Alex Eskin Maryam Mirzakhani - University of Chicagoeskin/luminy2012/slides.pdf · Alex Eskin Maryam Mirzakhani April 10, 2012. Polygons and flat surfaces ... • Dictionary Ergodic

1 / 35

The SL(2, R) action on Moduli space

Alex Eskin Maryam Mirzakhani

April 10, 2012

Polygons and flat surfaces

Polygons and flat

surfaces

• Polygons

• Rational polygons

• Why are rational

polygons easier?

• Properties of flat

surfaces

• The SL(2,R) action

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

2 / 35

Polygons

3 / 35

Let P be a polygon (not necessarily convex).

We consider billiard trajectories inside P .

Problem. Compute the asymptotics as T → ∞ of the number of

(cylinders of) periodic trajectories of length at most T .

In general: Seems very difficult. Even for a triangle it is not known if

periodic trajectories always exist. Best known upper bound is that the

growth rate is subexponential.

Polygons

3 / 35

Let P be a polygon (not necessarily convex).

We consider billiard trajectories inside P .

Problem. Compute the asymptotics as T → ∞ of the number of

(cylinders of) periodic trajectories of length at most T .

In general: Seems very difficult. Even for a triangle it is not known if

periodic trajectories always exist. Best known upper bound is that the

growth rate is subexponential.

Polygons

3 / 35

Let P be a polygon (not necessarily convex).

We consider billiard trajectories inside P .

Problem. Compute the asymptotics as T → ∞ of the number of

(cylinders of) periodic trajectories of length at most T .

In general: Seems very difficult. Even for a triangle it is not known if

periodic trajectories always exist. Best known upper bound is that the

growth rate is subexponential.

Polygons

3 / 35

Let P be a polygon (not necessarily convex).

We consider billiard trajectories inside P .

Problem. Compute the asymptotics as T → ∞ of the number of

(cylinders of) periodic trajectories of length at most T .

In general: Seems very difficult. Even for a triangle it is not known if

periodic trajectories always exist. Best known upper bound is that the

growth rate is subexponential.

Rational polygons

4 / 35

Standing Assumption: P is rational (i.e. all angles are rational multiples

of π).

Theorem (H. Masur) There exist constants c1 > 0 and c2 > 0depending on P such that as T → ∞ the number N(P, T ) of

(cylinders of) periodic trajectories of period at most T satisfies

c1T2 < N(P, T ) < c2T

2.

Goal. Convert the upper and lower bounds in this theorem to an

asymptotic formula, and compute the constant.

Rational polygons

4 / 35

Standing Assumption: P is rational (i.e. all angles are rational multiples

of π).

Theorem (H. Masur) There exist constants c1 > 0 and c2 > 0depending on P such that as T → ∞ the number N(P, T ) of

(cylinders of) periodic trajectories of period at most T satisfies

c1T2 < N(P, T ) < c2T

2.

Goal. Convert the upper and lower bounds in this theorem to an

asymptotic formula, and compute the constant.

Rational polygons

4 / 35

Standing Assumption: P is rational (i.e. all angles are rational multiples

of π).

Theorem (H. Masur) There exist constants c1 > 0 and c2 > 0depending on P such that as T → ∞ the number N(P, T ) of

(cylinders of) periodic trajectories of period at most T satisfies

c1T2 < N(P, T ) < c2T

2.

Goal. Convert the upper and lower bounds in this theorem to an

asymptotic formula, and compute the constant.

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

P

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

SP

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

P

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

P S

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

P S

4π

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

P S

4π

Why are rational polygons easier?

5 / 35

Construction (Zemlyakov-Katok): Given a rational polygon P construct a

surface S such that billiard trajectories on P correspond to straight lines

on S.

P S

4π

A cylinder of periodic trajectories.

Properties of flat surfaces

6 / 35

• The flat metric is nonsingular outside of a finite number of conical

singularities (inherited from the vertices of the polygon).

• The flat metric has trivial holonomy, i.e. parallel transport along any

closed path brings a tangent vector to itself.

• In particular, all cone angles are integer multiples of 2π.

• By convention, the choice of the vertical direction (“direction to the

North”) will be considered as a part of the “ flat structure”.

For example, a surface obtained from a rotated polygon is

considered as a different flat surface.

• A conical singularity with the cone angle 2π · N has N outgoing

directions to the North.

Properties of flat surfaces

6 / 35

• The flat metric is nonsingular outside of a finite number of conical

singularities (inherited from the vertices of the polygon).

• The flat metric has trivial holonomy, i.e. parallel transport along any

closed path brings a tangent vector to itself.

• In particular, all cone angles are integer multiples of 2π.

• By convention, the choice of the vertical direction (“direction to the

North”) will be considered as a part of the “ flat structure”.

For example, a surface obtained from a rotated polygon is

considered as a different flat surface.

• A conical singularity with the cone angle 2π · N has N outgoing

directions to the North.

Properties of flat surfaces

6 / 35

• The flat metric is nonsingular outside of a finite number of conical

singularities (inherited from the vertices of the polygon).

• The flat metric has trivial holonomy, i.e. parallel transport along any

closed path brings a tangent vector to itself.

• In particular, all cone angles are integer multiples of 2π.

• By convention, the choice of the vertical direction (“direction to the

North”) will be considered as a part of the “ flat structure”.

For example, a surface obtained from a rotated polygon is

considered as a different flat surface.

• A conical singularity with the cone angle 2π · N has N outgoing

directions to the North.

Properties of flat surfaces

6 / 35

• The flat metric is nonsingular outside of a finite number of conical

singularities (inherited from the vertices of the polygon).

• The flat metric has trivial holonomy, i.e. parallel transport along any

closed path brings a tangent vector to itself.

• In particular, all cone angles are integer multiples of 2π.

• By convention, the choice of the vertical direction (“direction to the

North”) will be considered as a part of the “ flat structure”.

For example, a surface obtained from a rotated polygon is

considered as a different flat surface.

• A conical singularity with the cone angle 2π · N has N outgoing

directions to the North.

Properties of flat surfaces

6 / 35

• The flat metric is nonsingular outside of a finite number of conical

singularities (inherited from the vertices of the polygon).

• The flat metric has trivial holonomy, i.e. parallel transport along any

closed path brings a tangent vector to itself.

• In particular, all cone angles are integer multiples of 2π.

• By convention, the choice of the vertical direction (“direction to the

North”) will be considered as a part of the “ flat structure”.

For example, a surface obtained from a rotated polygon is

considered as a different flat surface.

• A conical singularity with the cone angle 2π · N has N outgoing

directions to the North.

The SL(2, R) action

7 / 35

A flat surface S is a union of polygons, S = P1 ∪ . . . Pn.

We regard each polygon Pi a subset of R2. The polygons Pi are glued

together along parallel sides. Each side is glued to exactly one other.

