Teichmüller dynamics and unique ergodicity via currents and Hodge theory

Curtis T. McMullen 1
  • 1 Mathematics Department, Harvard University, MA 02138-2901, Cambridge, USA
Curtis T. McMullen

Abstract

We present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

1 Introduction

Let g denote the moduli space of compact Riemann surfaces X of genus g, and let Ωgg denote the bundle of nonzero holomorphic 1-forms (X,ω). Any 1-form determines a horizontal foliation (ω) of X together with a transverse invariant measure. If this measure is unique up to scale, we say (ω) is uniquely ergodic.

The purpose of this note is to present a cohomological proof of the following important result of Masur:

Theorem 1.1.

Suppose the Teichmüller geodesic ray generated by shrinking the leaves of F(ω) is recurrent in moduli space Mg. Then the foliation F(ω) is uniquely ergodic.

The perspective we adopt is based on currents and Hodge theory. First, we introduce the convex cone P(ω) of closed, positive currents carried by (ω). These are the 1-forms ξ on X, with distributional coefficients, satisfying

dξ=0,ξβ=0andαξ0,

where ω=α+iβ. As we will see in Section 3, there is a natural bijection between such currents and transverse invariant measures for (ω).

The language of currents provides a useful bridge between foliations, differential forms and Hodge theory. Moreover, the closed currents P(ω) map injectively into H1(X,) when (ω) has a dense leaf, so unique ergodicity can be addressed at the level of cohomology.

In this language, our main result is:

Theorem 1.2.

Suppose X lies in a compact subset KMg, and the geodesic ray generated by (X,ω) spends at least time T in K. Then the closed, positive currents carried by F(ω) determine a convex cone

[P(ω)]H1(X,)

which meets the unit sphere in a set of diameter O(e-λ(K)T).

Here the unit sphere and diameter are defined using the Hodge norm on H1(X,), and λ(K)>0 depends only on K.

One can regard Theorem 1.2 as a quantitative refinement of Theorem 1.1. In the recurrent case we can take T=, [P(ω)] reduces to a single ray, and we obtain unique ergodicity (see Section 5).

Billiards

Theorem 1.2 also sheds light on the distribution of closed geodesics on (X,|ω|), and leads to new results on billiards in polygons. To illustrate this connection, in Section 6 we will show:

Theorem 1.3.

Consider a sequence of periodic billiard trajectories of slope sns on the golden L-shaped table. If the lengths of the golden continued fractions for sn tend to infinity, then the trajectories become uniformly distributed as n.

Three examples with sn0 are shown in Figure 1. Only the last sequence of trajectories is uniformly distributed.

Figure 1
Figure 1Figure 1Figure 1

Periodic billiard trajectories with slopes tending to zero.

Citation: Journal für die reine und angewandte Mathematik 2020, 768; 10.1515/crelle-2019-0037

An analogous statement holds for any lattice polygon and any 1-form generating a Teichmüller curve Vg. These applications were our original motivation for proving Theorem 1.2. A more complete development will appear in a sequel [Mc5].

The cone of positive currents

Here is a sketch of the proof of Theorem 1.2.

Let ξP(ω) be a closed, positive current carried by (ω) as above, with ξ0. The standard transverse measure for (ω) corresponds to the smooth current β=Im(ω). To compare the two, we first scale ξ so it has the form

ξ=β+δ,

where

Xδω=0.

Let Xt, t0, denote the Teichmüller geodesic ray generated by ω with X0=X. The natural flat connection on cohomology groups allows us to transport the Hodge norm from H1(Xt,) to a varying family of norms on H1(X,), which we denote by Xt.

In Section 3 we show that a cone condition of the form

δXtβXtC(K)

holds whenever XtK. On the other hand, in Section 4 we show that as t, the Hodge norm of β shrinks more rapidly than that of δ: there is a λ=λ(K)>0 such that

δXtβXteλTδXβX,

where T is the amount of time Xs spends in K for s[0,t].

Combining these bounds gives the stronger cone condition

δXβXe-λTC(K)

whenever XtK. This inequality says that the line through [ξ]=[β+δ] is exponentially close to the line through [β] in H1(X,), and Theorem 1.2 follows. (For more details, see Section 5.)

Conceptually, equation (1.1) follows from uniform contraction (over K) of the complementary period mapping

σ:g-1,

which records the Hodge structure on the part of H1(X,) orthogonal to ω, as X moves along a complex geodesic (see Section 4).

