Exponential mixing for the Teichmüller flow

Artur Avila; Sébastien Gouëzel; Jean-Christophe Yoccoz

Publications Mathématiques de l'IHÉS (2006)

  • Volume: 104, page 143-211
  • ISSN: 0073-8301

Abstract

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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 S L ( 2 , ℝ ) action in the moduli space has a spectral gap.

How to cite

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Avila, Artur, Gouëzel, Sébastien, and Yoccoz, Jean-Christophe. "Exponential mixing for the Teichmüller flow." Publications Mathématiques de l'IHÉS 104 (2006): 143-211. <http://eudml.org/doc/104218>.

@article{Avila2006,
abstract = {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 $SL(2,\mathbb \{R\})$ action in the moduli space has a spectral gap.},
author = {Avila, Artur, Gouëzel, Sébastien, Yoccoz, Jean-Christophe},
journal = {Publications Mathématiques de l'IHÉS},
language = {eng},
pages = {143-211},
publisher = {Springer},
title = {Exponential mixing for the Teichmüller flow},
url = {http://eudml.org/doc/104218},
volume = {104},
year = {2006},
}

TY - JOUR
AU - Avila, Artur
AU - Gouëzel, Sébastien
AU - Yoccoz, Jean-Christophe
TI - Exponential mixing for the Teichmüller flow
JO - Publications Mathématiques de l'IHÉS
PY - 2006
PB - Springer
VL - 104
SP - 143
EP - 211
AB - 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 $SL(2,\mathbb {R})$ action in the moduli space has a spectral gap.
LA - eng
UR - http://eudml.org/doc/104218
ER -

References

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  1. 1. A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, preprint (www.arXiv.org), to appear in Ann. Math. Zbl1136.37003MR2299743
  2. 2. A. Avila and M. Viana, Simplicity of Lyapunov spectra: proof of the Zorich–Kontsevich conjecture, to appear in Acta Math. Zbl1143.37001MR2316268
  3. 3. J. Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50. American Mathematical Society, Providence, RI, 1997. Zbl0882.28013MR1450400
  4. 4. J. Athreya, Quantitative recurrence and large deviations for Teichmüller geodesic flow, Geom. Dedicata, 119 (2006), 121-140 Zbl1108.32007MR2247652
  5. 5. V. Baladi, B. Vallée, Exponential decay of correlations for surface semi-flows without finite Markov partitions, Proc. Amer. Math. Soc., 133 (2005), 865-874 Zbl1055.37027MR2113938
  6. 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 Zbl1100.37002MR2220100
  7. 7. D. Dolgopyat, On decay of correlations in Anosov flows, Ann. Math. (2), 147 (1998), 357-390 Zbl0911.58029MR1626749
  8. 8. A. Eskin, H. Masur, Asymptotic formulas on flat surfaces, Ergod. Theory Dynam. Syst., 21 (2001), 443-478 Zbl1096.37501MR1827113
  9. 9. G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. Math. (2), 155 (2002), 1-103 Zbl1034.37003MR1888794
  10. 10. H. Hennion, Sur un théorème spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634 Zbl0772.60049MR1129880
  11. 11. S.P. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergod. Theory Dynam. Syst., 5 (1985), 257-271 Zbl0597.58024MR796753
  12. 12. M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678 Zbl1087.32010MR2000471
  13. 13. G.A. Margulis, A. Nevo, E.M. Stein, Analogs of Wiener’s ergodic theorems for semisimple Lie groups. II, Duke Math. J., 103 (2000), 233-259 Zbl0978.22006
  14. 14. S. Marmi, P. Moussa, J.-C. Yoccoz, The cohomological equation for Roth type interval exchange transformations, J. Amer. Math. Soc., 18 (2005), 823-872 Zbl1112.37002MR2163864
  15. 15. H. Masur, Interval exchange transformations and measured foliations, Ann. Math. (2), 115 (1982), 169-200 Zbl0497.28012MR644018
  16. 16. M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergod. Theory Dynam. Syst., 7 (1987), 267-288 Zbl0623.22008MR896798
  17. 17. G. Rauzy, Echanges d’intervalles et transformations induites, Acta Arith., 34 (1979), 315-328 Zbl0414.28018
  18. 18. W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math. (2), 115 (1982), 201-242 Zbl0486.28014MR644019
  19. 19. W. Veech, The Teichmüller geodesic flow, Ann. Math. (2), 124 (1986), 441-530 Zbl0658.32016MR866707
  20. 20. A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier, 46 (1996), 325-370 Zbl0853.28007MR1393518

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