P

P

P1 2

3

Suppose g ∈ SL(2, R), e.g. g =

(

1 1/30 1

)

. Since g acts on R2, we

may define

gS = gP1 ∪ . . . ∪ gPn,

with the same identifications of the sides as S.

The SL(2, R) action

7 / 35

A flat surface S is a union of polygons, S = P1 ∪ . . . Pn.

We regard each polygon Pi a subset of R2. The polygons Pi are glued

together along parallel sides. Each side is glued to exactly one other.

P

P

P1 2

3

Suppose g ∈ SL(2, R), e.g. g =

(

1 1/30 1

)

. Since g acts on R2, we

may define

gS = gP1 ∪ . . . ∪ gPn,

with the same identifications of the sides as S.

The SL(2, R) action

7 / 35

A flat surface S is a union of polygons, S = P1 ∪ . . . Pn.

We regard each polygon Pi a subset of R2. The polygons Pi are glued

together along parallel sides. Each side is glued to exactly one other.

P

P

P1 2

3

1 2

3gP

gP gP

g

P

P

P1 2

3

Suppose g ∈ SL(2, R), e.g. g =

(

1 1/30 1

)

. Since g acts on R2, we

may define

gS = gP1 ∪ . . . ∪ gPn,

with the same identifications of the sides as S.

The SL(2, R) action

7 / 35

A flat surface S is a union of polygons, S = P1 ∪ . . . Pn.

We regard each polygon Pi a subset of R2. The polygons Pi are glued

together along parallel sides. Each side is glued to exactly one other.

P

P

P1 2

3

1 2

3gP

gP gP

g

P

P

P1 2

3

Suppose g ∈ SL(2, R), e.g. g =

(

1 1/30 1

)

. Since g acts on R2, we

may define

gS = gP1 ∪ . . . ∪ gPn,

with the same identifications of the sides as S.

Holomorphic 1-forms

versus flat surfaces

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

• From flat to complex

structure

• From complex to flat

structure

• The (relative) period

map and local

coordinates

• Dictionary

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

8 / 35

Holomorphic 1-form associated to a flat structure

9 / 35

Consider the natural coordinate z in the complex plane. In this

coordinate the parallel translations which we use to identify the sides of

the polygon are represented as z′ = z + const.

Since this correspondence is holomorphic, our flat surface S with

punctured conical points has a natural complex structure. This complex

structure extends to the punctured points.

Consider now a holomorphic 1-form dz in the complex plane. The

coordinate z is not globally defined on the surface S. However, since the

changes of local coordinates are defined as z′ = z + const, we see that

dz = dz′. Thus, the holomorphic 1-form dz on C defines a holomorphic

1-form ω on S which in local coordinates has the form ω = dz.

The form ω has zeroes exactly at those points of S where the flat

structure has conical singularities.

Holomorphic 1-form associated to a flat structure

9 / 35

Consider the natural coordinate z in the complex plane. In this

coordinate the parallel translations which we use to identify the sides of

the polygon are represented as z′ = z + const.

Since this correspondence is holomorphic, our flat surface S with

punctured conical points has a natural complex structure. This complex

structure extends to the punctured points.

Consider now a holomorphic 1-form dz in the complex plane. The

coordinate z is not globally defined on the surface S. However, since the

changes of local coordinates are defined as z′ = z + const, we see that

dz = dz′. Thus, the holomorphic 1-form dz on C defines a holomorphic

1-form ω on S which in local coordinates has the form ω = dz.

The form ω has zeroes exactly at those points of S where the flat

structure has conical singularities.

Holomorphic 1-form associated to a flat structure

9 / 35

Consider the natural coordinate z in the complex plane. In this

coordinate the parallel translations which we use to identify the sides of

the polygon are represented as z′ = z + const.

Since this correspondence is holomorphic, our flat surface S with

punctured conical points has a natural complex structure. This complex

structure extends to the punctured points.

Consider now a holomorphic 1-form dz in the complex plane. The

coordinate z is not globally defined on the surface S. However, since the

changes of local coordinates are defined as z′ = z + const, we see that

dz = dz′. Thus, the holomorphic 1-form dz on C defines a holomorphic

1-form ω on S which in local coordinates has the form ω = dz.

The form ω has zeroes exactly at those points of S where the flat

structure has conical singularities.

Holomorphic 1-form associated to a flat structure

9 / 35

Consider the natural coordinate z in the complex plane. In this

coordinate the parallel translations which we use to identify the sides of

the polygon are represented as z′ = z + const.

Since this correspondence is holomorphic, our flat surface S with

punctured conical points has a natural complex structure. This complex

structure extends to the punctured points.

Consider now a holomorphic 1-form dz in the complex plane. The

coordinate z is not globally defined on the surface S. However, since the

changes of local coordinates are defined as z′ = z + const, we see that

dz = dz′. Thus, the holomorphic 1-form dz on C defines a holomorphic

1-form ω on S which in local coordinates has the form ω = dz.

The form ω has zeroes exactly at those points of S where the flat

structure has conical singularities.

Flat structure canonically defined by a holomor-

phic 1-form

10 / 35

Reciprocally a pair (Riemann surface M , holomorphic 1-form ω)

uniquely defines a flat structure.

• In the neighborhood of a point where ω is non-zero, there exists a

local coordinate z such that ω = dz. This coordinate is unique up to

translation z → z + c.

• If we use an atlas of charts using these coordinates in each chart,

we get transition functions which are translations.

• In a neighborhood of zero a holomorphic 1-form can be represented

as ζd dζ , where d is the degree of zero. The form ω has a zero of

degree d at a conical point with cone angle 2π(d + 1).

• The moduli space of pairs (complex structure, holomorphic 1-form) is

naturally stratified by the strata H(d1, . . . , dm) enumerated by

unordered partitions d1 + · · · + dm = 2g − 2.

• Any holomorphic 1-form corresponding to a fixed stratum

H(d1, . . . , dm) has exactly m zeroes; their degrees are d1, . . . , dm.

Flat structure canonically defined by a holomor-

phic 1-form

10 / 35

Reciprocally a pair (Riemann surface M , holomorphic 1-form ω)

uniquely defines a flat structure.

• In the neighborhood of a point where ω is non-zero, there exists a

local coordinate z such that ω = dz. This coordinate is unique up to

translation z → z + c.

• If we use an atlas of charts using these coordinates in each chart,

we get transition functions which are translations.

• In a neighborhood of zero a holomorphic 1-form can be represented

as ζd dζ , where d is the degree of zero. The form ω has a zero of

degree d at a conical point with cone angle 2π(d + 1).

• The moduli space of pairs (complex structure, holomorphic 1-form) is

naturally stratified by the strata H(d1, . . . , dm) enumerated by

unordered partitions d1 + · · · + dm = 2g − 2.

• Any holomorphic 1-form corresponding to a fixed stratum

H(d1, . . . , dm) has exactly m zeroes; their degrees are d1, . . . , dm.