Notes and references

Many of the ideas presented in Section 4 below were developed independently and earlier by Forni and others, with somewhat different aims and formulations. In particular, a version of Theorem 4.1 for strata is given in [AF, Theorem 4.2], and a variant of equation (A.1) is derived, by different means, in [Fo, Lemma 2.1]. The strategy to prove Theorem 1.1 is similar to the proof that (ω) is ergodic for almost every ωΩg sketched in [FM, Remark 60]. Here we use currents and the Hodge norm throughout, and exploit the cone condition given in Theorem 3.1.

Masur’s original proof of Theorem 1.1, which also applies to quadratic differentials, is given in [Mas, Theorem 1.1]; see also [Mc2]. The original argument works directly with dynamics and Anosov properties of the foliation of X. A strengthening of Theorem 1.1 is given in [Tr, Theorem 4].

A discussion of currents and foliations on general manifolds can be found in [Sul]; another instance of their use in the present setting is given in [Mc4, Section 2]. For more on the interaction between Hodge theory and Teichmüller theory, see e.g. [Ah, Roy, Fo, Mc1, Mo1, EKZ, FM, FMZ].

2 Background

We begin by recalling some basic results regarding the Hodge theory, foliations, geodesics in Teichmüller space, and the action of SL2() on the moduli space of holomorphic 1-forms. For more details, see e.g. [FLP, GH, Ga, IT, Nag, MT, Mo2].

The Hodge norm

Let X be a Riemann surface of genus g. The spaces of holomorphic and real harmonic 1-forms on X will be denoted by Ω(X) and 1(X), respectively.

By Hodge theory, the map sending a cohomology class to its harmonic representative provides an isomorphism H1(X,)1(X). These representatives, together with the Hodge star, give a rise to a natural inner product

α,βX=Xα*β

on H1(X,); and the associated Hodge norm is defined by

αX2=α,αX.

Similarly, the space Ω(X) carries a natural Hermitian form defined by

ω1,ω2X=i2Xω1ω¯2,

whose associated norm is given by

ωX2=X|ω|2.

These norms are compatible in the sense that the map ωReω gives a norm–preserving, real linear isomorphism

Ω(X)H1(X,).

Foliations and measures

Every nonzero ωΩ(X) determines a natural horizontal foliation (ω) of X. To describe this foliation, recall that

ω=α+iβ

is a linear combination of real harmonic forms satisfying *α=β.

The foliation (ω) has multipronged singularities at the zeros of ω. Away from these points, we can choose local coordinates such that ω=dz and the leaves of (ω) become horizontal lines in . In particular, the tangent space to (ω) is the kernel of β. Each leaf L of (ω) is naturally oriented by the condition α|L>0.

A transverse measure for (ω) is the specification of a Borel measure μτ0 on every smooth arc τX disjoint from Z(ω) and transverse to the leaves of the foliation. We require that μτ is compatible with restriction; that is, μσ=μτ|σ whenever στ. A transverse measure is invariant if it also compatible with the smooth maps between nearby transversal obtained by flowing along the leaves of (ω).

The standard transverse invariant measure for (ω) is defined by μτ=β|τ (where τ is oriented so the measure is positive.) There may be many others; for example, any closed leaf L of (ω) supports a transverse atomic measure μτ with mass one at each point of τL.

Moduli of Riemann surfaces

Fix a compact, oriented topological surface Σg of genus g2 with mapping–class group Modg. A point (X,f) in the associated Teichmüller space 𝒯g is specified by a Riemann surface of genus g together with an orientation–preserving marking homeomorphism f:ΣgX. By forgetting the marking, we obtain a natural map

𝒯g𝒯g/Modgg,

presenting 𝒯g as the orbifold universal cover of the moduli space g of Riemann surfaces of genus g.

The cotangent space to 𝒯g at X is naturally identified with the space Q(X) of holomorphic quadratic differentials on X, and the Teichmüller metric corresponds to the norm q=X|q|. To describe the tangent space, let M(X) denote the space of measurable Beltrami differentials on X with μ=supX|μ|<. The natural pairing

q,μ=Xqμ=Xq(z)μ(z)|dz|2

between Q(X) and M(X) then allows one to identify the tangent space TX𝒯g with the quotient space M(X)/Q(X).

For later reference, we note that when q=μ=1, we have

Req,μ1,and equality holds if and only if μ=q¯|q|.

This observation is an infinitesimal form of uniqueness of the Teichmüller mapping.

Moduli of forms

Consider the holomorphic vector bundle over 𝒯g whose fiber over X is Ω(X). Removing the zero section, we obtain the space Ω𝒯g of marked holomorphic 1-forms. The associated sphere bundle, whose fibers are

Ω1(X)={ωΩ(X):ωX=1},

will be denoted by Ω1𝒯g. Taking the quotient by Modg yields the corresponding bundles Ωg and Ω1g over moduli space.

