We consider the Teichmüller flow on the unit cotangent bundle of the moduli space of compact Riemann surfaces with punctures. We show that it is exponentially mixing for the Ratner class of observables. More generally, this result holds for the restriction of the Teichmüller flow to an arbitrary connected component of stratum. This result generalizes [AGY] which considered the case of strata of squares.
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Artur Avila, Maria João Resende, Exponential mixing for the Teichmüller flow in the space of quadratic differentials. Comment. Math. Helv. 87 (2012), no. 3, pp. 589–638DOI 10.4171/CMH/263