Flat structure canonically defined by a holomor-

phic 1-form

10 / 35

Reciprocally a pair (Riemann surface M , holomorphic 1-form ω)

uniquely defines a flat structure.

• In the neighborhood of a point where ω is non-zero, there exists a

local coordinate z such that ω = dz. This coordinate is unique up to

translation z → z + c.

• If we use an atlas of charts using these coordinates in each chart,

we get transition functions which are translations.

• In a neighborhood of zero a holomorphic 1-form can be represented

as ζd dζ , where d is the degree of zero. The form ω has a zero of

degree d at a conical point with cone angle 2π(d + 1).

• The moduli space of pairs (complex structure, holomorphic 1-form) is

naturally stratified by the strata H(d1, . . . , dm) enumerated by

unordered partitions d1 + · · · + dm = 2g − 2.

• Any holomorphic 1-form corresponding to a fixed stratum

H(d1, . . . , dm) has exactly m zeroes; their degrees are d1, . . . , dm.

Flat structure canonically defined by a holomor-

phic 1-form

10 / 35

Reciprocally a pair (Riemann surface M , holomorphic 1-form ω)

uniquely defines a flat structure.

• In the neighborhood of a point where ω is non-zero, there exists a

local coordinate z such that ω = dz. This coordinate is unique up to

translation z → z + c.

• If we use an atlas of charts using these coordinates in each chart,

we get transition functions which are translations.

• In a neighborhood of zero a holomorphic 1-form can be represented

as ζd dζ , where d is the degree of zero. The form ω has a zero of

degree d at a conical point with cone angle 2π(d + 1).

• The moduli space of pairs (complex structure, holomorphic 1-form) is

naturally stratified by the strata H(d1, . . . , dm) enumerated by

unordered partitions d1 + · · · + dm = 2g − 2.

• Any holomorphic 1-form corresponding to a fixed stratum

H(d1, . . . , dm) has exactly m zeroes; their degrees are d1, . . . , dm.

Flat structure canonically defined by a holomor-

phic 1-form

10 / 35

Reciprocally a pair (Riemann surface M , holomorphic 1-form ω)

uniquely defines a flat structure.

• In the neighborhood of a point where ω is non-zero, there exists a

local coordinate z such that ω = dz. This coordinate is unique up to

translation z → z + c.

• If we use an atlas of charts using these coordinates in each chart,

we get transition functions which are translations.

• In a neighborhood of zero a holomorphic 1-form can be represented

as ζd dζ , where d is the degree of zero. The form ω has a zero of

degree d at a conical point with cone angle 2π(d + 1).

• The moduli space of pairs (complex structure, holomorphic 1-form) is

naturally stratified by the strata H(d1, . . . , dm) enumerated by

unordered partitions d1 + · · · + dm = 2g − 2.

• Any holomorphic 1-form corresponding to a fixed stratum

H(d1, . . . , dm) has exactly m zeroes; their degrees are d1, . . . , dm.

The (relative) period map and local coordinates

11 / 35

For a path γ ∈ S = (M,ω) we denote hol(γ) =∫

γω. Informally, the

real and imaginary parts of hol(γ) are how far “east” and “north” one

travels along γ.

Coordinates on H(α). Let Σ denote the set of singularities (aka

zeroes). Choose a basis γ1, . . . γn for the relative homology group

H1(S, Σ, Z). Then the map Φ : H(α) → (R2)n ≈ Cn given by

Φ(S) = (hol(γ1), . . . , hol(γn))

is a local coordinate system on H(α).

Measure. Let λ be the measure on H(α) which is the pullback of

Lebesque measure on (R2)n. Then λ is well defined, and is invariant

under the SL(2, R) action.

Thm(Masur, Veech) λ(H1(α)) < ∞.

The (relative) period map and local coordinates

11 / 35

For a path γ ∈ S = (M,ω) we denote hol(γ) =∫

γω. Informally, the

real and imaginary parts of hol(γ) are how far “east” and “north” one

travels along γ.

Coordinates on H(α). Let Σ denote the set of singularities (aka

zeroes). Choose a basis γ1, . . . γn for the relative homology group

H1(S, Σ, Z). Then the map Φ : H(α) → (R2)n ≈ Cn given by

Φ(S) = (hol(γ1), . . . , hol(γn))

is a local coordinate system on H(α).

Measure. Let λ be the measure on H(α) which is the pullback of

Lebesque measure on (R2)n. Then λ is well defined, and is invariant

under the SL(2, R) action.

Thm(Masur, Veech) λ(H1(α)) < ∞.

The (relative) period map and local coordinates

11 / 35

For a path γ ∈ S = (M,ω) we denote hol(γ) =∫

γω. Informally, the

real and imaginary parts of hol(γ) are how far “east” and “north” one

travels along γ.

Coordinates on H(α). Let Σ denote the set of singularities (aka

zeroes). Choose a basis γ1, . . . γn for the relative homology group

H1(S, Σ, Z). Then the map Φ : H(α) → (R2)n ≈ Cn given by

Φ(S) = (hol(γ1), . . . , hol(γn))

is a local coordinate system on H(α).

Measure. Let λ be the measure on H(α) which is the pullback of

Lebesque measure on (R2)n. Then λ is well defined, and is invariant

under the SL(2, R) action.

Thm(Masur, Veech) λ(H1(α)) < ∞.

The (relative) period map and local coordinates

11 / 35

For a path γ ∈ S = (M,ω) we denote hol(γ) =∫

γω. Informally, the

real and imaginary parts of hol(γ) are how far “east” and “north” one

travels along γ.

Coordinates on H(α). Let Σ denote the set of singularities (aka

zeroes). Choose a basis γ1, . . . γn for the relative homology group

H1(S, Σ, Z). Then the map Φ : H(α) → (R2)n ≈ Cn given by

Φ(S) = (hol(γ1), . . . , hol(γn))

is a local coordinate system on H(α).

Measure. Let λ be the measure on H(α) which is the pullback of

Lebesque measure on (R2)n. Then λ is well defined, and is invariant

under the SL(2, R) action.

Thm(Masur, Veech) λ(H1(α)) < ∞.

12 / 35

flat structure (including a choice complex structure and a choice

of the vertical direction) of a holomorphic 1-form ω

conical point zero of degree dwith a cone angle 2π(d + 1) of the holomorphic 1-form ω

(in local coordinates ω = ξd dξ)

side ~vj of a polygon relative period∫ Pj+1

Pjω

family of flat surfaces sharing stratum H(d1, . . . , dm) in the

the same cone angles moduli space of holomorphic 1-forms

2π(d1 + 1), . . . , 2π(dm + 1)

coordinates in the family: coordinates in H(d1, . . . , dm) :

vectors ~vi corresponding

to an independent set of cohomology class of ω in

edges in a triangulation H1(S, P1, . . . , Pm; C)

Ergodic Theory

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

13 / 35

The Birkhoff Ergodic Theorem

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

14 / 35

Let X be a topological space, and let T : X → X be a map,

which preserves a measure ν on X . We assume ν(X) = 1(so ν is a “probability measure”).

Definition ν is called ergodic if for any T -invariant subset

E ⊂ X , either ν(E) = 0 or ν(E) = 1.

Theorem (Birkhoff) Suppose ν is ergodic. Then for any

f ∈ L1(X, ν) and ν-almost all x ∈ X ,

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

The Birkhoff Ergodic Theorem

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

14 / 35

Let X be a topological space, and let T : X → X be a map,

which preserves a measure ν on X . We assume ν(X) = 1(so ν is a “probability measure”).

Definition ν is called ergodic if for any T -invariant subset

E ⊂ X , either ν(E) = 0 or ν(E) = 1.

Theorem (Birkhoff) Suppose ν is ergodic. Then for any

f ∈ L1(X, ν) and ν-almost all x ∈ X ,

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

What if we want to know what happens for all x?

A big problem

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

15 / 35

Fact. Let P ⊂ H1(α) denote the flat surfaces which arise

from unfolding a polygon. Then

λ(P ) = 0.

A big problem

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

15 / 35

Fact. Let P ⊂ H1(α) denote the flat surfaces which arise

from unfolding a polygon. Then

λ(P ) = 0.

Using general ergodic theorems (A. Nevo) one can prove

theorems like:

Theorem (Masur and E., Veech) Let N(S, T ) denote the

number of cylinders of closed geodesics on S of period at

most T . There exists a constant bα such that for λ-almost all

S ∈ H1(α), as T → ∞,

N(S, T ) ∼ πbαT 2,

A big problem

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

15 / 35

Fact. Let P ⊂ H1(α) denote the flat surfaces which arise

from unfolding a polygon. Then

λ(P ) = 0.

Using general ergodic theorems (A. Nevo) one can prove

theorems like:

Theorem (Masur and E., Veech) Let N(S, T ) denote the

number of cylinders of closed geodesics on S of period at

most T . There exists a constant bα such that for λ-almost all

S ∈ H1(α), as T → ∞,

N(S, T ) ∼ πbαT 2,

but this says nothing about polygons.

Uniquely ergodic systems

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

16 / 35

Definition A map T : X → X is called uniquely ergodic if

there exists a unique invariant measure ν.

Proposition If T : X → X is uniquely ergodic and X is

compact, then for any f ∈ C(X) and for all x ∈ X ,

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

Proof of Proposition

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

17 / 35

Proof. Let νN be the measure such that

νN (f) =1

N

N−1∑

n=0

f(T nx).

Then

νN (f T ) − νN (f) =1

N(f(TNx) − f(x)). (1)

Let ν∞ be any weak-* limit of a subsequence of the νN . Then

(1) implies that ν∞ is T -invariant. Then unique ergodicity

implies ν∞ = ν. This is the same as νN → ν, i.e.

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

Proof of Proposition

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

17 / 35

Proof. Let νN be the measure such that

νN (f) =1

N

N−1∑

n=0

f(T nx).

Then

νN (f T ) − νN (f) =1

N(f(TNx) − f(x)). (1)

Let ν∞ be any weak-* limit of a subsequence of the νN . Then

(1) implies that ν∞ is T -invariant. Then unique ergodicity

implies ν∞ = ν. This is the same as νN → ν, i.e.

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

Proof of Proposition

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

17 / 35

Proof. Let νN be the measure such that

νN (f) =1

N

N−1∑

n=0

f(T nx).

Then

νN (f T ) − νN (f) =1

N(f(TNx) − f(x)). (1)

Let ν∞ be any weak-* limit of a subsequence of the νN . Then

(1) implies that ν∞ is T -invariant. Then unique ergodicity

implies ν∞ = ν. This is the same as νN → ν, i.e.

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

Proof of Proposition

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

17 / 35

Proof. Let νN be the measure such that

νN (f) =1

N

N−1∑

n=0

f(T nx).

Then

νN (f T ) − νN (f) =1

N(f(TNx) − f(x)). (1)

Let ν∞ be any weak-* limit of a subsequence of the νN . Then

(1) implies that ν∞ is T -invariant. Then unique ergodicity

implies ν∞ = ν. This is the same as νN → ν, i.e.

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

Proof of Proposition

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

17 / 35

Proof. Let νN be the measure such that

νN (f) =1

N

N−1∑

n=0

f(T nx).

Then

νN (f T ) − νN (f) =1

N(f(TNx) − f(x)). (1)

Let ν∞ be any weak-* limit of a subsequence of the νN . Then

(1) implies that ν∞ is T -invariant. Then unique ergodicity

implies ν∞ = ν. This is the same as νN → ν, i.e.

limN→∞

1

N

N−1∑

n=0

f(T nx) =

∫

X

f dν

Unipotent Flows and Ratner’s Theorem

18 / 35

Let G be a semisimple Lie group with finite center, and let Γ ⊂ G be a

lattice. Let U ⊂ G be a unipotent one-parameter subgroup. Then Uacts on X = G/Γ by left multiplication.

Theorem (Ratner)

(i) Any ergodic U -invariant measure on X is homogeneous, i.e. is

L-invariant measure supported on a closed orbit of a subgroup L,

with U ⊆ L ⊆ G.

(ii) For any x ∈ X , Ux = Lx for some subgroup L, with U ⊆ L ⊆ G.

In particular, Ux is a homogeneous submanifold of X .

(iii) “Any orbit is uniformly distributed in its closure”.

Unipotent Flows and Ratner’s Theorem

18 / 35

Let G be a semisimple Lie group with finite center, and let Γ ⊂ G be a

lattice. Let U ⊂ G be a unipotent one-parameter subgroup. Then Uacts on X = G/Γ by left multiplication.

Theorem (Ratner)

(i) Any ergodic U -invariant measure on X is homogeneous, i.e. is

L-invariant measure supported on a closed orbit of a subgroup L,

with U ⊆ L ⊆ G.

(ii) For any x ∈ X , Ux = Lx for some subgroup L, with U ⊆ L ⊆ G.

In particular, Ux is a homogeneous submanifold of X .

(iii) “Any orbit is uniformly distributed in its closure”.

This theorem makes it possible to control all orbits of U .

Unipotent Flows and Ratner’s Theorem

18 / 35

Let G be a semisimple Lie group with finite center, and let Γ ⊂ G be a

lattice. Let U ⊂ G be a unipotent one-parameter subgroup. Then Uacts on X = G/Γ by left multiplication.

Theorem (Ratner)

(i) Any ergodic U -invariant measure on X is homogeneous, i.e. is

L-invariant measure supported on a closed orbit of a subgroup L,

with U ⊆ L ⊆ G.