Dynamics and geodesics

There is a natural action of SL2() on Ω𝒯g, characterized by the property that, for L=(abcd)SL2() we have

L(X,ω)=(Y,η)

if and only if there is a map f:XY, compatible with markings, such that

f*(η)=(1i)(abcd)(ReωImω).

The orbits (Xt,ωt)=at(X,ω) of the diagonal group

A={at=(e-t00et):t}SL2()

project to Teichmüller geodesics in 𝒯g, parameterized by arclength, and satisfying

dXtdt=[-ω¯tωt]M(Xt)/Q(Xt)

for all t. For t>0, the natural affine map ft:(X,|ω|)(Xt,|ωt|) is area–preserving and shrinks the leaves of (ω) by a factor of e-t. In particular, if all the leaves of (ω) are closed, then [Xt] converges to a stable Riemann surface in g by pinching these closed curves.

Dynamics and cohomology

The vector bundle H1𝒯g with fibers H1(X,) is both trivial and flat with respect to the connection provided by the marking isomorphisms H1(X,)H1(Σg,). The same is true when it is pulled back to Ω𝒯g. Over this space we have H1=WW, where the splitting

H1(X,)=W(X,ω)W(X,ω)

on each fiber is obtained by taking the image of the direct sum

Ω(X)=(ω)(ω)

under the map ωReω. Since W(X,ω) is *-invariant, the summands in (2.5) are also orthogonal with respect to the symplectic form. Thus the following result is immediate from (2.3):

Proposition 2.1.

The sub-bundles W and W are flat over any SL2(R) orbit in ΩTg.

Put differently, we have W(L(X,ω))=W(X,ω) when both are identified with subspaces of H1(Σg,), and similarly for W(X,ω).

3 Currents and cones

In this section we describe the connection between measured foliations and closed, positive currents. We then show that the shape of the convex cone in cohomology determined by these currents is uniformly controlled over compact subsets of moduli space. This control can be expressed in terms of the Hodge norm as a reverse Cauchy–Schwarz inequality, which we state as follows.

Theorem 3.1.

Let KMg be a compact set. Then for any closed, positive current ξ carried by the horizontal foliation F(ω) of a holomorphic 1-form ωΩ(X) with XK, we have

ωXξXC(K)|Xωξ|.

Here C(K)>0 is a constant depending only on K. Geometrically, this results says that the length of the current ξ in the metric |ω| controls the Hodge norm of its harmonic representative.

Currents and foliations

Recall that a 1-dimensional current ξ on X is an element of the dual of the space of smooth 1-forms (see e.g. [dR], [GH, Section 3.1]). Since X is oriented, currents on X can be thought of as forms with distributional coefficients. The current ξ is closed if dξ=0; equivalently, if Xξdf=0 for every smooth function f on X. Any closed current determines a cohomology class [ξ]H1(X,).

Given a nonzero holomorphic form ω=α+iβ, let

P(ω)={currents ξ:dξ=0,ξβ=0 and αξ0}.

The final positivity condition means

X(fα)ξ0

for all smooth f0 on X; in particular, ξ is required to satisfy infinitely many linear inequalities. We refer to P(ω) as the space of closed, positive currents carried by (ω). It is a closed, convex cone in the natural topology on currents.

Proposition 3.2.

There is a natural bijection between the closed, positive currents carried by F(ω) and its transverse invariant measures.

Proof.

A transverse invariant measure determines a current ξP(ω) by integration along the (oriented) leaves of (ω) locally weighted by μτ. Conversely, if ξP(ω), then the fact that ξβ=0 implies ξ is locally a distributional multiple of β; the fact that ξ is closed implies it is locally the pullback of a distribution on a transversal τ; and positivity implies this distribution is a measure μτ0. ∎

The cohomology class of a measured foliation

Each transverse invariant measure determines a cohomology class, by the correspondence μτξ[ξ]H1(X,). Recall that the foliation (ω) is minimal if each of its leaves is dense in X.

Proposition 3.3.

If F(ω) is minimal, its transverse invariant measures are determined by their cohomology classes. Equivalently, the natural map P(ω)H1(X,R) is injective.

Proof.

