Hölder cocycles and ergodic integrals for translation flows on flat surfaces

The main results announced in this note are an asymptotic expansion for ergodic integrals of 
translation flows on flat surfaces of higher genus (Theorem 1) 
and a limit theorem for such flows (Theorem 2). 
Given an abelian differential on a compact oriented surface, 
consider the space $\mathfrak B^+$ of Holder cocycles over the corresponding vertical flow that are 
invariant under holonomy by the horizontal flow. 
Cocycles in $\mathfrak B^+$ are closely related to G.Forni's invariant distributions for 
translation flows [10]. Theorem 1 states that ergodic integrals of Lipschitz functions are approximated 
by cocycles in $\mathfrak B^+$ up to an error that grows more slowly than any power of time. Theorem 2 is obtained using the renormalizing action of the Teichmuller flow on the space $\mathfrak B^+$. 
A symbolic representation of translation flows as suspension flows over Vershik's automorphisms allows one to construct cocycles in $\mathfrak B^+$ explicitly. 
Proofs of Theorems 1, 2 are given in [5].


Hölder cocycles over translation flows.
Let ρ ≥ 2 be an integer, let M be a compact oriented surface of genus ρ, and let ω be a holomorphic one-form on M . Denote by m = i(ω ∧ ω)/2 the area form induced by ω and assume that m(M ) = 1.
Let h + t be the vertical flow on M (i.e., the flow corresponding to ℜ(ω)); let h − t be the horizontal flow on M (i.e., the flow corresponding to ℑ(ω)). The flows h + t , h − t preserve the area m. Take x ∈ M , t 1 , t 2 ∈ R + and assume that the closure of the set does not contain zeros of the form ω. The set (1) is then called an admissible rectangle and denoted Π(x, t 1 , t 2 ). Let C be the semi-ring of admissible rectangles. Consider the linear space B + of Hölder cocyles Φ + (x, t) over the vertical flow h + t that are invariant under horizontal holonomy. More precisely, a function Φ + (x, t) : M × R → R belongs to the space B + if it satisfies: (1) Φ + (x, t + s) = Φ + (x, t) + Φ + (h + t x, s); (2) There exists t 0 > 0, θ > 0 such that |Φ + (x, t)| ≤ t θ for all x ∈ M and all t ∈ R satisfying |t| < t 0 ; (3) If Π(x, t 1 , t 2 ) is an admissible rectangle, then Φ + (x, t 1 ) = Φ + (h − t2 x, t 1 ). For example, a cocycle Φ + 1 defined by Φ + 1 (x, t) = t belongs to B + . In the same way define the space of B − of Hölder cocyles Φ − (x, t) over the horizontal flow h − t which are invariant under vertical holonomy, and set Φ − 1 (x, t) = t.
Given Φ + ∈ B + , Φ − ∈ B − , a finitely additive measure Φ + ×Φ − on the semi-ring C of admissible rectangles is introduced by the formula and is an invariant distribution in the sense of G. Forni [9], [10]. For instance, m Φ − 1 = m. An R-linear pairing between B + and B − is given, for Φ + ∈ B + , Φ − ∈ B − , by the formula Take an abelian differential X = (M, ω). The space B + X = B + (M, ω) can be mapped to H 1 (M, R) in the following way. A continuous closed curve γ on M is called rectangular if It is easy to show that if γ is homologous to γ ′ , then Φ + (γ) = Φ + (γ ′ ). Using a similar construction for B − X = B − (M, ω), we obtain maps . For a generic abelian differential, the image of B + X underǏ + X is the strictly unstable space of the Konstevich-Zorich cocycle over the Teichmüller flow.
The Teichmüller flow g s on M κ sends the modulus of a pair (M, ω) to the modulus of the pair (M, ω ′ ), where ω ′ = e s ℜ(ω) + ie −s ℑ(ω); the new complex structure on M is uniquely determined by the requirement that the form ω ′ be holomorphic. As shown by Veech, the space M κ need not be connected; let H be a connected component of M κ .
Let H 1 (H) be the fibre bundle over H whose fibre at a point (M, ω) is the cohomology group H 1 (M, R). The bundle H 1 (H) carries the Gauss-Manin connection which declares continuous integer-valued sections of our bundle to be flat and is uniquely defined by that requirement. Parallel transport with respect to the Gauss-Manin connection along the orbits of the Teichmüller flow yields a cocycle over the Teichmüller flow, called the Kontsevich-Zorich cocycle and denoted A = A KZ .
Let P be a g s -invariant ergodic probability measure on H. By definition, the Kontsevich-Zorich cocycle A KZ satisfies the assumptions of the Oseledets Theorem with respect to P. For X ∈ H, X = (M, ω), denote by E u X ⊂ H 1 (M, R) the space spanned by vectors corresponding to the positive Lyapunov exponents of A KZ , and by E s X ⊂ H 1 (M, R) the space spanned by vectors corresponding to the negative Lyapunov exponents of A KZ . As before, let B + X , B − X be the spaces of Hölder cocycles corresponding to the vertical and the horizontal flows of X.
The pairing , is nondegenerate and is taken by the isomorphisms I + X , I − X to the cup-product in the cohomology group H 1 (M, R).
Remark. In particular, if P is the Masur-Veech "smooth" measure [17], [18], then almost surely, with respect to P, we have Remark. The isomorphismsǏ + X ,Ǐ − X are analogues of G. Forni's isomorphism between his space of invariant distributions and the unstable space of the Kontsevich-Zorich cocycle.
Remark. Cocycles in B + can be interpreted, in the spirit of Bonahon [4], as finitely-additive holonomy-invariant Hölder transverse measures on oriented measured foliations and also as finitely-additive invariant measures for interval exchange transformations. See the preprint [5] for details.
Consider the inverse isomorphisms be the distinct positive Lyapunov exponents of the Kontsevich-Zorich cocycle A KZ , and let be the corresponding Oseledets decomposition at X.
. Then for any ε > 0 the cocycle Φ + satisfies the Hölder condition with exponent θ i − ε and for any