(ii) For any x ∈ X , Ux = Lx for some subgroup L, with U ⊆ L ⊆ G.

In particular, Ux is a homogeneous submanifold of X .

(iii) “Any orbit is uniformly distributed in its closure”.

This theorem makes it possible to control all orbits of U .

Theorems are false if one replaces U by a 1-parameter diagonalizable

subgroup (e.g. orbit closures can be Cantor sets).

Applications of the theory of Unipotent Flows

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

19 / 35

Uses work of many people, including Dani, Margulis, Ratner,

Mozes, Shah and others.

• Oppenheim Conjecture∗ (Margulis)

• Quantitative Oppenheim Conjecture∗∗ (Margulis-Mozes-E.)

• Some cases of Manin’s Conjecture on asymptotics of the

number of rational points of bounded height

(Macourant-Gorodnik-Oh)

• Some cases of Mazur’s Conjectures on values of

L-functions in the middle of the critical strip (Vatsal,

Cornut-Vatsal)

• Progress toward the Andre-Oort Conjecture (Clozel-Ullmo,

Klingler-Yafaev)

Applications of the theory of Unipotent Flows

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

• The Birkhoff Ergodic

Theorem

• A big problem

• Uniquely ergodic

systems

• Proof of Proposition

• Unipotent Flows and

Ratner’s Theorem

• Applications of the

theory of Unipotent

Flows

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

19 / 35

Uses work of many people, including Dani, Margulis, Ratner,

Mozes, Shah and others.

• Oppenheim Conjecture∗ (Margulis)

• Quantitative Oppenheim Conjecture∗∗ (Margulis-Mozes-E.)

• Some cases of Manin’s Conjecture on asymptotics of the

number of rational points of bounded height

(Macourant-Gorodnik-Oh)

• Some cases of Mazur’s Conjectures on values of

L-functions in the middle of the critical strip (Vatsal,

Cornut-Vatsal)

• Progress toward the Andre-Oort Conjecture (Clozel-Ullmo,

Klingler-Yafaev)

• Distribution of gaps of n1/2α mod 1 (Elkies-McMullen).

• The distribution of free path length in the periodic Lorentz

gas (Marklof-Strombergsson).

Many others...

Moduli space and the space

of lattices

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

• The piecewise linear

structure on H(α).

• Analogy with the

space of lattices

• Oversimplified idea of

proof of Ratner’s

Theorem

• Unipotent flows in

Moduli space

The SL(2, R) action

Further Problems

20 / 35

The piecewise linear structure on H(α).

21 / 35

Fix a (symplectic) basis for γ1, . . . , γn for H1(M, Σ, Z). The local

coordinates are

Ψ((M,ω)) = (

∫

γ1

ω, . . . ,

∫

γn

ω) ∈ Cn ≈ (R2)n

which we write as a 2 by n matrix x. The action of

g =

(

a bc d

)

∈ SL(2, R) in these coordinates is:

x =

(

x1 . . . xn

y1 . . . yn

)

→ gx =

(

a bc d

) (

x1 . . . xn

y1 . . . yn

)

A(g, x),

where A(g, x) ∈ Sp(2g, Z) ⋉ Rk is the Kontsevich-Zorich cocycle.

The piecewise linear structure on H(α).

21 / 35

Fix a (symplectic) basis for γ1, . . . , γn for H1(M, Σ, Z). The local

coordinates are

Ψ((M,ω)) = (

∫

γ1

ω, . . . ,

∫

γn

ω) ∈ Cn ≈ (R2)n

which we write as a 2 by n matrix x. The action of

g =

(

a bc d

)

∈ SL(2, R) in these coordinates is:

x =

(

x1 . . . xn

y1 . . . yn

)

→ gx =

(

a bc d

) (

x1 . . . xn

y1 . . . yn

)

A(g, x),

where A(g, x) ∈ Sp(2g, Z) ⋉ Rk is the Kontsevich-Zorich cocycle.

A(g, x) is change of basis one needs to perform to return the point gxto the fundamental domain.

Analogy with the space of lattices

22 / 35

Let Ωn denote the space of (unimodular) lattices in Rn. We have

Ωn ≈ SL(n, R)/SL(n, Z). Given a basis for the lattice, we can

represent a lattice x in Ωn as an n by n matrix, whose columns are the

basis vectors of the lattice.

The (left-multiplication) action of g ∈ SL(n, R) on Ωn in these

coordinates, sending x → gx, can be written in these coordinates as:

x =

x11 . . . x1n...

...

xn1 . . . xnn

→ g

x11 . . . x1n...

...

xn1 . . . xnn

A(g, x)

where A(g, x) ∈ SL(n, Z) is the change of basis.

Analogy with the space of lattices

22 / 35

Let Ωn denote the space of (unimodular) lattices in Rn. We have

Ωn ≈ SL(n, R)/SL(n, Z). Given a basis for the lattice, we can

represent a lattice x in Ωn as an n by n matrix, whose columns are the

basis vectors of the lattice.

The (left-multiplication) action of g ∈ SL(n, R) on Ωn in these

coordinates, sending x → gx, can be written in these coordinates as:

x =

x11 . . . x1n...

...

xn1 . . . xnn

→ g

x11 . . . x1n...

...

xn1 . . . xnn

A(g, x)

where A(g, x) ∈ SL(n, Z) is the change of basis. The key difference is

that the action of A(g, x) by right multiplication preserves right-invariant

distance and right-invariant vector fields.

Oversimplified idea of proof of Ratner’s Theorem

23 / 35

Suppose ν is a U -invariant measure, and not supported on an orbit of

U . We can find x, y in the support of ν which are close together, and not

in the same U -orbit. There exists a time t such that d(utx, uty) = 1.

x

u yt

y

u xt

There exists δ > 0 (independent of t and x, y) such that for

s ∈ [(1 − δ)t, t], 1/2 ≤ d(usx, usy) ≤ 1, and also the difference

between usx and usy is in the direction of some right-invariant vector

field V . By Birkhoff, usx | (1 − δ)t ≤ s ≤ t equidistributes in G/Γ.

Therefore ν-almost every point in G/Γ has a “friend” in the direction Vapproximately distance 1 away. With some work, one can show that ν is

V -invariant.

Oversimplified idea of proof of Ratner’s Theorem

23 / 35

Suppose ν is a U -invariant measure, and not supported on an orbit of

U . We can find x, y in the support of ν which are close together, and not

in the same U -orbit. There exists a time t such that d(utx, uty) = 1.

u x(1−δ) t

u yt

y

u xtx

u y(1−δ) t

V

There exists δ > 0 (independent of t and x, y) such that for

s ∈ [(1 − δ)t, t], 1/2 ≤ d(usx, usy) ≤ 1, and also the difference

between usx and usy is in the direction of some right-invariant vector

field V . By Birkhoff, usx | (1 − δ)t ≤ s ≤ t equidistributes in G/Γ.