It suffices to show that the values of Cξ for CH1(X,) determine μτ(τ) for every transversal τ. By minimality μτ has no atoms. Let L be a dense leaf of (ω). Then Lτ is dense in τ, so we can find an increasing sequence of subarcs τ1τ2τ3τ such that τn has full measure in τ and each τn has endpoints in L. By adding a piece of L to connect these endpoints, we obtain a closed loop Cn with [Cn]H1(X,). Using these loops, we find

μτ(τ)=limμτ(τn)=limCnξ.

Since the integrals above only depend on [ξ]H1(X,), the proof is complete. ∎

Corollary 3.4 (Katok).

If F(ω) is minimal, it carries at most g=g(X) mutually singular, ergodic, transverse invariant measures.

Proof.

The cohomology classes of these measures are linearly independent by the previous result; and since the corresponding currents are given by integration along the leaves of the same foliation (ω), they lie in a Lagrangian subspace of H1(X,), which has dimension g. ∎

See e.g. [Ka, Theorem 1], [V1, Theorem 0.5] and [Fi, Theorem 1.29] for other perspectives on Corollary 3.4.

Proof of Theorem 3.1.

We will first prove a cone inequality for a single form

ω=α+iβΩ(X),

XK, normalized so that ωX=1. By definition (3.2) we have |ωξ|=αξ. Thus our goal is to prove an inequality of the form

ξXC(K)Xαξ

for all ξP(ω).

Choose a sequence of smooth, closed 1-forms δ1,,δ2g that represent an orthonormal basis for H1(X,) with respect to the inner product (2.1). We may assume that these forms all vanish on an open neighborhood U of Z(ω). Since the smooth area form αβ0 only vanishes at the zeros of ω, there is a constant M>0 such that

|β*δi|Mαβ

pointwise on X, for i=1,2,,2g.

Now for any ξP(ω), the current ξ is locally a limit of smooth currents of the form fnβ with fn0. Since (3.4) implies that

|(fnβ)*δi|Mα(fnβ)

as measures on X, in the limit we obtain

|ξ*δi|Mαξ,

and hence

ξ,δiXMXαξ

for all i. This implies that

ξX2=12gξ,δiX22gM2(Xαξ)2,

and taking the square–root of both sides yields the desired inequality (3.3).

We now allow the form (X,ω) to move in Ω1g. Since the zero set Z(ω) and the Hodge norm vary continuously with the form, it is easy to extend the argument just given to obtain a uniform cone constant in a neighborhood of (X,ω). By properness of the projection map Ω1gg, we can then obtain a uniform cone constant C(K) for all unit–norm forms (X,ω) with X in a given compact set Kg. But once the cone inequality (3.1) holds for ω it also holds for all the positive real multiples of ω, so the proof is complete. ∎

4 The complementary period mapping

In this section we study the variation of the Hodge norm along a geodesic in 𝒯g. Its relationship to the period mapping τ:𝒯gg will be presented at the end.

Norms

Let Xt be the geodesic in 𝒯g generated by a holomorphic 1-form

ω0=α0+iβ0Ω1(X0).

Since all the Riemann surfaces in 𝒯g are marked by Σg, we have a natural isomorphism

H1(X0,)H1(Xt,)

for all t. Using this identification, we can transport the Hodge norms on Xt to a varying family of norms Xt on the fixed vector space H1(X0,).

It is easy to see that

α0Xt=etandβ0Xt=e-t

for all t. The next result says that the Hodge norm of any cohomology class orthogonal to these moves more slowly.

Theorem 4.1.

Fix a compact set KMg. Then there is a constant λ=λ(K)>0 such that for all [δ0]H1(X0,R) with δ0X0=1 and

X0δ0α0=X0δ0β0=0,

and all t>0, we have

eλT-tδ0Xtet-λT.

Here T=|{s[0,t]:XsK}|.

Proof.

Let (Xt,ωt)=at(X0,ω0), and write

ωt=αt+iβt.

Then

dXtdt=X˙t=[-ω¯tωt]

by equation (2.4).

Let ηtΩ1(Xt) be the unique unit norm 1-form such that

[δ0]δ0Xt=[Reηt]

under the identification (4.1). By assumption, we have η0,ω0X0=0, and thus

ηt,ωtXt=0

for all t by Proposition 2.1.

Define a function κ on Ωg by

κ(X,ω)=sup{|η2,ω¯ω|:ηΩ1(X) and η,ωX=0}.

(The brackets denote the natural pairing (2.2) between tangent and cotangent vectors to 𝒯g at X.) Since η=ω is excluded by the orthogonality condition, we have κ(X,ω)<1 (see equation (2.2)). It can also be readily verified that κ is continuous on Ω1g, and hence

λ=λ(K)=1-sup{κ(X,ω):XK}>0.

