PUBLICATIONS MATHÉMATIQUES DE L'IHES

Online first
Exponential mixing for the Teichmüller flow
Publications Mathématiques de l'IHÉS, Volume 104 (2006), pp. 143-211
PDF

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,ℝ) action in the moduli space has a spectral gap.

Received:
Online First:
Published online:
DOI: 10.1007/s10240-006-0001-5 
@article{PMIHES_2006__104__143_0,
     author = {Artur Avila and S\'ebastien Gou\"ezel and Jean-Christophe Yoccoz},
     title = {Exponential mixing for the {Teichm\"uller} flow},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {143--211},
     year = {2006},
     publisher = {Springer},
     volume = {104},
     doi = {10.1007/s10240-006-0001-5},
     mrnumber = {2264836},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-006-0001-5/}
}
TY  - JOUR
AU  - Artur Avila
AU  - Sébastien Gouëzel
AU  - Jean-Christophe Yoccoz
TI  - Exponential mixing for the Teichmüller flow
JO  - Publications Mathématiques de l'IHÉS
PY  - 2006
SP  - 143
EP  - 211
VL  - 104
PB  - Springer
UR  - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-006-0001-5/
DO  - 10.1007/s10240-006-0001-5
LA  - en
ID  - PMIHES_2006__104__143_0
ER  - 
%0 Journal Article
%A Artur Avila
%A Sébastien Gouëzel
%A Jean-Christophe Yoccoz
%T Exponential mixing for the Teichmüller flow
%J Publications Mathématiques de l'IHÉS
%D 2006
%P 143-211
%V 104
%I Springer
%U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-006-0001-5/
%R 10.1007/s10240-006-0001-5
%G en
%F PMIHES_2006__104__143_0
Artur Avila; Sébastien Gouëzel; Jean-Christophe Yoccoz. Exponential mixing for the Teichmüller flow. Publications Mathématiques de l'IHÉS, Volume 104 (2006), pp. 143-211. doi: 10.1007/s10240-006-0001-5

1. A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, preprint (www.arXiv.org), to appear in Ann. Math. | MR

2. A. Avila and M. Viana, Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture, to appear in Acta Math. | MR

3. J. Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50. American Mathematical Society, Providence, RI, 1997. | Zbl | MR

4. J. Athreya, Quantitative recurrence and large deviations for Teichmüller geodesic flow, Geom. Dedicata, 119 (2006), 121-140 | Zbl | MR

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 | Zbl | MR

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 | MR

7. D. Dolgopyat, On decay of correlations in Anosov flows, Ann. Math. (2), 147 (1998), 357-390 | Zbl | MR

8. A. Eskin, H. Masur, Asymptotic formulas on flat surfaces, Ergod. Theory Dynam. Syst., 21 (2001), 443-478 | Zbl | MR

9. G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. Math. (2), 155 (2002), 1-103 | Zbl | MR

10. H. Hennion, Sur un théorème spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634 | Zbl | MR

11. S.P. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations, Ergod. Theory Dynam. Syst., 5 (1985), 257-271 | Zbl | MR

12. M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678 | Zbl | MR

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 | Zbl

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 | Zbl | MR

15. H. Masur, Interval exchange transformations and measured foliations, Ann. Math. (2), 115 (1982), 169-200 | Zbl | MR

16. M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergod. Theory Dynam. Syst., 7 (1987), 267-288 | Zbl | MR

17. G. Rauzy, Echanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328 | Zbl | EuDML

18. W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. Math. (2), 115 (1982), 201-242 | Zbl | MR

19. W. Veech, The Teichmüller geodesic flow, Ann. Math. (2), 124 (1986), 441-530 | Zbl | MR

20. A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier, 46 (1996), 325-370 | Numdam | Zbl | MR | EuDML

Cited by Sources: