Measures on Good Pants

MIXING OF THE FRAME FLOW

3.4 Measures on Good Pants

Let ˜µbe the measure on well-connected pairs of tripods given by dµ(T˜ (z),T¯(w), γ) =bγ(T(z),T¯(w))dλT(z,w, γ),

where λT(z,w, γ) is the product of the Liouville measure Λon the first two terms and the counting measure on γ. Sinceb_γ(T(z),T¯(w)) has compact support, so ˜µ is finite. To each pair of (,r) well-connected tripods, we can associate a pair of pantsΠ. We denoteΠ = π(T(z),T¯(w), γ). In the previous section, we proved that π maps well-connected pairs of tripods to good pairs of pants. So the projection induces a measure on good pants by µ= π∗µ˜. We will show that this measure µis a real (positive) solution to the system of inequalities in Claim 3.4.1.

To any framezwe can associate a bipodB(z) = (z, ω(z)), and likewise to any frame zwe associate the “anti-bipod” ¯B(z) = (z,ω(z))¯ . We say that

B(z),B(w), γ¯ ₀, γ₁ is a well connected pair of bipods along the pair of segmentγ₀andγ₁if

aγ₀(z,w)aγ₁(ω(z),ω¯(w)) > 0.

Let δ be the closed geodesic that is homotopic toγ₀∪γ₁. For any third connec- tionγ₂ such thataγ₂(ω(z), ω(w)) > 0, we can get a pair of well-connected tripods (T(z),T¯(w), γ₀, γ₁, γ₂) from the bipods. Denote by fδ(T(z),T¯(w), γ₀, γ₁) the pre- dicted feet. Then, by Lemma 3.3.2, no matter what the third connection γ₂ is, we have

d(f_δ(T(z),T¯(w), γ₀, γ₁),feetδ(T(z),T¯(w), γ)) ≤ De^−r4

for some constant D > 0. Since fδ doesn’t rely on the third connectionγ₂, so we can definefδ on well connected pairs of bipods by:

fδ(B(z),B(w), γ¯ ₀, γ₁) =fδ(T(z),T¯(w), γ₀, γ₁).

LetSδ be the set of well connected bipods (B(z),B¯(w), γ₀, γ₁) such thatγ₀∪γ₁is homotopic toδ. The set ofS_δ carries the natural measure λB which is the product of the Liouville measures on the first two terms and the counting measure on the last two terms. Let Cδ be the set of well-connected tripods (T(z),T¯(w), γ) for which γ₀∪γ₁ is homotopic to δ, and let χ : Cδ → Sδ be the forgetting map, so

χ(T(z),T¯(w), γ₀, γ₁, γ₂) = (B(z),B(w), γ¯ ₀, γ₁). Then by definition, we have

∂ µ

N²(δ) = (feetδ)∗(µ˜

^Cδ). (3.21)

AsfδisDe^−r4-close to feetδ, so it suffices to prove that(fδ)∗(µ|˜Cδ)andA_∗((fδ)∗(µ|˜Cδ)) are _R⁹2-equivalent when Ris large. Let fδ = (f_δ¹, f_δ²), where f¹ is the first projec- tion and f² is the second projection. By Remark 1.3.3, it is enough to show that (f_δ¹)∗((µ˜|_Cδ) and A_∗((f_δ¹)∗(µ˜|_Cδ)) are _R⁶₂-equivalent. Since fδ is well defined on well connected bipods, we have (f_δ¹)∗(µ|˜C_δ) = (f_δ¹)∗(χ∗(µ|˜C_δ)).

We will consider two natural measures onSδ. The first one is χ∗(µ|˜C_δ). The other isνδ, defined on Sδby

dνδ(B(z),B¯(w), γ₀, γ₁) =a_γ

0(z,w)a_γ

1(ω(z),ω(w))dλ¯ B(z,w, γ₀, γ₁),

where we recall that λB(z,w, γ₀, γ₁) is the product of the Liouville measure on the first two terms and the counting measure on the last two. Then by Corollary 3.0.6, the two measures satisfy the fundamental inequality

dχ∗(µ|˜Cδ)

dνδ(B(z),B(w), γ¯ ₀, γ₁) − 1 Λ(F(M))

< De^−qr2. (3.22)

Now we study the symmetry onSδ. Choose a pointx₀onδ, and denoteF the fiber ofx₀inN(δ). ThenF is isometric toSⁿ⁻². Let Abe the monodromy ofδ. ThenA acts onF, then−2 dimensional sphere. We can choose an(n−1)-frame inFsuch that under this frame the monodromy Aacts onF as the following block matrix

* . . . . . . . . . . ,

cosθ₁ sinθ₁ · · · 0 0

−sinθ₁ cosθ₁ · · · 0 0

... ... . . . ... ...

0 0 · · · cosθk sinθk

0 0 · · · −sinθk cosθk

+ / / / / / / / / / / -

for someθ₁,· · ·, θk, wherek = ⁿ⁻₂¹. LetC(A)be the isometry group of N(δ)such that each element preservesF and it acts onF as the following matrix

* . . . . . . . . . . ,

cosϕ₁ sinϕ₁ · · · 0 0

−sinϕ₁ cosϕ₁ · · · 0 0

... ... . . . ... ...

