Exponential mixing for the Teichmüller flow in the space of quadratic differentials (Q439891)

The authors study the Teichmüller flow on the unit cotangent bundle \({\mathcal O}\) of the moduli space associated with a cofinite Riemann surface. Their main result establishes that for observables in the Ratner class, this flow is exponentially mixing with respect to the Liouville-like measure, obtained by pushing down the canonical (infinite) Liouville measure (which is invariant under the \(\mathrm{SL}_2({\mathbb R})\)-action on \({\mathcal O}\)) to a finite measure on the quotient of \({\mathcal O}\) by the associated modular group. Note that it is well known that this type of mixing property is equivalent to the so-called ``spectral gap property'' for the \(\mathrm{SL}_2({\mathbb R})\)-action on \({\mathcal O}\). The paper then continues by extending this main result to each connected component of the associated stratum (which is an analytic orbifold invariant under the Teichmüller flow). The paper finishes by discussing some consequences for interval exchange transformations with involution as well as for half-translation surfaces.