Teichmüller Space
Let S = Sg ,n = surface of genus g , with n punctures. (3g − 3 + n > 0)
Define Teichmüller space T = Tg ,n by:
The Weil-Petersson geodesic flow is ergodic
T
Amie Wilkinson
(with Keith Burns and Howard Masur)
= { marked conformal structures on S}/conformal equivalence
= { marked hyperbolic structures on S}/isometry
Marked means each curve in S “has a name.”
Formally, an element of T is represented by a pair (X , f ), where
X = Riemann surface, and
f:S →X
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is a marking homeomorphism. Equivalent definition:
T
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Properties of T
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Properties of T
Homeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).
“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).
Complex structure is natural, but not obvious.
“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).
Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
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= {discrete, faithful rep’n ρ : π1 (S) → PSL(2, R)}/conjugacy
Homeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).
“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).
Complex structure is natural, but not obvious.
“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).
Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
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Properties of T
Properties of
of TT
Properties
Homeomorphic to
to aa ball
ball of
of dimension
dimension 6g
6g −
− 66 +
+ 2n.
2n.
Homeomorphic
Real analytic
analytic manifold
manifold (embeds
(embeds in
in real
real representation
representation variety).
variety).
Real
“Fuchsian uniformization”
uniformization” is
is mechanism.
mechanism.
“Fuchsian
Complex analytic
analytic manifold
manifold (embeds
(embeds in
in complex
complex representation
representation variety).
variety).
Complex
Complex
structure
is
natural,
but
not
obvious.
Complex structure is natural, but not obvious.
“Quasifuchsian uniformization”
uniformization” is
is mechanism.
mechanism.
“Quasifuchsian
Example: SS =
= punctured
punctured torus,
torus, (g
(g ,, n)
n) =
= (1,
(1, 1).
1).
Example:
Conformal
structure
on
S=
lattice
in
C
(up
to multiplication
multiplication by
by λ
λ∈
∈ C)
C)
Conformal structure on S= lattice in C (up to
Homeomorphic to
to aa ball
ball of
of dimension
dimension 6g
6g −
− 66 +
+ 2n.
2n.
Homeomorphic
Real analytic
analytic manifold
manifold (embeds
(embeds in
in real
real representation
representation variety).
variety).
Real
“Fuchsian uniformization”
uniformization” isis mechanism.
mechanism.
“Fuchsian
Complex analytic
analytic manifold
manifold (embeds
(embeds in
in complex
complex representation
representation variety).
variety).
Complex
Complex
structure
is
natural,
but
not
obvious.
Complex structure is natural, but not obvious.
“Quasifuchsian uniformization”
uniformization” isis mechanism.
mechanism.
“Quasifuchsian
Example: SS =
= punctured
punctured torus,
torus, (g
(g,, n)
n) =
= (1,
(1, 1).
1).
Example:
Conformal
structure
on
S=
lattice
in
C
(up
to multiplication
multiplication by
by λλ ∈
∈ C)
C)
Conformal structure on S= lattice in C (up to
T ∼
=H
T ∼
=H
same structure, different marking!
z
0
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Chicago)
0
1
Ergodicity of
of WP
WP flow
flow
Ergodicity
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Properties of T
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13
63 // 32
Fenchel-Nielsen coordinates on T
Let σ = maximal collection of disjoint simple closed curves on S:
Homeomorphic to a ball of dimension 6g − 6 + 2n.
Real analytic manifold (embeds in real representation variety).
“Fuchsian uniformization” is mechanism.
Complex analytic manifold (embeds in complex representation variety).
Complex structure is natural, but not obvious.
“Quasifuchsian uniformization” is mechanism.
Example: S = punctured torus, (g , n) = (1, 1).
Conformal structure on S= lattice in C (up to multiplication by λ ∈ C)
Fenchel-Nielsen coordinates {�α , τα }α∈σ . For X ∈ T :
�α (X ) = hyperbolic length of α on X ;
different
τα (X ) = ”twist parameter”
H
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼
= H; σ = {α}
0
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�α (x + yi) �
1
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1
,
y
τα (x + yi) �
Ergodicity of WP flow
x
y
as y → ∞.
