# Teichmüller dynamics and unique ergodicity via currents and Hodge theory

Curtis T. McMullen

# Abstract

We present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

Funding statement: Research supported in part by the NSF.

## A Appendix: Variation of the Hodge norm

We will show that a classical result of Ahlfors gives:

## Theorem A.1.

Fix a cohomology class C H 1 ( Σ g , R ) . Consider any ( X , ω ) Ω T g such that [ Re ω ] = C . Then for any variation of X T g , the Hodge norm of C satisfies

(A.1) ( C X 2 ) ˙ = - 2 Re ω 2 , X ˙ .

Here a variation in X is described by a smooth path X ( t ) in 𝒯 g with X ( 0 ) = X . We use the shorthand X ˙ = X ( 0 ) , and adopt a similar convention for other quantities that depend on t. Note that the quadratic differential ω 2 represents a cotangent vector to 𝒯 g at X, so it pairs naturally with the tangent vector X ˙ as in equation (2.2).

## Proof.

Fix a standard symplectic basis ( a 1 , , a g ) , ( b 1 , , b g ) for H 1 ( Σ g , ) . The associated Siegel period matrix for X is defined by

τ i j = b j ω i ,

where ( ω 1 , , ω g ) is the basis for Ω ( X ) characterized by

a i ω j = δ i j .

The matrix σ i j = Im τ i j is symmetric and positive-definite, and the norm of a general form η = 1 g s i ω i Ω ( X ) is given by

η X 2 = X | η | 2 = s t σ s ¯ .

Since equation (A.1) is homogeneous, we can assume ω X = C X = 1 . We can then choose a symplectic basis such that ω 1 = ω . With this normalization, we have

(A.2) C , a i = Re a i ω 1 = δ i 1 .

Now consider a variation X ( t ) of X. Then ω i , τ i j and σ i j vary as well. By Ahlfors’ variational formula [Ah, equation (7)], we have

(A.3) τ ˙ i j = - 2 i ω i ω j , X ˙ .

Let ω ( t ) = s i ( t ) ω i ( t ) be the unique form in Ω ( X ( t ) ) satisfying [ Re ω ( t ) ] = C . Then

( C X 2 ) ˙ = ( s t σ s ¯ ) ˙ .

Since ω ( 0 ) = ω 1 , we have s i ( 0 ) = δ i 1 . By equation (A.2) we also have Re s i = δ i 1 , and hence Re s ˙ = 0 . Using the fact that σ t = σ , this gives

s ˙ t σ s ¯ + s t σ s ¯ ˙ = 2 ( Re ( s ˙ ) t σ s ¯ ) = 0 ,

and therefore

( s t σ s ¯ ) ˙ = s σ ˙ s ¯ = σ ˙ 11 .

Formula (A.1) then follows directly from Ahlfors variational formula (A.3). ∎

Here is an equivalent formulation, used Section 4:

## Corollary A.2.

For any nonzero C H 1 ( Σ g , R ) , we have

( log C X ) ˙ = - Re ω 2 , X ˙ ,

where [ Re ω ] = C C X .

### Notes and references

A variant of Theorem A.1, with a different proof, is given in [Fo, Lemma 2.1 ]. A precursor to (A.3) appears in [Ra, (7)], where the factor 1 / 2 π i should be replaced by 1. Ahlfors’ formula (A.3) is sometimes stated without the factor - 2 i , which results from the identity d z d z ¯ = - 2 i | d z | 2 .

# Acknowledgements

I would like to thank J. Chaika and G. Forni for useful discussions and references.

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Revised: 2019-10-03
Published Online: 2019-12-08
Published in Print: 2020-11-01

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