0 0 · · · cosϕk sinϕk

0 0 · · · −sinϕk cosϕk

+ / / / / / / / / / / -

for some 0 ≤ ϕ₁,· · · , ϕk < 2π.Then Acommutes with all elements of C(A). As before, we denote by At the parallel transport of a distancetalongδ. Then, for any

t ∈ R, At commutes with all elements ofC(A). Let I(A) = C(A)× {At : 0 ≤ t <

l(δ)}, thenI(A)is an isometry group on N(δ). We have the following inequality aγ₀(z,w)aγ₁(ω(z),ω¯(w)) =ag(γ₀) g(z),g(w)

ag(γ₁) g(ω(z)),g(ω¯(w)), for allg ∈ I(A), as the affinity functionais left invariant. Also,I(A)naturally acts onSδas isometry, so the group action leaves invariant the measureλ(B). It follows thatI(A)will leave the measureνB invariant.

On the other hand, the predicted feet f_δ is equivariant under the group action of I(A); that is, for eachg ∈I(A), we have

fδ

B(g(z)),B(g(w)),¯ g(γ₀),g(γ₁)

=g

fδ(B(z),B(w), γ¯ ₀, γ₁) .

It follows from the above two observations that the measure (fδ)∗νB is invariant under the I(A)action. By projection, we get that the measure (f_δ¹)∗νB on N(δ) is invariant under the I(A)action. Since I(δ) may not be transitive on N(δ), it is not necessary that (f_δ¹)∗νB is a multiple of the Euclidean measure on N(δ). However, for anyv ∈ N(δ), the groupI(A)acts transitively onI(A)·v, so, when restricted on the orbitI(A)·v, the measure(f_δ¹)∗νB is a multiple of the Euclidean measure. By the definition ofC(A)andI(A), the orbitI(A)·vis a torusT^mfor some 2 ≤ m ≤ k. Notice that the antipodal mapOis inC(A). SoA = A₁·Ois also inI(A). Hence, when restricted on a orbit I(A) ·v, the measure A_∗((f_δ¹)∗νB) is a multiple of the Euclidean measure. Denoteβδ = (f_δ¹)∗(χ∗(µ|˜Cδ)), and denoteλto be the Euclidean measure on the orbitI(A)·v. Then by Equation (3.22), when restricted onI(A)·v, there exists some constantKδ such that

Kδ ≤

dβδ

dλ

≤ Kδ(1+D₂e^−qR2), whereD₂ > 0 is some constant.

The following lemma shows that any C⁰ measure on the torus T^m that is close to the Euclidean measure is obtained by pull-back the Euclidean measure by a diffeomorphism that isC⁰-close to the identity.

Lemma 3.4.2. Letg :R^m → Rbe aC⁰function onR^m that is well defined on the quotientT^m =R^m/Z^m, and such that:

1. For some0 < δ≤ ¹

4, we have

1−δ ≤ g(x)¯ ≤ 1+δ

for allx¯ ∈R^m.

2. The following equality holds:

Z

[0,1]^m

g(x)d¯ x¯ =1, wheredx¯ = dx₁· · ·dxm is the volume form.

Then we can find aC¹diffeomorphismh :T^m →T^m such that:

1. g(x)d¯ x¯ = h^∗(dx), where¯ h^∗(dx)¯ is the pull-back of the volume form by the diffeomorphismh.

2. There exists some constantD > 0such that the inequality

||h(x¯)− x|| ≤¯ Dδ holds for everyx¯ ∈R^m.

SinceKδ ≤

dβδ

dλ

≤ Kδ(1+D₂e^−qR2)and the Euclidean measureλis invariant under A, we have

Kδ ≤

dA^∗(βδ) dλ

≤ Kδ(1+D₂e^−qR2).

By Lemma 3.4.2, when restricted on a orbit I(A)· v, the measure βδ is De^−qR4- equivalent with the measure K₁λ for some K₁ > 0 and some constant D > 0.

Similarly, the measureA^∗(β_δ)isDe^−qR4-equivalent with the measureK₂λfor some K₂ > 0. AsA^∗(βδ) and βδ have the same total measure on the orbit I(A)·v, so we can chooseK₂ = K₁. Then by Proposition A.2.2, βδ andA^∗(βδ) are 2De^−qR4- equivalent measures on the orbitI(A)·v. Notice that N(δ)is the union of all orbits SI(A)· v. As 2De^−qR4 will be universally less than _R¹2 when R is large enough, the two measures βδandA^∗(βδ) are _R¹2-equivalent measures on N(δ). Recall that

βδ = (f_δ¹)∗(χ∗(µ˜|_Cδ)).So we have shown that

(f_δ)∗(µ˜|_Cδ)andA_∗((f_δ)∗(µ˜|_Cδ)

are _R¹2-equivalent measures onN(δ). From equation 3.21, we have A_∗(∂ µ)and∂ µ

are _R¹2-equivalent.

This just shows that µis a real positive solution to the linear system. Since both measuresµand∂ µare atomic, the linear system has only finitely many inequalities.

Then the standard rationalization procedure implies that this system has a positive integral solution. The existence of the surfaceS just follows from Claim 3.4.1.

A p p e n d i x A