Let N(t)=logδ0Xt. Computing the variation of the Hodge norm (see the Appendix, Corollary A.2), we find that

N(t)=-Reηt,X˙t,

and hence

|N(t)||ηt,X˙t|=|ηt,ω¯tωt|κ(X,ωt).

This shows that

|N(t)|<1

for all t, and that

|N(t)|1-λ

whenever XtK. Since N(0)=0, this implies that |N(t)|t-λT for all t, and then exponentiation yields the theorem above. ∎

Conceptual framework

The idea behind the proof above can be expressed as follows. First, the Hodge norm on H1(X,) provides the same information as the period matrix τij(X), which is recorded by the holomorphic period map to Siegel space,

τ:𝒯gg.

Second, the choice of a 1-form ω=α+iβ on X determines a natural splitting

H1(X,)=WW,

where W=*W is spanned by α and β. Third, the form ω generates a holomorphic, isometric complex geodesic

F:𝒯g,

related to the real geodesic by Xt=F(ie2t). The splitting (4.2) is constant over this geodesic (see Proposition 2.1), and accordingly the period map τF can be written as

×g-1g,

where the two factors of the product ×g-1 record the Hodge structures on W and W respectively. Indeed, with a suitable choice of coordinates on the first factor, we can write τF(s)=(s,σ(s)).

It is then straightforward to show, using Ahlfors variational formula, that the complementary period map

σ:g-1

is a contraction for the Kobayashi metric. In fact, we have

Dσ(s)=κ(Xs,ωs)<1

for all s, and the upper bound can be replaced by 1-λ(K)<1 provided XsKg. To complete the proof, one need only observe that the rate of change of σ(s) controls the rate at which the Hodge norm varies for a class in W (cf. [Mc3, Proposition 3.1]).

For more details and similar discussions, see e.g. [Ah], [Roy], [Mc1, Theorem 4.2], [Mc3, Section 3], the Appendix, and the works [Fo, AF, FMZ] on ergodic averages and Lyapunov exponents.

5 Unique ergodicity with bounds

With the previous results in place, it is now easy to prove Theorems 1.1 and 1.2.

Narrowing the cone

We begin with Theorem 1.2. Let Kg be a compact set, let ω=α+iβ be a 1-form on XK with ωX=1, and let Xt be the Teichmüller ray generated by (X,ω). Let

P1(ω)={ξP(ω):β,ξ=1}.

Let T denote the amount of time that Xt spends in K for t0, and let C(K),λ(K)>0 be the constants provided by Theorems 3.1 and 4.1.

Theorem 5.1.

We have ξ-βXC(K)e-λ(K)T for all ξP1(ω).

Proof.

Let (Xt,ωt)=at(X,ω), and write ωt=αt+iβt. The Teichmüller mapping ft:XXt provides a natural identification between H1(Xt,) and H1(X,), under which we have

[αt]=e-t[α0]and[βt]=et[β0]

by equation (2.3). Since ωtXt=βtXt=1, this gives

β0Xt=e-t.

Recall that α,ξ=0 by the definition (3.2) of P(ω). Since β,ξ=1, we can write

ξ=β+δ,where Xωδ=0.

The Teichmüller mapping ft transports ξ to a current ξtP(ωt) which we can similarly write as

ξt=e-tβt+δt,where Xtωtδt=0.

By Theorem 3.1, we then have

δtXtC(K)|Xtωtξt|=C(K)e-t

whenever XtK.

Note that the cohomology classes [ξt] and [δt] do not depend on t. The first is constant because we use ft to identify cohomology groups as t varies, and the second is constant because the span of [αt] and [βt] does not depend on t (cf. Proposition 2.1).

Suppose s=sup{t0:XtK} is finite. Then we also have

δsXsδXe-s+λ(K)T,

by Theorem 4.1. Setting t=s in (5.1) and combining these inequalities gives

δXC(K)e-λ(K)T.

Since δ=ξ-β, the proof is complete. The case s= is similar. ∎

Theorem 1.2 is then equivalent to:

Corollary 5.2.

The diameter of the intersection of [P(ω)]H1(X,R) with the unit sphere in the Hodge norm is bounded by 2C(K)e-λ(K)T.

Proof.

