the weil-petersson geodesic ﬂow is ergodic t s · 2013-05-14 · the weil-petersson geodesic...

The Weil-Petersson geodesic flow is ergodic

Amie Wilkinson(with Keith Burns and Howard Masur)

University of Chicago

INdAM Lectures

Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 1 / 32

Teichmuller Space

Let S = Sg ,n = surface of genus g , with n punctures. (3g − 3 + n > 0)

Define Teichmuller space T = Tg ,n by:

T = { 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 ), whereX = Riemann surface, and

f : S → X

is a marking homeomorphism. Equivalent definition:

T = {discrete, faithful rep’n ρ : π1(S) → PSL(2,R)}/conjugacy


Properties of THomeomorphic 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)















0 1

z

T ∼= H

Wilkinson (Northwestern University) Ergodicity of WP flow Palis Birthday Conference 3 / 13Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 5 / 32









0 1

same structure, different marking!

T ∼= H

Wilkinson (Northwestern University) Ergodicity of WP flow Palis Birthday Conference 3 / 13Wilkinson (University of Chicago) Ergodicity of WP flow INdAM Lectures 6 / 32





0 1

H

different


Fenchel-Nielsen coordinates on TLet σ = maximal collection of disjoint simple closed curves on S :

Fenchel-Nielsen coordinates {�α, τα}α∈σ. For X ∈ T :

�α(X ) = hyperbolic length of α on X ; τα(X ) = ”twist parameter”

For any σ, these give global coordinates on T .

Example: once-punctured torus: T ∼= H; σ = {α}

�α(x + yi) � 1

y, τα(x + yi) � x

yas y → ∞.


Fenchel-Nielsen coordinates on TLet σ = maximal collection of disjoint simple closed curves on S :Fenchel-Nielsen coordinates on T : “angle/action”Let σ = maximal collection of disjoint simple closed curves on S :

α1

α2

α3


Wilkinson (Northwestern University) Ergodicity of WP flow Palis Birthday Conference 4 / 13





�α(x + yi) � 1


yas y → ∞.







�α(x + yi) � 1


yas y → ∞.







�α(x + yi) � 1


yas y → ∞.







�α(x + yi) � 1


yas y → ∞.


The Weil-Petersson metric

Theorem: [Wolpert] ∃ symplectic form ω on T s.t. for any maximalcollection σ:

ω =�

α∈σd�α ∧ dτα

Together with the almost complex structure J, this determines a (Kahler)Riemannian metric:

gWP(v ,w) = ω(v , Jw), (for v ,w ∈ TXT )

called the Weil-Petersson metric.

Example: once-punctured torus: T ∼= H, standard complex structure:

ω = d�α ∧ dτα � dx ∧ dy

y3; g2

WP � dx2 + dy2

y3, as y → ∞.


Moduli space (where the curves have no name...)

Define moduli space M = Mg ,n by:

M = {conformal structures on S}/conformal equivalence

= {hyperbolic structures on S}/isometry

Mapping class group MCG = MCGg ,n = Diff+(S)/Diff0(S)

MCG acts (virtually) freely on T and is the (orbifold) fundamental groupof M:

M = T /MCG

In punctured torus example, MCG = SL(2,Z) and M = punctured spherewith 2 cone points.


Properties of the WP metricIncomplete: 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 → ∞)

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.


























The WP Metric and deformation of hyperbolic structuresThe WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:

Limit set for fuchsian (hyperbolic) punctured torus

The WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:

Figure: Limit set for Fuchsian (hyperbolic) punctured torus

Theorem (McMullen):

d2

dt2dimH(Λt)|t=0 =

1

3

�X0�2WP

area(X0).


The WP Metric and deformation of hyperbolic structuresThe WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:

Limit set for quasifuchsian (projective) punctured torus (McMullen)

Theorem (McMullen):

d2

dt2dimH(Λt)|t=0 =

1

3

�X0�2WP

area(X0).

The WP metric carries information about infinitesimal deformations ofhyperbolic structures. For example:

Figure: Limit set for Fuchsian (hyperbolic) punctured torus

Theorem (McMullen):

d2

dt2dimH(Λt)|t=0 =

1

3

�X0�2WP

area(X0).


The geodesic flow

M = Riemannian manifold. T 1M = unit tangent bundle to M. Naturalflow ϕt : T 1M → T 1M called the geodesic flow:

Starting point: the geodesic flow

S = closed surface, Riemannian metric. T 1S = unit tangent bundle to S .Natural flow ϕt : T 1S → T 1S called the geodesic flow:

S

v

ϕt(v)

distance t

Basic question: is there a dense geodesic on S?

Stronger: is there a dense ϕt -orbit in T 1S?