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Fenchel-Nielsen
Fenchel-Nielsen coordinates
coordinates on
on T
T : “angle/action”
Fenchel-Nielsen coordinates on T
Let
Let σ
σ=
= maximal
maximal collection
collection of
of disjoint
disjoint simple
simple closed
closed curves
curves on
on S:
S:
Let σ = maximal collection of disjoint simple closed curves on S:
α3
α1
α2
Fenchel-Nielsen coordinates
coordinates {�
{�αα ,, τταα }}α∈σ
For X
X ∈
∈T
T ::
α∈σ .. For
Fenchel-Nielsen
�α (X ) = hyperbolic length of α on X ;
Fenchel-Nielsen coordinates {�α , τα }α∈σ . For X ∈ T :
τα (X ) = ”twist parameter”
�α (X ) = hyperbolic length of α on X ;
τα (X ) = ”twist parameter”
For any σ, these give global coordinates on T .
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼
= H; σ = {α}
Example: once-punctured torus: T ∼
= H; σ = {α}
�α (x + yi) �
1
,
y
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x
y
τα (x + yi) �
as y → ∞.
Ergodicity of WP flow
Ergodicity of WP flow
�α (x + yi) �
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1
,
y
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x
y
τα (x + yi) �
as y → ∞.
Ergodicity of WP flow
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Fenchel-Nielsen coordinates on T
Let σ = maximal collection of disjoint simple closed curves on S:
Fenchel-Nielsen coordinates on T
Fenchel-Nielsen coordinates {�α , τα }α∈σ . For X ∈ T :
Fenchel-Nielsen coordinates {�α , τα }α∈σ . For X ∈ T :
�α (X ) = hyperbolic length of α on X ;
Let σ = maximal collection of disjoint simple closed curves on S:
τα (X ) = ”twist parameter”
�α (X ) = hyperbolic length of α on X ;
τα (X ) = ”twist parameter”
For any σ, these give global coordinates on T .
For any σ, these give global coordinates on T .
Example: once-punctured torus: T ∼
= H; σ = {α}
Example: once-punctured torus: T ∼
= H; σ = {α}
�α (x + yi) �
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,
y
τα (x + yi) �
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x
y
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�α (x + yi) �
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1
,
y
τα (x + yi) �
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x
y
as y → ∞.
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The Weil-Petersson metric
Moduli space (where the curves have no name...)
Theorem: [Wolpert] ∃ symplectic form ω on T s.t. for any maximal
collection σ:
�
ω=
d�α ∧ dτα
Define moduli space M = Mg ,n by:
M = {conformal structures on S}/conformal equivalence
α∈σ
Together with the almost complex structure J, this determines a (Kähler)
Riemannian metric:
Mapping class group MCG = MCGg ,n = Diff+ (S)/Diff0 (S)
MCG acts (virtually) freely on T and is the (orbifold) fundamental group
of M:
(for v , w ∈ TX T )
gWP (v , w ) = ω(v , Jw ),
= {hyperbolic structures on S}/isometry
called the Weil-Petersson metric.
M = T / MCG
Example: once-punctured torus: T ∼
= H, standard complex structure:
ω = d�α ∧ dτα �
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dx ∧ dy
;
y3
2
gWP
�
dx 2 + dy 2
,
y3
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In punctured torus example, MCG = SL(2, Z) and M = punctured sphere
with 2 cone points.
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Properties of the WP metric
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Properties of the WP metric
Incomplete: X ∈ T can go to infinity along a geodesic in time �
�(X )1/2
(where �(X ) = length of shortest curve on X ).
Incomplete: X ∈ T can go to infinity along a geodesic in time � �(X )1/2
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞
nor are they bounded away from 0, except in sporadic cases.
(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Negatively curved: Sectional curvatures are not bounded away from −∞
nor are they bounded away from 0, except in sporadic cases.
(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Geodesically convex: ∀X , Y ∈ T , ∃ unique geodesic in T from X to Y .
Geodesically convex: ∀X , Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.
Almost every geodesic on M is defined for all time and is recurrent.
MCG-invariant: descends to a metric on M, of finite volume.
Almost every geodesic on M is defined for all time and is recurrent.
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Properties of the WP metric
Properties of the WP metric
Incomplete: X ∈ T can go to infinity along a geodesic in time �
�(X )1/2
(where �(X ) = length of shortest curve on X ).