Therefore ν-almost every point in G/Γ has a “friend” in the direction Vapproximately distance 1 away. With some work, one can show that ν is

V -invariant.

Oversimplified idea of proof of Ratner’s Theorem

23 / 35

Suppose ν is a U -invariant measure, and not supported on an orbit of

U . We can find x, y in the support of ν which are close together, and not

in the same U -orbit. There exists a time t such that d(utx, uty) = 1.

u x(1−δ) t

u yt

y

u xtx

u y(1−δ) t

V

There exists δ > 0 (independent of t and x, y) such that for

s ∈ [(1 − δ)t, t], 1/2 ≤ d(usx, usy) ≤ 1, and also the difference

between usx and usy is in the direction of some right-invariant vector

field V . By Birkhoff, usx | (1 − δ)t ≤ s ≤ t equidistributes in G/Γ.

Therefore ν-almost every point in G/Γ has a “friend” in the direction Vapproximately distance 1 away. With some work, one can show that ν is

V -invariant.

Oversimplified idea of proof of Ratner’s Theorem

23 / 35

Suppose ν is a U -invariant measure, and not supported on an orbit of

U . We can find x, y in the support of ν which are close together, and not

in the same U -orbit. There exists a time t such that d(utx, uty) = 1.

u x(1−δ) t

u yt

y

u xtx

u y(1−δ) t

V

There exists δ > 0 (independent of t and x, y) such that for

s ∈ [(1 − δ)t, t], 1/2 ≤ d(usx, usy) ≤ 1, and also the difference

between usx and usy is in the direction of some right-invariant vector

field V . By Birkhoff, usx | (1 − δ)t ≤ s ≤ t equidistributes in G/Γ.

Therefore ν-almost every point in G/Γ has a “friend” in the direction Vapproximately distance 1 away. With some work, one can show that ν is

V -invariant.

Unipotent flows in Moduli space

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

• The piecewise linear

structure on H(α).

• Analogy with the

space of lattices

• Oversimplified idea of

proof of Ratner’s

Theorem

• Unipotent flows in

Moduli space

The SL(2, R) action

Further Problems

24 / 35

This proof does not work in H1(α). The problem is that in

order to tell e.g. the distance between utx and uty one has to

apply the cocycle A(ut, x), and then one loses control. Thus

“polynomial divergence” which is the basis of the theory of

unipotent flows fails in moduli space, except for a few special

cases. This seems to be a very serious problem.

In a few very special cases, e.g. branched covers of Veech

surfaces, one does have polynomial divergence, and one can

prove the analogue of Ratner’s theorem (Marklof-Morris-E,

Calta-Wortman).

Major progress was made by McMullen (2003), who was able

to classify the invariant measures and orbit closures for the

SL(2, R) action in genus 2. This is done by a very clever

reduction to the homogeneous case.

Unipotent flows in Moduli space

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

• The piecewise linear

structure on H(α).

• Analogy with the

space of lattices

• Oversimplified idea of

proof of Ratner’s

Theorem

• Unipotent flows in

Moduli space

The SL(2, R) action

Further Problems

24 / 35

This proof does not work in H1(α). The problem is that in

order to tell e.g. the distance between utx and uty one has to

apply the cocycle A(ut, x), and then one loses control. Thus

“polynomial divergence” which is the basis of the theory of

unipotent flows fails in moduli space, except for a few special

cases. This seems to be a very serious problem.

In a few very special cases, e.g. branched covers of Veech

surfaces, one does have polynomial divergence, and one can

prove the analogue of Ratner’s theorem (Marklof-Morris-E,

Calta-Wortman).

Major progress was made by McMullen (2003), who was able

to classify the invariant measures and orbit closures for the

SL(2, R) action in genus 2. This is done by a very clever

reduction to the homogeneous case.

Unipotent flows in Moduli space

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

• The piecewise linear

structure on H(α).

• Analogy with the

space of lattices

• Oversimplified idea of

proof of Ratner’s

Theorem

• Unipotent flows in

Moduli space

The SL(2, R) action

Further Problems

24 / 35

This proof does not work in H1(α). The problem is that in

order to tell e.g. the distance between utx and uty one has to

apply the cocycle A(ut, x), and then one loses control. Thus

“polynomial divergence” which is the basis of the theory of

unipotent flows fails in moduli space, except for a few special

cases. This seems to be a very serious problem.

In a few very special cases, e.g. branched covers of Veech

surfaces, one does have polynomial divergence, and one can

prove the analogue of Ratner’s theorem (Marklof-Morris-E,

Calta-Wortman).

Major progress was made by McMullen (2003), who was able

to classify the invariant measures and orbit closures for the

SL(2, R) action in genus 2. This is done by a very clever

reduction to the homogeneous case.

The SL(2, R) action

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

25 / 35

The main theorem

26 / 35

Let P = AU =

(

∗ ∗0 ∗

)

⊂ SL(2, R).

Definition An ergodic SL(2, R)-invariant probability measure ν on

H1(α) is called affine if in local coordinates it is the restriction to H1(α)of the Lebesgue measure on a complex subspace of H1(M, Σ, C).

Definition A submanifold of H1(α) is called affine if it is the support of

an affine measure. (So in particular, it is closed, SL(2, R)-invariant, andin local coordinates it is a linear subspace).

Theorem (joint work with Maryam Mirzakhani)

(i) Any ergodic P -invariant measure on H1(α) is SL(2, R)-invariant andaffine.

(ii) For any x, Px = SL(2, R)x is an affine submanifold.

(iii) “Any P -orbit is uniformly distributed in its closure”.

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

27 / 35

Theorem (from previous slide)

(i) Any ergodic P -invariant measure on H1(α) is

SL(2, R)-invariant and affine.

(ii) For any x, Px = SL(2, R)x is an affine submanifold.

(iii) “Any P -orbit is uniformly distributed in its closure”.

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

27 / 35

Theorem (from previous slide)

(i) Any ergodic P -invariant measure on H1(α) is

SL(2, R)-invariant and affine.

(ii) For any x, Px = SL(2, R)x is an affine submanifold.

(iii) “Any P -orbit is uniformly distributed in its closure”.

In genus 2, the SL(2, R) case of statements (i) and (ii) is due

to Curt McMullen (who also does the classification in genus 2).

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

27 / 35

Theorem (from previous slide)

(i) Any ergodic P -invariant measure on H1(α) is

SL(2, R)-invariant and affine.

(ii) For any x, Px = SL(2, R)x is an affine submanifold.

(iii) “Any P -orbit is uniformly distributed in its closure”.

In genus 2, the SL(2, R) case of statements (i) and (ii) is due

to Curt McMullen (who also does the classification in genus 2).

