 
Summary: EXPONENTIAL MIXING FOR THE TEICHM ¨ULLER FLOW
ARTUR AVILA, S´EBASTIEN GOU¨EZEL AND JEANCHRISTOPHE YOCCOZ
Abstract. We study the dynamics of the Teichm¨uller flow in the moduli space of Abelian differ
entials (and more generally, its restriction to any connected component of a stratum). We show
that the (MasurVeech) absolutely continuous invariant probability measure is exponentially mix
ing for the class of H¨older observables. A geometric consequence is that the SL(2, R) action in the
moduli space has a spectral gap.
Contents
1. Introduction 1
2. Statements of the results 3
3. The Veech flow 9
4. Reduction to recurrence estimates 13
5. A distortion estimate 20
6. Proof of the recurrence estimates 27
7. Exponential mixing for expanding semiflows 29
8. Exponential mixing for hyperbolic semiflows 43
Appendix A. A simple distortion estimate 45
Appendix B. Spectral gap 46
References 49
1. Introduction
