba k , whichis obviously simple. The proof is complete. We are using here the fact that in the Rauzy graphthere are both a - and b -cycles starting at every vertex. (cid:3) Proof of Lemma 11.1. Let W be the admissible word in the Rauzy graph given by Lemma 11.2.By construction, the matrix S η of the substitution η = η ( W ) has all entries strictly positive, hence η ( i ) contains all letters j , for any i ≤ m . We can always replace the substitution η by its positivepower η k , since η corresponds to a loop in the Rauzy graph. Note that, for every i ≤ m , theword η ( i ) is a concatenation of all words η ( j ), j ≤ m , in some order, maybe with repetitions,all of which begin with c . Therefore, for every i ≤ m , the word η ( i ) contains every ζ ( j ), j ≤ m ,followed by another ζ ( j ′ ), also starting with c . It follows that u j := η ( j ) is a good return wordfor ζ := η , for every j ≤ m . The population vector ~ℓ ( η ( j )) is the j -th column vector of S η . Asis well-known, the matrices corresponding to Rauzy operations are invertible, which implies thatthe columns of S η span R m . Finally, note that if W is simple, then q = W W W is simple as well,and it has all the desired properties. The proof is complete. (cid:3) HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 41 The only remaining, key condition to check is (4.5). It will be derived from a variant of thethe exponential estimate for return times of the Teichm¨uller flow into compact sets. For largecompact sets of special form, this estimate is due to Athreya [1], whereas in the general form itwas established in [8] and independently in [4]. We will mostly use the same notation as above,but indicate the correspondence with the notation of [8]. The symbolic coding of the Rauzy-Veech induction map on the space of interval exchange transformations used in [8] correspondsto the symbolic coding of the Teichm¨uller flow as a suspension flow over the shift on the space ofMarkov compacta: indeed, the Rauzy-Veech expansion precisely identifies an interval exchangetransformation with a Bratteli-Vershik automorphism (cf. [9]).The symbol ∆( R ) stands for the space of interval exchange transformations whose permutationlies in a given Rauzy class R (fixed and omitted from notation); the symbolic space Ω + is theone-sided topological Markov chain over a countable alphabet that realizes the symbolic codingof the Rauzy-Veech induction map; the space Ω is its natural extension, the corresponding two-sided topological Markov chain. The space Ω can also be viewed as the phase space of the naturalextension of the Rauzy-Veech induction, that is, the space of sequences of interval exchangetransformations ordered by nonpositive integers:( λ (0) , π (0)) , ( λ ( − , π ( − , . . . , ( λ ( − k ) , π ( − k )) , . . . )where ( λ ( n ) , π ( n )) is the image of ( λ ( n − , π ( n − λ ( n ) = A n λ ( n + 1) | ( A n λ ( n + 1) | , where A n is the renormalization matrix A n from Section 4. The symbol | λ | stands for the sum ofcoordinates of a vector λ . In other words, the induction map that takes ( λ ( n ) , π ( n )) to ( λ ( n +1) , π ( n + 1)) consists in applying the matrix A − n and normalizing to unit length. Following [8],we also introduce the “non-normalized” lengths Λ( n ) inductively by the rule λ (0) = Λ(0) , Λ( n ) = A n +1 Λ( n + 1) for n < . Informally, log | Λ( − n ) | is the “Teichm¨uller time” corresponding to the discrete normalization“Rauzy-Veech” time n .Let q = q . . . q k be a simple word admissible in the Rauzy class, such that the resulting Rauzy-Veech renormalization matrix has all entries positive. We further let Ω q = [ q . q ]. The symbol P denotes the push-forward of the Masur-Veech measure. For a ∈ Ω q , let ℓ q ( a ) be the return timeto the cylinder set [ q , q ] in the negative direction, i.e. ℓ q ( a ) = min { n ≥ a − n − k +1 . . . a − n = a − n +1 . . . a − n + k = q } . Further, denote by L q ( a ) the corresponding “Teichm¨uller time”: L q ( a ) = log | Λ( − ℓ q ( a )) | . Proposition 11.3. There exists ǫ > such that for any a + ∈ Ω + q we have Z Ω exp( ǫL q ( a )) d P ( a | a + ) < + ∞ . Comparing the definitions, we see that this is equivalent to (4.5), so