The most difficult part is the proof of (i). Then one proves

(i) =⇒ (iii) =⇒ (ii). These proofs rely on the amenability

of P , and the adaptation of some techniques of Margulis (joint

work with Amir Mohammadi). In particular, one needs the

following:

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

27 / 35

Theorem (from previous slide)

(i) Any ergodic P -invariant measure on H1(α) is

SL(2, R)-invariant and affine.

(ii) For any x, Px = SL(2, R)x is an affine submanifold.

(iii) “Any P -orbit is uniformly distributed in its closure”.

In genus 2, the SL(2, R) case of statements (i) and (ii) is due

to Curt McMullen (who also does the classification in genus 2).

The most difficult part is the proof of (i). Then one proves

(i) =⇒ (iii) =⇒ (ii). These proofs rely on the amenability

of P , and the adaptation of some techniques of Margulis (joint

work with Amir Mohammadi). In particular, one needs the

following:

Proposition Any stratum H(α) contains at most countably

many affine submanifolds.

Notes on the proofs

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

28 / 35

The proof of (i) uses extensively entropy and conditional

measure techniques developed in the context of homogeneous

spaces (Margulis-Tomanov, Einsiedler-Katok-Lindenstrass).

Some of the ideas came from discussions with Amir

Mohammadi.

But the main strategy is to replace polynomial divergence by

the “exponential drift” idea of Benoist-Quint.

Notes on the proofs

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

28 / 35

The proof of (i) uses extensively entropy and conditional

measure techniques developed in the context of homogeneous

spaces (Margulis-Tomanov, Einsiedler-Katok-Lindenstrass).

Some of the ideas came from discussions with Amir

Mohammadi.

But the main strategy is to replace polynomial divergence by

the “exponential drift” idea of Benoist-Quint.

At one point we need to use a result about the isometric

subspace of the cocycle (joint work with A. Avila and

M. Moller) to avoid a problem with relative homology.

Random Walks

29 / 35

Let µ be a finitely supported probability measure on on G. Then µdefines a random walk on G/Γ (where in one step, one moves from

x ∈ G/Γ to gx with probability µ(g)).

Definition A measure ν on G/Γ is called µ-stationary if µ ∗ ν = ν,where

µ ∗ ν =

∫

G

(gν) dµ(g).

Theorem (Benoist-Quint) Suppose the support of µ is compact and

Zariski dense in G. Then any µ-stationary measure on G/Γ is

homogeneous.

Random Walks

29 / 35

Let µ be a finitely supported probability measure on on G. Then µdefines a random walk on G/Γ (where in one step, one moves from

x ∈ G/Γ to gx with probability µ(g)).

Definition A measure ν on G/Γ is called µ-stationary if µ ∗ ν = ν,where

µ ∗ ν =

∫

G

(gν) dµ(g).

Theorem (Benoist-Quint) Suppose the support of µ is compact and

Zariski dense in G. Then any µ-stationary measure on G/Γ is

homogeneous.

The proof introduces a beautiful new idea called “exponential drift” which

can, in this context, substitute for polynomial divergence. In their

argument, the Birkhoff ergodic theorem is replaced by the martingale

convergence theorem.

A Theorem of Furstenberg

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

30 / 35

Definition A measure µ on G is called admissible if it is

compactly supported and absolutely continuous with respect

to the Haar measure on G.

Theorem (Furstenberg) (special case) Suppose µ is an

admissible measure on SL(2, R). Then for any “reasonable”

X , there is a 1 to 1 correspondence between µ-stationarymeasures on X and P -invariant measures on X .

Thus we can rephrase our statement (i) as:

Theorem Suppose µ is an admissible measure on SL(2, R).Then any µ-stationary measure ν on H1(α) is SL(2, R)invariant and affine.

The scheme of the proof

31 / 35

In order to use the the Benoist-Quint exponential drift argument, we

must show that the Zariski closure (more precisely the algebraic hull) of

the Kontsevich-Zorich cocycle is semisimple.

Step 1 We use an entropy argument (together with some ideas from

Benoist-Quint) to show that any P -invariant measure ν on H1(α) is in

fact SL(2, R) invariant.

Step 2 By some results of Forni, for an SL(2, R)-invariant measure ν,

the Kontsevich-Zorich cocycle over the SL(2, R) action is semisimple.

Step 3 We pick an admissible measure µ on SL(2, R) and consider the

associated random walk. By a result of Guivarc‘h and Raugi, the cocycle

remains semisimple in the random walk context.

Step 4 We can now use the Benoist-Quint exponential drift method to

show that the measure ν is affine.

The scheme of the proof

31 / 35

In order to use the the Benoist-Quint exponential drift argument, we

must show that the Zariski closure (more precisely the algebraic hull) of

the Kontsevich-Zorich cocycle is semisimple.

Step 1 We use an entropy argument (together with some ideas from

Benoist-Quint) to show that any P -invariant measure ν on H1(α) is in

fact SL(2, R) invariant.

Step 2 By some results of Forni, for an SL(2, R)-invariant measure ν,

the Kontsevich-Zorich cocycle over the SL(2, R) action is semisimple.

Step 3 We pick an admissible measure µ on SL(2, R) and consider the

associated random walk. By a result of Guivarc‘h and Raugi, the cocycle

remains semisimple in the random walk context.

Step 4 We can now use the Benoist-Quint exponential drift method to

show that the measure ν is affine.

The scheme of the proof

31 / 35

In order to use the the Benoist-Quint exponential drift argument, we

must show that the Zariski closure (more precisely the algebraic hull) of

the Kontsevich-Zorich cocycle is semisimple.

Step 1 We use an entropy argument (together with some ideas from

Benoist-Quint) to show that any P -invariant measure ν on H1(α) is in

fact SL(2, R) invariant.

Step 2 By some results of Forni, for an SL(2, R)-invariant measure ν,

the Kontsevich-Zorich cocycle over the SL(2, R) action is semisimple.

Step 3 We pick an admissible measure µ on SL(2, R) and consider the

associated random walk. By a result of Guivarc‘h and Raugi, the cocycle

remains semisimple in the random walk context.

Step 4 We can now use the Benoist-Quint exponential drift method to

show that the measure ν is affine.

The scheme of the proof

31 / 35

In order to use the the Benoist-Quint exponential drift argument, we

must show that the Zariski closure (more precisely the algebraic hull) of

the Kontsevich-Zorich cocycle is semisimple.

Step 1 We use an entropy argument (together with some ideas from

Benoist-Quint) to show that any P -invariant measure ν on H1(α) is in

fact SL(2, R) invariant.

Step 2 By some results of Forni, for an SL(2, R)-invariant measure ν,

the Kontsevich-Zorich cocycle over the SL(2, R) action is semisimple.

Step 3 We pick an admissible measure µ on SL(2, R) and consider the

associated random walk. By a result of Guivarc‘h and Raugi, the cocycle

remains semisimple in the random walk context.

