×

zbMATH — the first resource for mathematics

Exponential mixing for the Teichmüller flow. (English) Zbl 1263.37051
From the text: We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the action in the moduli space has a spectral gap.
This paper has two main parts: first we obtain exponential recurrence estimates, and then, using ideas first introduced by D. Dolgopyat [Ann. Math. (2) 147, 357–390 (1998; Zbl 0911.58029)] and developed in [V. Baladi and B. Vallée, Proc. Am. Math. Soc. 133, No. 3, 865–874 (2005; Zbl 1055.37027)], we obtain exponential mixing. The proof of exponential recurrence uses an “induction on the complexity” scheme. Intuitively, the dynamics at “infinity” of the Teichmüller flow can be partially described by the dynamics in simpler (lower dimensional) connected components of strata, and we obtain estimates by induction all the way from the simplest of the cases. A simpler version of this scheme was used to show some combinatorial richness of the Teichmüller flow in the proof of the Zorich-Kontsevich conjecture [A. Avila and M. Viana, Acta Math. 198, No. 1, 1–56 (2007; Zbl 1143.37001)]. The recurrence estimates thus obtained are close to optimal.
It should be noted that our work does not use the \(\mathrm{SL}(2,\mathbb R)\) action for the estimates, and can be used to obtain new proofs of some previously known results which used to depend on the \(\mathrm{SL}(2,\mathbb R)\) action. In the other direction, however, it was pointed out to us by Bufetov that our main theorem has an important new corollary for the \(\mathrm{SL}(2,\mathbb R)\) action. It regards the nature of the corresponding unitary representation of \(\mathrm{SL}(2,\mathbb R)\) on the space \(L^2_0(\cdot)\) of \(L^2\) zero-average functions.

MSC:
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37A25 Ergodicity, mixing, rates of mixing
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, preprint (www.arXiv.org), to appear in Ann. Math. · Zbl 1136.37003
[2] A. Avila and M. Viana, Simplicity of Lyapunov spectra: proof of the Zorich–Kontsevich conjecture, to appear in Acta Math. · Zbl 1143.37001
[3] J. Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50. American Mathematical Society, Providence, RI, 1997. · Zbl 0882.28013
[4] J. Athreya, Quantitative recurrence and large deviations for Teichmüller geodesic flow, Geom. Dedicata, 119 (2006), 121–140. · Zbl 1108.32007 · doi:10.1007/s10711-006-9058-z
[5] V. Baladi and B. Vallée, Exponential decay of correlations for surface semi-flows without finite Markov partitions, Proc. Amer. Math. Soc., 133 (2005), 865–874. · Zbl 1055.37027 · doi:10.1090/S0002-9939-04-07671-3
[6] A. Bufetov, Decay of correlations for the Rauzy–Veech–Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials, J. Amer. Math. Soc., 19 (2006), 579–623. · Zbl 1100.37002 · doi:10.1090/S0894-0347-06-00528-5
[7] D. Dolgopyat, On decay of correlations in Anosov flows, Ann. Math. (2), 147 (1998), 357–390. · Zbl 0911.58029 · doi:10.2307/121012
[8] A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergod. Theory Dynam. Syst., 21 (2001), 443–478. · Zbl 1096.37501 · doi:10.1017/S0143385701001225
[9] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. Math. (2), 155 (2002), 1–103. · Zbl 1034.37003 · doi:10.2307/3062150
[10] H. Hennion, Sur un théorème spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118 (1993), 627–634. · Zbl 0772.60049
[11] S. P. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergod. Theory Dynam. Syst., 5 (1985), 257–271. · Zbl 0597.58024 · doi:10.1017/S0143385700002881
[12] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631–678. · Zbl 1087.32010 · doi:10.1007/s00222-003-0303-x
[13] G. A. Margulis, A. Nevo, and E. M. Stein, Analogs of Wiener’s ergodic theorems for semisimple Lie groups. II, Duke Math. J., 103 (2000), 233–259. · Zbl 0978.22006 · doi:10.1215/S0012-7094-00-10323-7
[14] S. Marmi, P. Moussa, and J.-C. Yoccoz, The cohomological equation for Roth type interval exchange transformations, J. Amer. Math. Soc., 18 (2005), 823–872. · Zbl 1112.37002 · doi:10.1090/S0894-0347-05-00490-X
[15] H. Masur, Interval exchange transformations and measured foliations, Ann. Math. (2), 115 (1982), 169–200. · Zbl 0497.28012 · doi:10.2307/1971341
[16] M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergod. Theory Dynam. Syst., 7 (1987), 267–288. · Zbl 0623.22008 · doi:10.1017/S0143385700004004
[17] G. Rauzy, Echanges d’intervalles et transformations induites, Acta Arith., 34 (1979), 315–328. · Zbl 0414.28018
[18] W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math. (2), 115 (1982), 201–242. · Zbl 0486.28014 · doi:10.2307/1971391
[19] W. Veech, The Teichmüller geodesic flow, Ann. Math. (2), 124 (1986), 441–530. · Zbl 0658.32016 · doi:10.2307/2007091
[20] A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier, 46 (1996), 325–370. · Zbl 0853.28007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.