Incomplete: X ∈ T can go to infinity along a geodesic in time � �(X )1/2
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞
nor are they bounded away from 0, except in sporadic cases.
(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Negatively curved: Sectional curvatures are not bounded away from −∞
nor are they bounded away from 0, except in sporadic cases.
(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Geodesically convex: ∀X , Y ∈ T , ∃ unique geodesic in T from X to Y .
Geodesically convex: ∀X , Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.
Almost every geodesic on M is defined for all time and is recurrent.
MCG-invariant: descends to a metric on M, of finite volume.
Almost every geodesic on M is defined for all time and is recurrent.
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Properties of the WP metric
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The WP Metric and deformation of hyperbolic structures
Incomplete: X ∈ T can go to infinity along a geodesic in time �
�(X )1/2
The WP metric carries information about infinitesimal deformations of
hyperbolic structures. For example:
(where �(X ) = length of shortest curve on X ).
Negatively curved: Sectional curvatures are not bounded away from −∞
nor are they bounded away from 0, except in sporadic cases.
(In punctured torus case, curvature at x + yi is � −y as y → ∞)
Limit set for fuchsian (hyperbolic) punctured torus
Geodesically convex: ∀X , Y ∈ T , ∃ unique geodesic in T from X to Y .
MCG-invariant: descends to a metric on M, of finite volume.
Almost every geodesic on M is defined for all time and is recurrent.
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The WP Metric and deformation of hyperbolic structures
The WP metric carries information about infinitesimal deformations of
hyperbolic structures. For example:
The geodesic flow
Starting point: the geodesic
M = Riemannian
manifold. T 1 M = flow
unit tangent bundle to M. Natural
1
1
flow ϕt : T S M
→ Tsurface,
M called
themetric.
geodesic
= closed
Riemannian
T 1 S =flow:
unit tangent bundle to S.
Natural flow ϕt : T 1 S → T 1 S called the geodesic flow:
S
ϕt (v )
v
distance t
Basic question: is there a dense geodesic on S?
Stronger:
there aadense
ϕt -orbit
in T 1 S?on M?
Basic question:
is isthere
dense
geodesic
Stronger yet: is almost every orbit (w.r.t. volume) equidistributed in T 1 S?
Limit set for quasifuchsian (projective) punctured torus (McMullen)
Stronger: is there a dense ϕt -orbit in T 1 M?
Theorem (McMullen):
Stronger yet: is almost every orbit (w.r.t. volume) equidistributed in
T 1 M?
d2
1 �Ẋ0 �2WP
dim
(Λ
)|
=
.
t
t=0
H
dt 2
3 area(X0 )
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Conservative partially hyperbolic dynamics
August 22, 2010
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E. Hopf (1939): Yes, in the special case where M is a closed, negatively
curved surface.
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The geodesic flow in negative curvature
The geodesic flow in negative curvature
Negative curvature on M creates hyperbolicity orthogonal to the direction
Negative curvature on S creates hyperbolicity orthogonal to the direction
of the geodesic flow:
of the geodesic flow:
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The geodesic flow in negative curvature
The geodesic flow in negative curvature
Negative curvature on M creates hyperbolicity orthogonal to the direction
Negative curvature on S creates hyperbolicity orthogonal to the direction
of the geodesic flow:
of the geodesic flow:
ϕt (v )
S
S
v
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Conservative partially hyperbolic dynamics
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Conservative partially hyperbolic dynamics
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The geodesic flow in negative curvature
The geodesic flow in negative curvature
The geodesic flow in negative curvature
The geodesic flow in negative curvature
Negative curvature on M creates hyperbolicity orthogonal to the direction
Negative curvature on S creates hyperbolicity orthogonal to the direction
of the geodesic flow:
of the geodesic flow:
Negative curvature
curvature on
on S
M creates
createshyperbolicity
hyperbolicityorthogonal
orthogonalto
tothe
thedirection
direction
Negative
of
the
geodesic
flow:
of the geodesic flow:
S
ϕt (v )
v
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Conservative partially hyperbolic dynamics
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August 22, 2010
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Conservative partially hyperbolic dynamics
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4 / 19
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Hyperbolicity
Hyperbolicity
The Hopf argument for ergodicity
Theorem (E. Hopf): If M is a closed negatively curved surface, then the
geodesic flow is ergodic (with respect to Liouville volume): for almost
every v ∈ T 1 M, the ϕt -orbit of v is equidistributed in T 1 M:
1
u
c
s
If
compact and
negatively
then
If M
S isis negatively
curved,
then Tcurved,
(T 1 S) =
E u T⊕(T
E c M)
⊕ E=s : E ⊕ E ⊕ E :
Ec
v
1
T
ϕt (v )
Es
Eu
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h(ϕt (v )) dt
0
T →±∞
−→
h± (v ) :=
(Hadamard) The geodesic flow is hyperbolic. E u = unstable bundle, E s =
stable bundle. E u is tangent to unstable foliation W u , and E s is tangent
to stable foliation W s . E c is tangent to orbit foliation O
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T
1
Vol(T 1 M)
�
h dVol,
T 1S
∀h ∈ C 0 (T 1 S).