Approximation of weakly Lipschitz functions.
The space of Lipschitz functions is not invariant under h + t , and a larger function space Lip + w (M, ω) of weakly Lipschitz functions is introduced as follows. A bounded measurable function f belongs to Lip + w (M, ω) if there exists a constant C, depending only on f , such that for any admissible rectangle Π(x, t 1 , t 2 ) we have By definition, the space Lip + w (M, ω) contains all Lipschitz functions on M and is invariant under h + t . We denote by Lip + w,0 (M, ω) the subspace of Lip + w (M, ω) of functions whose integral with respect to m is 0.
For any f ∈ Lip + w (M, ω) and any Φ − ∈ B − , the integral M f dm Φ − can be defined as the limit of Riemann sums.
If the pairing , induces an isomorphism between B + and the dual (B − ) * , then one can assign to a function f ∈ Lip + w (M, ω) the cocycle Φ + f ∈ B + by the formula Theorem 1. Let P be a g s -invariant ergodic probability measure on H. For any ε > 0 there exists a constant C ε depending only on P such that for P-almost every X ∈ H, any f ∈ Lip + w (X), any x ∈ M and any T > 0 we have Consider the case in which the Lyapunov spectrum of the Kontsevich-Zorich cocycle is simple in restriction to the space E u (as, by the Avila-Viana theorem [2], is the case with the Masur-Veech smooth measure). Let l 0 = dimE u and let 1 = θ 1 > θ 2 > · · · > θ l0 be the corresponding simple expanding Lyapunov exponents.

ALEXANDER I. BUFETOV
Noting that by definition we also have m Φ − 1 = m, we derive from Theorem 1 the following corollary.
Corollary 1. Assume that P is an invariant ergodic probability measure for the Teichmüller flow such that, with respect to P, all positive Lyapunov exponents of the Kontsevich-Zorich cocycle are simple.
Then for any ε > 0 there exists a constant C ε depending only on P such that for P-almost every X ∈ H, any f ∈ Lip + w (X), any x ∈ X and any T > 0 we have Remark. If P is the Masur-Veech smooth measure on H, then the work of G.Forni [9], [10], [11] and S. Marmi, P. Moussa, J.-C. Yoccoz [16] implies that the left-hand side is bounded for any f ∈ C 1+ε (M ) (in fact, for any f in the Sobolev space H 1+ε ). In particular, if f ∈ C 1+ε (M ) and Φ + f = 0, then f is a coboundary.

