## 1 Introduction

Let *X* of genus *g*, and let
*X*
together with a transverse invariant measure. If this measure is unique up to scale, we say *uniquely ergodic*.

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

*Suppose the Teichmüller geodesic ray generated by shrinking the leaves of *

The perspective we adopt is based on currents and Hodge theory.
First, we introduce the convex cone *X*, with distributional coefficients, satisfying

where

The language of currents provides a useful bridge between foliations, differential forms and Hodge theory.
Moreover, the closed currents

In this language, our main result is:

*Suppose X lies in a compact subset *

*which meets the unit sphere in a set of diameter *

Here the unit sphere and diameter are defined using the Hodge norm on *K*.

One can regard Theorem 1.2 as a quantitative refinement of Theorem 1.1.
In the recurrent case we can take

### Billiards

Theorem 1.2 also sheds light on the distribution of closed geodesics on

*Consider a sequence of periodic billiard trajectories of slope ${s}_{n}\mathrm{\to}s$ on the golden L-shaped table.
If the lengths of the golden continued fractions for ${s}_{n}$ tend to infinity,
then the trajectories become uniformly distributed as $n\mathrm{\to}\mathrm{\infty}$.*

Three examples with

An analogous statement holds for any lattice polygon and any 1-form generating a
Teichmüller curve

### The cone of positive currents

Here is a sketch of the proof of Theorem 1.2.

Let

where

Let

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

holds whenever

where *T* is the amount of time *K* for

Combining these bounds gives the stronger cone condition

whenever

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

which records the Hodge structure on the part of *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

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

### 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

By Hodge theory, the map sending a cohomology class to its harmonic representative
provides an isomorphism

on *Hodge norm* is defined by

Similarly, the space

whose associated norm is given by

These norms are compatible in the sense that
the map

### Foliations and measures

Every nonzero *horizontal foliation*
*X*. To describe this
foliation, recall that

is a linear combination of real harmonic forms satisfying

The foliation *L* of

A *transverse measure* for *invariant* if it also compatible with the smooth maps between nearby transversal obtained
by flowing along the leaves of

The standard transverse invariant measure for *L* of

### Moduli of Riemann surfaces

Fix a compact, oriented topological surface *Teichmüller space*
*g* together with an orientation–preserving *marking* homeomorphism

presenting *moduli space*
*g*.

The cotangent space to *X* is naturally
identified with the space *X*,
and the Teichmüller metric corresponds to the norm *X* with

between

For later reference, we note that when

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

### Moduli of forms

Consider the holomorphic vector bundle over *X* is

will be denoted by

### Dynamics and geodesics

There is a natural action of

if and only if there is a map

The orbits

project to Teichmüller geodesics in

for all

### Dynamics and cohomology

The vector bundle

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

under the map

*The sub-bundles W and *

Put differently, we have

## 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.

*Let *

Here *K*. Geometrically, this results says that the
length of the current ξ in the metric

### 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 *f* on *X*.
Any closed current determines a cohomology class

Given a nonzero holomorphic form

The final positivity condition means

for all smooth *X*;
in particular, ξ is required to satisfy infinitely many linear inequalities.
We refer to *closed, positive currents* carried by

*There is a natural bijection between the closed, positive currents carried by *

A transverse invariant measure determines a current

### The cohomology class of a measured foliation

Each transverse invariant measure determines a cohomology class, by the correspondence
*minimal* if each of its leaves is dense in *X*.

*If *

It suffices to show that the values of *L* be a dense leaf of *L*. By adding a piece of *L* to connect these endpoints, we obtain a closed loop

Since the integrals above only depend on

*If *

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 *g*.
∎

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

We will first prove a cone inequality for a single form

for all

Choose a sequence of smooth, closed 1-forms *U* of

pointwise on *X*, for

Now for any

as measures on *X*, in the limit we obtain

and hence

for all *i*. This implies that

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

We now allow the form *X* in a given compact set

## 4 The complementary period mapping

In this section we study the variation of the Hodge norm along a geodesic in

### Norms

Let

Since all the Riemann surfaces in

for all

It is easy to see that

for all

*Fix a compact set *

*and all *

*Here *

Let

Then

by equation (2.4).

Let

under the identification (4.1). By assumption, we have

for all

Define a function κ on

(The brackets denote the natural pairing (2.2) between tangent and cotangent vectors to *X*.)
Since

Let

and hence

This shows that

for all *t*, and that

whenever *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 *period map* to Siegel space,

Second, the choice of a 1-form *X* determines a natural splitting

where *complex geodesic*

related to the real geodesic by

where the two factors of the product *W* and

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

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

for all *s*, and the upper bound can be replaced by

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

Let *T* denote the amount of time that *K* for

*We have *

Let

by equation (2.3).
Since

Recall that

The Teichmüller mapping

By Theorem 3.1, we then have

whenever

Note that the cohomology classes *t*. The first is constant
because we use *t* varies, and the second is constant because
the span of *t* (cf. Proposition 2.1).

Suppose

by Theorem 4.1. Setting

Since

Theorem 1.2 is then equivalent to:

*The diameter of the intersection of *

Consider any

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

## 6 Billiards and equidistribution

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

### The golden table

Let *P* denote the symmetric *L*-shaped polygon *P* together by horizontal and vertical translations, we obtain a holomorphic 1-form

Its quotient

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

### 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
*golden continued fraction*,

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

and similarly for each subsequent *length*

Consider a sequence of periodic slopes *P* at slope *X*, *P*
it suffices to prove equidistribution of *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

Let *X*;
dividing through by the length of

Our goal is to show that *X* naturally decomposes into a pair
of horizontal cylinders

for some

Now we use the fact that *K* is comparable
to

and hence *X* as

### Sample slopes

The first two examples in Figure 1 depict periodic billiard trajectories in *P*
for the sequence of slopes *P*, and they all
lie in

# A Appendix: Variation of the Hodge norm

We will show that a classical result of Ahlfors gives:

*Fix a cohomology class *

*C*satisfies

Here a variation in *X* is described by a smooth path *t*.
Note that the quadratic differential *X*,
so it pairs naturally with the tangent vector

Fix a standard symplectic basis *Siegel period matrix* for *X*
is defined by

where

The matrix

Since equation (A.1) is homogeneous, we can assume

Now consider a variation *X*. Then

Let

Since

and therefore

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

Here is an equivalent formulation, used Section 4:

*For any nonzero *

*where *

## Notes and references

A variant of Theorem A.1,
with a different proof, is given in
[Fo, Lemma 2.1

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

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