Idea of proof (“Hopf Argument”): Fix h ∈ C 0 (T 1 M). Birkhoff/von
Neumann Ergodic Theorems imply that for a.e. v :
where E c is tangent to ϕt orbits, E u is expanded under Dϕt , and E s is
contracted under Dϕt .
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�
26 / 32
5 / 19
1
T →±∞ T
lim
�
T
h(ϕt (v )) dt
0
exist and are equal. Moreover, h+ is constant along W s leaves, h− is
constant along W u leaves, and h± are constant along ϕt orbits (O leaves).
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The Hopf argument, continued
The Hopf argument, continued
h− is constant along W u manifolds.
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A typical W u manifold.
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The Hopf argument, continued
Ergodicity of WP flow
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The Hopf argument, continued
h− is const. along O manifolds (orbits of ϕ).
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h− is const along the O leaf through a.e. point.
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The Hopf argument, continued
The Hopf argument, continued
h− is const along the O saturate of a typical W u manifold.
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h− is const along the O saturate of a typical W u manifold.
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The Hopf argument, continued
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The Hopf argument, continued
h+ is constant along W s manifolds.
On this surface, h− is constant, and h− = h+ a.e.
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Ergodicity of WP flow
The Hopf argument, continued
The Hopf argument, continued
h+ is constant along W s manifolds.
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Ergodicity of WP flow
h+ is constant along the W s -saturate of this surface.
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The Hopf argument, continued
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The Hopf argument, continued
h+ is constant in a box ⇒ locally constant ⇒ constant.
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Hence ϕ is ergodic.
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Ergodicity of geodesic flows
Main Result
Anosov (1960’s): M = closed, negatively curved manifold, any dimension
⇒ geodesic flow is ergodic (new ingredient: absolute continuity).
Theorem: [Burns, Masur, W] For any (g , n), the WP geodesic flow on
T 1 T / MCG is ergodic and has finite metric entropy. Almost every stable
manifold (horosphere) in T 1 T is smooth and large.
Question of when ergodicity holds is still open for:
Previous results on WP geodesic flow:
• Closed, nonpositively curved surfaces.
Transitivity: there exist dense geodesics [Brock-Masur-Minsky, also
Pollicott-Weiss-Wolpert for (1, 1) case].
• Complete, negatively curved surfaces of finite volume.
Infinite topological entropy: there exist compact invariant sets with
unboundedly large entropy [BMM].
• Incomplete, negatively curved surfaces.
Ergodic closing lemma: periodic measures are dense among ergodic
probability measures [Hamenstädt] .
As a very special case of the latter, consider the WP geodesic flow on
T 1 T / MCG: is it ergodic?
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The proof
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Input from Teichmüller theory:
• Asymptotics for WP metric and covariant derivative: Wolpert
gives precise asymptotics to order 2 in special coordinates near ∂T .
• Asymptotics for higher derivatives of WP metric: (after McMullen).
∃ totally real embedding of T into complex manifold (quasifuchsian
space) where ω extends to a holomorphic form with bounded primitive.
Asymptotics obtained from Cauchy integral formula.
The proof has three main ingredients:
Teichmüller theory
Differential geometry component:
Differential geometry
• Analyze Jacobi equation (using Teich input) to bound first and second
derivatives of geodesic flow in terms of distance to ∂T .
Smooth ergodic theory
Smooth ergodic theory:
• Modified Hopf argument: key property is absolute continuity. One
novelty: Stable manifolds are complete! (thanks to geodesic convexity).
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