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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
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References:
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