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

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.

@article{PMIHES_2006__104__143_0,
     author = {Avila, Artur and Gou\"ezel, S\'ebastien and Yoccoz, Jean-Christophe},
     title = {Exponential mixing for the {Teichm\"uller} flow},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {143--211},
     publisher = {Springer},
     volume = {104},
     year = {2006},
     doi = {10.1007/s10240-006-0001-5},
     mrnumber = {2264836},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-006-0001-5/}
}
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
SP  - 143
EP  - 211
VL  - 104
PB  - Springer
UR  - http://www.numdam.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 Avila, Artur
%A Gouëzel, Sébastien
%A Yoccoz, Jean-Christophe
%T Exponential mixing for the Teichmüller flow
%J Publications Mathématiques de l'IHÉS
%D 2006
%P 143-211
%V 104
%I Springer
%U http://www.numdam.org/articles/10.1007/s10240-006-0001-5/
%R 10.1007/s10240-006-0001-5
%G en
%F PMIHES_2006__104__143_0
Avila, Artur; Gouëzel, Sébastien; Yoccoz, Jean-Christophe. Exponential mixing for the Teichmüller flow. Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 143-211. doi : 10.1007/s10240-006-0001-5. http://www.numdam.org/articles/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. | MR | Zbl

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

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 | MR | Zbl

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 | MR | Zbl

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

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

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

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

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

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

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 | MR | Zbl

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

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

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

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

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

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

Cité par Sources :