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:
Fix a cohomology class . Consider any such that . Then for any variation of , the Hodge norm of C satisfies
Here a variation in X is described by a smooth path in with . We use the shorthand , and adopt a similar convention for other quantities that depend on t. Note that the quadratic differential represents a cotangent vector to at X, so it pairs naturally with the tangent vector as in equation (2.2).
Fix a standard symplectic basis for . The associated Siegel period matrix for X is defined by
where is the basis for characterized by
The matrix is symmetric and positive-definite, and the norm of a general form is given by
Since equation (A.1) is homogeneous, we can assume . We can then choose a symplectic basis such that . With this normalization, we have
Now consider a variation of X. Then , and vary as well. By Ahlfors’ variational formula [Ah, equation (7)], we have
Let be the unique form in satisfying . Then
Since , we have . By equation (A.2) we also have , and hence . Using the fact that , this gives
Here is an equivalent formulation, used Section 4:
For any nonzero , we have
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 should be replaced by 1. Ahlfors’ formula (A.3) is sometimes stated without the factor , which results from the identity .
I would like to thank J. Chaika and G. Forni for useful discussions and references.
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