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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.

References

[Ah] L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, Analytic functions, Princeton University Press, Princeton (1960), 45–66. 10.1515/9781400876709-004Search in Google Scholar

[AF] J. S. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards, Duke Math. J. 144 (2008), no. 2, 285–319. 10.1215/00127094-2008-037Search in Google Scholar

[dR] G. de Rham, Differentiable manifolds. Forms, currents, harmonic forms, Grundlehren Math. Wiss. 266, Springer, Berlin 1984. 10.1007/978-3-642-61752-2Search in Google Scholar

[EKZ] A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 207–333. 10.1007/s10240-013-0060-3Search in Google Scholar

[FLP] A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces. Séminaire Orsay, With an English summary, Astérisque 66, Société Mathématique de France, Paris 1979. Search in Google Scholar

[Fi] J. Fickenscher, Self-inverses, Lagrangian permutations and minimal interval exchange transformations with many ergodic measures, Commun. Contemp. Math. 16 (2014), no. 1, Article ID 1350019. 10.1142/S0219199713500193Search in Google Scholar

[Fo] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), no. 1, 1–103. 10.2307/3062150Search in Google Scholar

[FM] G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn. 8 (2014), no. 3–4, 271–436. 10.3934/jmd.2014.8.271Search in Google Scholar

[FMZ] G. Forni, C. Matheus and A. Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle, Ergodic Theory Dynam. Systems 34 (2014), 353–408. 10.1017/etds.2012.148Search in Google Scholar

[Ga] F. P. Gardiner, Teichmüller theory and quadratic differentials, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York 1987. Search in Google Scholar

[GH] P. Griffiths and J. Harris, Principles of algebraic geometry, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York 1978. Search in Google Scholar

[IT] Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer, Tokyo 1992. 10.1007/978-4-431-68174-8Search in Google Scholar

[Ka] A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR 211 (1973), 775–778. Search in Google Scholar

[Le] A. Leutbecher, Über die Heckeschen Gruppen G ( λ ) , Abh. Math. Semin. Univ. Hambg. 31 (1967), 199–205. 10.1007/BF02992399Search in Google Scholar

[Mas] H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), no. 3, 387–442. 10.1215/S0012-7094-92-06613-0Search in Google Scholar

[MT] H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems. Vol. 1A, North-Holland, Amsterdam (2002), 1015–1089. 10.1016/S1874-575X(02)80015-7Search in Google Scholar

[Mc1] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857–885. 10.1090/S0894-0347-03-00432-6Search in Google Scholar

[Mc2] C. T. McMullen, Diophantine and ergodic foliations on surfaces, J. Topol. 6 (2013), no. 2, 349–360. 10.1112/jtopol/jts033Search in Google Scholar

[Mc3] C. T. McMullen, Entropy on Riemann surfaces and the Jacobians of finite covers, Comment. Math. Helv. 88 (2013), no. 4, 953–964. 10.4171/CMH/308Search in Google Scholar

[Mc4] C. T. McMullen, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 1–39. 10.24033/asens.2237Search in Google Scholar

[Mc5] C. T. McMullen, Modular symbols for Teichmüller curves, preprint (2019). 10.1515/crelle-2021-0019Search in Google Scholar

[Mo1] M. Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), no. 2, 327–344. 10.1090/S0894-0347-05-00512-6Search in Google Scholar

[Mo2] M. Möller, Affine groups of flat surfaces, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13, European Mathematical Society, Zürich (2009), 369–387. 10.4171/055-1/11Search in Google Scholar

[Nag] S. Nag, The complex analytic theory of Teichmüller spaces, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York 1988. Search in Google Scholar

[Ra] H. E. Rauch, On the transcendental moduli of algebraic Riemann surfaces, Proc. Natl. Acad. Sci. USA 41 (1955), 42–49. 10.1073/pnas.41.1.42Search in Google Scholar PubMed PubMed Central

[Roy] H. L. Royden, Invariant metrics on Teichmüller space, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York (1974), 393–399. 10.1016/B978-0-12-044850-0.50036-8Search in Google Scholar

[Sul] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. 10.1007/BF01390011Search in Google Scholar

[Tr] R. Treviño, On the ergodicity of flat surfaces of finite area, Geom. Funct. Anal. 24 (2014), no. 1, 360–386. 10.1007/s00039-014-0269-4Search in Google Scholar

[V1] W. A. Veech, Interval exchange transformations, J. Analyse Math. 33 (1978), 222–272. 10.1007/BF02790174Search in Google Scholar

[V2] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. 10.1007/BF01388890Search in Google Scholar

Received: 2019-04-01
Revised: 2019-10-03
Published Online: 2019-12-08
Published in Print: 2020-11-01

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