Step 4 We can now use the Benoist-Quint exponential drift method to

show that the measure ν is affine.

The scheme of the proof

31 / 35

In order to use the the Benoist-Quint exponential drift argument, we

must show that the Zariski closure (more precisely the algebraic hull) of

the Kontsevich-Zorich cocycle is semisimple.

Step 1 We use an entropy argument (together with some ideas from

Benoist-Quint) to show that any P -invariant measure ν on H1(α) is in

fact SL(2, R) invariant.

Step 2 By some results of Forni, for an SL(2, R)-invariant measure ν,

the Kontsevich-Zorich cocycle over the SL(2, R) action is semisimple.

Step 3 We pick an admissible measure µ on SL(2, R) and consider the

associated random walk. By a result of Guivarc‘h and Raugi, the cocycle

remains semisimple in the random walk context.

Step 4 We can now use the Benoist-Quint exponential drift method to

show that the measure ν is affine.

Application to Billiards

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

32 / 35

Theorem For any rational polygon P , there exists c = c(P )such that

limt→∞

1

t

∫ t

0

N(P, es)e−2s ds = c.

Application to Billiards

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

• The main theorem

• Comments

• Notes on the proofs

• Random Walks

• A Theorem of

Furstenberg

• The scheme of the

proof

• Application to Billiards

Further Problems

32 / 35

Theorem For any rational polygon P , there exists c = c(P )such that

limt→∞

1

t

∫ t

0

N(P, es)e−2s ds = c.

We would like to show that

lims→∞

N(P, es)e−2s = c,

but such a result requires proving the analogue of our main

theorem for U instead of P = AU . This seems beyond the

range of the current methods.

Further Problems

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

• Towards a

classification of

Teichmuller curves

• McMullen’s theorem

in genus 2

33 / 35

Towards a classification of Teichmuller curves

34 / 35

First constructions: Thurston and Veech.

Towards a classification of Teichmuller curves

34 / 35

First constructions: Thurston and Veech.

Genus 2: Classification problem is done.

Towards a classification of Teichmuller curves

34 / 35

First constructions: Thurston and Veech.

Genus 2: Classification problem is done.

• In H(2) there is one infinite family of non-square tiled surfaces

(construction done independently by Calta and McMullen). These

consist of pairs (M,ω) where the curve M has a Jacobian which

admits real multiplication by a quadratic number field, and ω is an

eigenform.

• In H(1, 1) the only primitive Teichmuller curve is the regular 10-gon.

(McMullen, using a theorem of Moller)

Towards a classification of Teichmuller curves

34 / 35

First constructions: Thurston and Veech.

Genus 2: Classification problem is done.

• In H(2) there is one infinite family of non-square tiled surfaces

(construction done independently by Calta and McMullen). These

consist of pairs (M,ω) where the curve M has a Jacobian which

admits real multiplication by a quadratic number field, and ω is an

eigenform.

• In H(1, 1) the only primitive Teichmuller curve is the regular 10-gon.

(McMullen, using a theorem of Moller)

Genus 3 or higher: Classification problem is open.

Towards a classification of Teichmuller curves

34 / 35

First constructions: Thurston and Veech.

Genus 2: Classification problem is done.

• In H(2) there is one infinite family of non-square tiled surfaces

(construction done independently by Calta and McMullen). These

consist of pairs (M,ω) where the curve M has a Jacobian which

admits real multiplication by a quadratic number field, and ω is an

eigenform.

• In H(1, 1) the only primitive Teichmuller curve is the regular 10-gon.

(McMullen, using a theorem of Moller)

Genus 3 or higher: Classification problem is open.

• There are infinite families in genus 3 and 4 constructed by McMullen

(coming from Prym varieties)

• There is a family found by Bouw and Moller (with finitely many curves

in each genus). This family contains the families previously found by

Veech and Ward.

• Finiteness results in H(3, 1) and H(4) by Bainbridge and Moller.

McMullen’s theorem in genus 2

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

• Towards a

classification of

Teichmuller curves

• McMullen’s theorem

in genus 2

35 / 35

Theorem (McMullen) In genus 2:

• The closure of any SL(2, R) orbit is an affine submanifold.

• Besides the Teichmuller curves, the only (non-obvious)

affine submanifolds are the sets of curves whose Jacobian

admits real multiplication by a quadratic number field, with

the given holomorphic form as eigenform.

• The only SL(2, R)-invariant measures are affine.

The classification of affine submanifolds beyond genus 2 is

completely open.

McMullen’s theorem in genus 2

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

• Towards a

classification of

Teichmuller curves

• McMullen’s theorem

in genus 2

35 / 35

Theorem (McMullen) In genus 2:

• The closure of any SL(2, R) orbit is an affine submanifold.

• Besides the Teichmuller curves, the only (non-obvious)

affine submanifolds are the sets of curves whose Jacobian

admits real multiplication by a quadratic number field, with

the given holomorphic form as eigenform.

• The only SL(2, R)-invariant measures are affine.

The classification of affine submanifolds beyond genus 2 is

completely open.

McMullen’s theorem in genus 2

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

• Towards a

classification of

Teichmuller curves

• McMullen’s theorem

in genus 2

35 / 35

Theorem (McMullen) In genus 2:

• The closure of any SL(2, R) orbit is an affine submanifold.

• Besides the Teichmuller curves, the only (non-obvious)

affine submanifolds are the sets of curves whose Jacobian

admits real multiplication by a quadratic number field, with

the given holomorphic form as eigenform.

• The only SL(2, R)-invariant measures are affine.

The classification of affine submanifolds beyond genus 2 is

completely open.

McMullen’s theorem in genus 2

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

• Towards a

classification of

Teichmuller curves

• McMullen’s theorem

in genus 2

35 / 35

Theorem (McMullen) In genus 2:

• The closure of any SL(2, R) orbit is an affine submanifold.

• Besides the Teichmuller curves, the only (non-obvious)

affine submanifolds are the sets of curves whose Jacobian

admits real multiplication by a quadratic number field, with

the given holomorphic form as eigenform.

• The only SL(2, R)-invariant measures are affine.

The classification of affine submanifolds beyond genus 2 is

completely open.

McMullen’s theorem in genus 2

Polygons and flat

surfaces

Holomorphic 1-forms

versus flat surfaces

Ergodic Theory

Moduli space and the

space of lattices

The SL(2, R) action

Further Problems

• Towards a

classification of

Teichmuller curves

• McMullen’s theorem

in genus 2

35 / 35

Theorem (McMullen) In genus 2:

• The closure of any SL(2, R) orbit is an affine submanifold.

• Besides the Teichmuller curves, the only (non-obvious)

affine submanifolds are the sets of curves whose Jacobian

admits real multiplication by a quadratic number field, with

the given holomorphic form as eigenform.

• The only SL(2, R)-invariant measures are affine.

The classification of affine submanifolds beyond genus 2 is

completely open.