Consider any ξP(ω) with ξX=1. Then r=ξ,β>0 by Theorem 3.1, and r1 since βX=1. The preceding result then gives

ξ-βXr-1ξ-βXC(K)e-λ(K)T,

where the first inequality comes from the fact that both ξ and β lie on the unit sphere. ∎

Unique ergodicity: Proof of Theorem 1.1

Suppose the geodesic ray Xt generated by (X,ω) is recurrent. Then Xt spends an infinite amount of time in some fixed compact set Kg, and thus [P1(ω)]H1(X,) is a single point by the preceding result. By recurrence, (ω) has no cylinders or loops of saddle connections; if it did, they would pinch and force Xt to infinity. Hence (ω) has a dense leaf, which implies the map P(ω)H1(X,) is injective, by Proposition 3.3. Thus P1(ω) itself is a single point, and hence (ω) is uniquely ergodic. ∎

6 Billiards and equidistribution

In this section we prove Theorem 1.3 on equidistribution of billiards.

Figure 2
Figure 2

The golden table P.

Citation: Journal für die reine und angewandte Mathematik 2020, 768; 10.1515/crelle-2019-0037

The golden table

Let γ=12(1+5) be the golden ratio, and let P denote the symmetric L-shaped polygon P shown in Figure 2, whose short and long sides have lengths 1 and γ respectively. By gluing parallel edges of P together by horizontal and vertical translations, we obtain a holomorphic 1-form (X,ω)=(P,dz)/Ωg, g=2, whose stabilizer is the lattice

Γ=SL(X,ω)=(01-10),(1γ01)SL2().

Its quotient V=/Γ is the (2,5,) hyperbolic orbifold.

By a well-known result of Veech, the fact that Γ is a lattice implies every billiard trajectory in P is either periodic or uniformly distributed [V2]. (The same result holds for the regular n-gon, and P is closely related to the case n=5.)

Continued fractions

Recall that the cusps of a Fuchsian group are the fixed points of its parabolic elements. The slopes s of periodic trajectories for P are essentially the same as the cusps of Γ.1 Since the cusps form a single orbit Γ, every periodic slope can be written as a finite golden continued fraction,

s=[a1,a2,a3,,aN]=a1γ+1a2γ+1a3γ+1aNγ.

This expression can be computed recursively and made unique by requiring that

x-a1γ(-γ2,γ2],

and similarly for each subsequent ai. We refer to the number of integers ai as the length N=N(s).

Proof of Theorem 1.3.

Consider a sequence of periodic slopes sns. As usual (cf. [MT]), a billiard trajectory in P at slope sn corresponds to a closed leaf Ln of the foliation (ωn) of X, ωn=(1+isn)-1ω, and to prove equidistribution of trajectories in P it suffices to prove equidistribution of Ln in X. We may assume that s itself is a periodic direction, otherwise equidistribution is immediate from unique ergodicity at slope s. In fact, since Γ acts transitively on periodic slopes, we may assume s=0 and ωnω.

Let ω=α+iβ. Each closed leaf LnX determines a current of integration on X; dividing through by the length of Ln in the |ω|-metric, we obtain a sequence of bounded currents ξnP(ωn) such that β,ξnX1. Pass to a subsequence such that ξnξP(ω).

Our goal is to show that ξ=β. To this end, note that X naturally decomposes into a pair of horizontal cylinders A1A2, corresponding to the two rectangles in Figure 2. Using the fact that the slope of Ln tends to zero, it is easy to see that Ln is equidistributed in each cylinder individually. This implies that ξ|Ai is a multiple of β|Ai for i=1,2, and thus

ξ=c1(β|A1)+c2(β|A2)

for some c1,c20.

Now we use the fact that N(sn). Let KV2 be the compact set obtained by deleting a small neighborhood of the cusp of the Teichmüller curve V=/Γ. The length of the continued fraction N(sn) essentially counts the number of times the Teichmüller ray generated by (X,ωn) makes an excursion into the cusp; hence the amount of time Tn it spends in K is comparable to N(sn). Applying Theorem 1.2, we find that

ξn-βnX0,

and hence [ξ]=[β] in H1(X,). Using (6.1), we can then conclude that ξ=β as currents, and hence Ln becomes uniformly distributed on X as n. ∎

Sample slopes

The first two examples in Figure 1 depict periodic billiard trajectories in P for the sequence of slopes sn=1nγ. In the first example the trajectories start near the right endpoint of the bottom edge of P, and they all lie in A2; in the second example, they start near the left endpoint, and they converge to a limiting measure that is not uniform (it assigns too much mass to A1). The third example is uniformly distributed; it corresponds to the sequence of slopes with continued fractions sn=[0,n,1,1,,1] with N(sn)=n+3.

A Appendix: Variation of the Hodge norm

We will show that a classical result of Ahlfors gives:

Theorem A.1.

Fix a cohomology class CH1(Σg,R). Consider any (X,ω)ΩTg such that [Reω]=C. Then for any variation of XTg, the Hodge norm of C satisfies

(CX2)˙=-2Reω2,X˙.