Stronger yet: is almost every orbit (w.r.t. volume) equidistributed in T 1S?

Amie Wilkinson (Northwestern University) Conservative partially hyperbolic dynamics August 22, 2010 3 / 19

Basic question: is there a dense geodesic on M?

Stronger: is there a dense ϕt-orbit in T 1M?

Stronger yet: is almost every orbit (w.r.t. volume) equidistributed inT 1M?

E. Hopf (1939): Yes, in the special case where M is a closed, negativelycurved surface.


The geodesic flow in negative curvature

Negative curvature on M creates hyperbolicity orthogonal to the directionof the geodesic flow:

The geodesic flow in negative curvatureNegative curvature on S creates hyperbolicity orthogonal to the directionof the geodesic flow:

S

v

ϕt(v)






S






S






vϕt(v)



Hyperbolicity

If M is compact and negatively curved, then T (T 1M) = Eu ⊕ E c ⊕ E s :HyperbolicityIf S is negatively curved, then T (T 1S) = Eu ⊕ E c ⊕ E s :

vϕt(v)

E u

E c

E s


where E c is tangent to ϕt orbits, Eu is expanded under Dϕt , and E s iscontracted under Dϕt .

(Hadamard) The geodesic flow is hyperbolic. Eu = unstable bundle, E s =stable bundle. Eu is tangent to unstable foliation Wu, and E s is tangentto stable foliation Ws . E c is tangent to orbit foliation O


The Hopf argument for ergodicity

Theorem (E. Hopf): If M is a closed negatively curved surface, then thegeodesic flow is ergodic (with respect to Liouville volume): for almostevery v ∈ T 1M, the ϕt-orbit of v is equidistributed in T 1M:

1

T

� T

0h(ϕt(v)) dt

T→±∞−→ 1

Vol(T 1M)

�

T 1Sh dVol, ∀h ∈ C 0(T 1S).

Idea of proof (“Hopf Argument”): Fix h ∈ C 0(T 1M). Birkhoff/vonNeumann Ergodic Theorems imply that for a.e. v :

h±(v) := limT→±∞

1

T

� T

0h(ϕt(v)) dt

exist and are equal. Moreover, h+ is constant along Ws leaves, h− isconstant along Wu leaves, and h± are constant along ϕt orbits (O leaves).


The Hopf argument, continued

h− is constant along Wu manifolds.



A typical Wu manifold.



h− is const. along O manifolds (orbits of ϕ).



h− is const along the O leaf through a.e. point.



h− is const along the O saturate of a typical Wu manifold.



h− is const along the O saturate of a typical Wu manifold.



On this surface, h− is constant, and h− = h+ a.e.



h+ is constant along Ws manifolds.



h+ is constant along Ws manifolds.



h+ is constant along the Ws -saturate of this surface.



h+ is constant in a box ⇒ locally constant ⇒ constant.



Hence ϕ is ergodic.


Ergodicity of geodesic flows

Anosov (1960’s): M = closed, negatively curved manifold, any dimension⇒ geodesic flow is ergodic (new ingredient: absolute continuity).

Question of when ergodicity holds is still open for:

• Closed, nonpositively curved surfaces.

• Complete, negatively curved surfaces of finite volume.

• Incomplete, negatively curved surfaces.

As a very special case of the latter, consider the WP geodesic flow onT 1T /MCG: is it ergodic?


Main Result

Theorem: [Burns, Masur, W] For any (g , n), the WP geodesic flow onT 1T /MCG is ergodic and has finite metric entropy. Almost every stablemanifold (horosphere) in T 1T is smooth and large.

Previous results on WP geodesic flow:

Transitivity: there exist dense geodesics [Brock-Masur-Minsky, alsoPollicott-Weiss-Wolpert for (1, 1) case].

Infinite topological entropy: there exist compact invariant sets withunboundedly large entropy [BMM].

Ergodic closing lemma: periodic measures are dense among ergodicprobability measures [Hamenstadt] .


The proof

The proof has three main ingredients:

Teichmuller theory

Differential geometry

Smooth ergodic theory


Input from Teichmuller theory:

• Asymptotics for WP metric and covariant derivative: Wolpertgives 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 (quasifuchsianspace) where ω extends to a holomorphic form with bounded primitive.Asymptotics obtained from Cauchy integral formula.

Differential geometry component:

• Analyze Jacobi equation (using Teich input) to bound first and secondderivatives of geodesic flow in terms of distance to ∂T .

Smooth ergodic theory:

• Modified Hopf argument: key property is absolute continuity. Onenovelty: Stable manifolds are complete! (thanks to geodesic convexity).


the weil-petersson geodesic ﬂow is ergodic t s · 2013-05-14 · the weil-petersson geodesic...

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