Limit Theorems for Translation Flows.
3.1. Time integrals as random variables. As before, (M, ω) is an abelian differential, and h + t , h − t are, respectively, its vertical and horizontal flows. Take τ ∈ [0, 1], s ∈ R, a real-valued f ∈ Lip + w,0 (M, ω) and introduce the function For any fixed τ ∈ [0, 1] the formula (9) yields a real-valued random variable S[f, s; τ ] : (M, m) → R whose expectation, by definition, is zero. Our first aim is to estimate the growth of its variance as s → ∞. Without loss of generality, one may take τ = 1.

3.2.
The growth rate of the variance. Let P be an invariant ergodic probability measure for the Teichmüller flow such that, with respect to P, the second Lyapunov exponent θ 2 of the Kontsevich-Zorich cocycle is positive and simple (recall that, as Veech and Forni showed, the first one, θ 1 = 1, is always simple [20,10] and that, by the Avila-Viana theorem [2], the second one is simple for the Masur-Veech smooth measure).
For an abelian differential X = (M, ω), denote by E + 2,X the one-dimensional subspace in H 1 (M, R) corresponding to the second Lyapunov exponent θ 2 , and let B + 2,X = I + X (E + 2,X ). Similarly, denote by E − 2,X the one-dimensional subspace in H 1 (M, R) corresponding to the Lyapunov exponent −θ 2 , and let B − 2,X = I − X (E − 2,X ). Recall that the space H 1 (M, R) is endowed with the Hodge norm | · | H ; the isomorphisms I ± X take the Hodge norm to a norm on B ± X ; slightly abusing notation, we denote the latter norm by the same symbol. Introduce a multiplicative cocycle H 2 (s, X) over the Teichmüller flow g s by taking v ∈ E + 2,X , v = 0, and setting By definition, we have lim s→∞ log H2(s,X) s = θ 2 . such that the following is true for P-almost all X ∈ H. If f ∈ Lip + w,0 (X) satisfies m Φ − 2 (f ) = 0, then for all s ≥ s 0 (X) we have
By definition, D + 2 (X ′ ) is a Borel probability measure on the space C[0, 1]; it is a compactly supported measure as its support consists of equibounded Hölder functions with exponent θ 2 /θ 1 − ε.
Consider By definition, the diagram Let d LP be the Lévy-Prohorov metric on M (see, e.g., [3]).
Theorem 2. Let P be a g s -invariant ergodic probability measure on H such that the second Lyapunov exponent of the Kontsevich-Zorich cocycle is positive and simple with respect to P. There exists a positive measurable function C : H ′ × H ′ → R + and a positive constant α depending only on P such that for P-almost every X ′ ∈ H ′ and any . Fix τ ∈ R and let m 2 (X ′ , τ ) be the distribution of the R-valued random variable .
Proposition 5. For P-almost any X ′ ∈ H ′ , the measure m 2 (X ′ , τ ) admits atoms for a dense set of τ ∈ R.
By definition, m 2 (X ′ ) is always compactly supported; the following Proposition shows, however, that the family {m 2 (X ′ ), X ′ ∈ H ′ } is in general not closed. Let δ 0 stand for the delta-measure at zero. Proposition 6. Let H be endowed with the Masur-Veech smooth measure. Then the measure δ 0 is an accumulation point for the set {m 2 (X ′ ), X ′ ∈ H ′ } in the weak topology.

A symbolic coding for translation flows.
By Vershik's Theorem [21], every ergodic automorphism of a Lebesgue probability space can be represented as a Vershik automorphism of a Markov compactum. For an interval exchange transformation, an explicit representation is obtained using Rohlin towers given by Rauzy-Veech induction (see [12]). Passing to Veech's zippered rectangles and their bi-infinite Rauzy-Veech expansions, one represents a minimal translation flow as a flow along the leaves of the asymptotic foliation of a bi-infinite Markov compactum. In this representation, the cocycles in B + become finitely-invariant measures on the asymptotic foliation of a Markov compactum.
Thus, after passage to a finite cover (namely, the Veech space of zippered rectangles), the moduli space of abelian differentials is represented as a space of Markov compacta. The Teichmüller flow and the Kontsevich-Zorich cocycle admit a simple description in terms of this symbolic representation, and the cocycles in B + are constructed explicitly. Theorems 1, 2 are then derived from their symbolic counterparts. Detailed proofs are given in [5].