Here a variation in X is described by a smooth path X(t) in 𝒯g with X(0)=X. We use the shorthand X˙=X(0), and adopt a similar convention for other quantities that depend on t. Note that the quadratic differential ω2 represents a cotangent vector to 𝒯g at X, so it pairs naturally with the tangent vector X˙ as in equation (2.2).

Proof.

Fix a standard symplectic basis (a1,,ag),(b1,,bg) for H1(Σg,). The associated Siegel period matrix for X is defined by

τij=bjωi,

where (ω1,,ωg) is the basis for Ω(X) characterized by

aiωj=δij.

The matrix σij=Imτij is symmetric and positive-definite, and the norm of a general form η=1gsiωiΩ(X) is given by

ηX2=X|η|2=stσs¯.

Since equation (A.1) is homogeneous, we can assume ωX=CX=1. We can then choose a symplectic basis such that ω1=ω. With this normalization, we have

C,ai=Reaiω1=δi1.

Now consider a variation X(t) of X. Then ωi, τij and σij vary as well. By Ahlfors’ variational formula [Ah, equation (7)], we have

τ˙ij=-2iωiωj,X˙.

Let ω(t)=si(t)ωi(t) be the unique form in Ω(X(t)) satisfying [Reω(t)]=C. Then

(CX2)˙=(stσs¯)˙.

Since ω(0)=ω1, we have si(0)=δi1. By equation (A.2) we also have Resi=δi1, and hence Res˙=0. Using the fact that σt=σ, this gives

s˙tσs¯+stσs¯˙=2(Re(s˙)tσs¯)=0,

and therefore

(stσs¯)˙=sσ˙s¯=σ˙11.

Formula (A.1) then follows directly from Ahlfors variational formula (A.3). ∎

Here is an equivalent formulation, used Section 4:

Corollary A.2.

For any nonzero CH1(Σg,R), we have

(logCX)˙=-Reω2,X˙,

where [Reω]=CCX.

Notes and references

A variant of Theorem A.1, with a different proof, is given in [Fo, Lemma 2.1]. A precursor to (A.3) appears in [Ra, (7)], where the factor 1/2πi should be replaced by 1. Ahlfors’ formula (A.3) is sometimes stated without the factor -2i, which results from the identity dzdz¯=-2i|dz|2.

Acknowledgements

I would like to thank J. Chaika and G. Forni for useful discussions and references.

References

  • [Ah]

    L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, Analytic functions, Princeton University Press, Princeton (1960), 45–66.

  • [AF]

    J. S. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards, Duke Math. J. 144 (2008), no. 2, 285–319.

  • [dR]

    G. de Rham, Differentiable manifolds. Forms, currents, harmonic forms, Grundlehren Math. Wiss. 266, Springer, Berlin 1984.

  • [EKZ]

    A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 207–333.

  • [FLP]

    A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces. Séminaire Orsay, With an English summary, Astérisque 66, Société Mathématique de France, Paris 1979.

  • [Fi]

    J. Fickenscher, Self-inverses, Lagrangian permutations and minimal interval exchange transformations with many ergodic measures, Commun. Contemp. Math. 16 (2014), no. 1, Article ID 1350019.

  • [Fo]

    G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), no. 1, 1–103.

  • [FM]

    G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn. 8 (2014), no. 3–4, 271–436.

  • [FMZ]

    G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle, Ergodic Theory Dynam. Systems 34 (2014), 353–408.

  • [Ga]

    F. P. Gardiner, Teichmüller theory and quadratic differentials, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York 1987.

  • [GH]

    P. Griffiths and J. Harris, Principles of algebraic geometry, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York 1978.

  • [IT]

    Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer, Tokyo 1992.

  • [Ka]

    A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR 211 (1973), 775–778.

  • [Le]

    A. Leutbecher, Über die Heckeschen Gruppen G ( λ ) {{G}(\lambda)}, Abh. Math. Semin. Univ. Hambg. 31 (1967), 199–205.

  • [Mas]

    H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), no. 3, 387–442.

  • [MT]

    H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems. Vol. 1A, North-Holland, Amsterdam (2002), 1015–1089.

  • [Mc1]

    C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857–885.

  • [Mc2]

    C. T. McMullen, Diophantine and ergodic foliations on surfaces, J. Topol. 6 (2013), no. 2, 349–360.

  • [Mc3]

    C. T. McMullen, Entropy on Riemann surfaces and the Jacobians of finite covers, Comment. Math. Helv. 88 (2013), no. 4, 953–964.

