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.