the remaining condition(d) in Theorem 4.1 will follow from Proposition 11.3. Proof of Proposition 11.3. We argue in much the same way as in Section 11 of [8] (compare withthe arguments involving the “bounded distortion” condition of [5]; see also Section 5.1 of [2]).Our main tool will be Lemma 16 from [8] whose formulation we now recall in our notation: Lemma 11.4. For any admissible word q such that all entries of the matrix A ( q ) are positive,there exist constants K ( q ) , p ( q ) , depending only on q and such that the following is true. Forany K ≥ K and any ( λ, π ) ∈ ∆( R ) , P (cid:0) ∃ n : ( λ ( − n ) , π ( − n )) ∈ [ q . q ] , | Λ( − n ) | < K ) | ( λ, π ) = ( λ (0) , π (0)) (cid:1) ≥ p ( q ) . Remark. In fact, the paper [8] proves this lemma for the coding of the Rauzy-Veech-Zorichinduction, that is, for the “Zorich acceleration” of the Rauzy-Veech induction, see [8] for details.However, the proof immediately transfers since it is stated in terms of the “Teichm¨uller time”log | Λ( − n ) | , which is invariant under the acceleration. To avoid potential problems with “over-lapping occurrences” we choose the word q to be simple in the Rauzy alphabet { a, b } , as we doin Lemma 11.2.The next lemma is a reformulation of Lemma 7 from [8]. Lemma 11.5. There exists K ∈ N and C = C ( R ) ≥ such that the following holds for any K > K : for all a + ∈ Ω + we have P (cid:0) | Λ( − | > K | a + (cid:1) ≤ CK . Define a random time k ( a ) to be the first moment n ≥ | Λ( − n )( a ) | > K . Notethat the map e σ ( a ) = σ − k ( a ) ( a )is invertible. Introduce a function η : Ω → N by the formula η ( a ) = (cid:20) log | Λ( − k ( a )) | log K (cid:21) . In other words, η ( a ) = n if K n ≤ | Λ( − k ( a )) | < K n +1 . Combining Lemmas 11.4 and 11.5 weobtain Proposition 11.6. For any a + ∈ Ω + and any r ∈ N we have P (cid:0) { a : η ( a ) = r, a − k ( a )+1 , . . . , a does not contain q | a + } (cid:1) ≤ CK r − · (1 − p ( q )) . HE H ¨OLDER PROPERTY FOR THE SPECTRUM OF TRANSLATION FLOWS 43 Now, take a large N and let n N ( a ) = min { n : η ( a ) + · · · + η ( e σ n a ) ≥ N } . Let also k n ( a ) = k ( a ) + · · · + k ( e σ n a ) . Then we have, by the definition of η :(11.3) K N ≤ | Λ( − k n N ( a ) ) | < K n N ( a )+ η ( a )+ ··· + η ( e σ n a ) . Clearly n N ( a ) ≤ N , and observe that for all a + ∈ Ω + , P ( B ( N ) | a + ) ≤ CK N , where B ( N ) = { a : η ( e σ n N ( a ) a ) > N } , by Lemma 11.5. Consider the set e Ω( N ) = { a : a − k nN ( a ) +1 , . . . , a does not contain the word q } ∩ { a : η ( e σ n N ( a ) a ) ≤ N } . Note that { a : L q ( a ) > N } ⊂ e Ω( N ) ∪ B ( N ) . It suffices, therefore, to prove that there exists ρ < P ( e Ω( N ) | a + ) ≤ ρ N . We have from Proposition 11.6, for any ℓ ∈ N and n , . . . , n ℓ ∈ N : P (cid:0) η ( a ) = n , η ( e σ a ) = n , . . . , η ( e σ ℓ a ) = n ℓ , a − k ℓ ( a )+1 , . . . , a does not contain q | a + (cid:1) ≤ (1 − p ( q )) ℓ (cid:18) CK (cid:19) ( n + ··· + n ℓ ) − ℓ . (11.4)Choose K ∈ N such that 1 − p ( q ) + CK < 1. We have P ( e Ω( N ) | a + ) ≤ N X e N = N P ( e Ω( N ) | a + & η ( a ) + · · · + η ( e σ n N ( ω ) a ) = e N ) ≤ N X e N = N (cid:18) − p ( q ) + CK (cid:19) e N ≤ ρ N , using (11.4) and the binomial formula in the last line, and Proposition 11.3 is proved. Thisconcludes the proof of Theorem 1.1 as well. (cid:3) References [1] Athreya, Jayadev S. Quantitative recurrence and large deviations for Teichm¨uller geodesic flow. Geom. Dedicata (2006), 121–140.[2] Avila, Artur; Delecroix, Vincent, Weak mixing directions in non-arithmetic Veech surfaces. 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Inst. Fourier(Grenoble) (1996), 325–370.[42] Zorich, Anton. Deviation for interval exchange transformations. Ergodic Theory Dynam. Systems (1997),no. 6, 1477–1499. Alexander I. Bufetov, Aix-Marseille Universit´e, CNRS, Centrale Marseille, I2M, UMR 737339 rue F. Joliot Curie Marseille FranceSteklov Mathematical Institute of RAS, MoscowInstitute for Information Transmission Problems, Moscow National Research University Higher School of Economics, Moscow E-mail address : [email protected] Boris Solomyak, Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel E-mail address ::