  • [Mc4]

    C. T. McMullen, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 1–39.

  • [Mc5]

    C. T. McMullen, Modular symbols for Teichmüller curves, preprint (2019).

  • [Mo1]

    M. Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), no. 2, 327–344.

  • [Mo2]

    M. Möller, Affine groups of flat surfaces, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13, European Mathematical Society, Zürich (2009), 369–387.

  • [Nag]

    S. Nag, The complex analytic theory of Teichmüller spaces, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York 1988.

  • [Ra]

    H. E. Rauch, On the transcendental moduli of algebraic Riemann surfaces, Proc. Natl. Acad. Sci. USA 41 (1955), 42–49.

  • [Roy]

    H. L. Royden, Invariant metrics on Teichmüller space, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York (1974), 393–399.

  • [Sul]

    D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255.

  • [Tr]

    R. Treviño, On the ergodicity of flat surfaces of finite area, Geom. Funct. Anal. 24 (2014), no. 1, 360–386.

  • [V1]

    W. A. Veech, Interval exchange transformations, J. Analyse Math. 33 (1978), 222–272.

  • [V2]

    W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583.

Footnotes

1

It can be shown that the cusps of Γ coincide with (5){}; see [Le, Satz 2], [Mc1, Theorem A.1].

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [Ah]

    L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, Analytic functions, Princeton University Press, Princeton (1960), 45–66.

  • [AF]

    J. S. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards, Duke Math. J. 144 (2008), no. 2, 285–319.

  • [dR]

    G. de Rham, Differentiable manifolds. Forms, currents, harmonic forms, Grundlehren Math. Wiss. 266, Springer, Berlin 1984.

  • [EKZ]

    A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 207–333.

  • [FLP]

    A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces. Séminaire Orsay, With an English summary, Astérisque 66, Société Mathématique de France, Paris 1979.

  • [Fi]

    J. Fickenscher, Self-inverses, Lagrangian permutations and minimal interval exchange transformations with many ergodic measures, Commun. Contemp. Math. 16 (2014), no. 1, Article ID 1350019.

  • [Fo]

    G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), no. 1, 1–103.

  • [FM]

    G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn. 8 (2014), no. 3–4, 271–436.

  • [FMZ]

    G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle, Ergodic Theory Dynam. Systems 34 (2014), 353–408.

  • [Ga]

    F. P. Gardiner, Teichmüller theory and quadratic differentials, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York 1987.

  • [GH]

    P. Griffiths and J. Harris, Principles of algebraic geometry, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York 1978.

  • [IT]

    Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer, Tokyo 1992.

  • [Ka]

    A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR 211 (1973), 775–778.

  • [Le]

    A. Leutbecher, Über die Heckeschen Gruppen G ( λ ) {{G}(\lambda)}, Abh. Math. Semin. Univ. Hambg. 31 (1967), 199–205.

  • [Mas]

    H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), no. 3, 387–442.

  • [MT]

    H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems. Vol. 1A, North-Holland, Amsterdam (2002), 1015–1089.

  • [Mc1]

    C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857–885.

  • [Mc2]

    C. T. McMullen, Diophantine and ergodic foliations on surfaces, J. Topol. 6 (2013), no. 2, 349–360.

  • [Mc3]

    C. T. McMullen, Entropy on Riemann surfaces and the Jacobians of finite covers, Comment. Math. Helv. 88 (2013), no. 4, 953–964.

  • [Mc4]

    C. T. McMullen, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 1–39.

  • [Mc5]

    C. T. McMullen, Modular symbols for Teichmüller curves, preprint (2019).

  • [Mo1]

    M. Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), no. 2, 327–344.

  • [Mo2]

    M. Möller, Affine groups of flat surfaces, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13, European Mathematical Society, Zürich (2009), 369–387.

  • [Nag]

    S. Nag, The complex analytic theory of Teichmüller spaces, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York 1988.

  • [Ra]

    H. E. Rauch, On the transcendental moduli of algebraic Riemann surfaces, Proc. Natl. Acad. Sci. USA 41 (1955), 42–49.

  • [Roy]

    H. L. Royden, Invariant metrics on Teichmüller space, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York (1974), 393–399.

  • [Sul]

    D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255.

  • [Tr]

    R. Treviño, On the ergodicity of flat surfaces of finite area, Geom. Funct. Anal. 24 (2014), no. 1, 360–386.

  • [V1]

    W. A. Veech, Interval exchange transformations, J. Analyse Math. 33 (1978), 222–272.

  • [V2]

    W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583.