Exponential mixing of the Teichmüller flow on affine invariant manifolds

The uploaded preprint proves that for any SL(2,R)-invariant ergodic probability measure supported on a component of a stratum in Q(S) or H(S), the Teichmüller flow is exponentially mixing, by building a symbolic coding on each affine invariant manifold and applying the Avila–Gouëzel–Yoccoz criterion once a “good” roof function with exponential tails is verified. The main theorem is stated explicitly (An SL(2,R)-invariant ergodic probability measure on Q(S) or H(S) is exponentially mixing for the Teichmüller flow) and the proof strategy (Sections 3–6) closely follows AGY with a coding based on train tracks and an entropy/pressure argument for exponential tails, culminating in Theorem 6.4 that asserts exponential mixing for the SL(2,R)-invariant measure on affine invariant manifolds . The reduction from quadratic to abelian strata via the orientation double cover is stated in the setup of Section 3, so symbolic coding can be restricted to abelian strata and then descends back to the quadratic case . The construction of a countable-state Markov shift, its one-sided version, the verification of a hyperbolic skew product structure and applicability of AGY’s mixing theorem are spelled out in Sections 3–6, including Proposition 6.3 and the final application of AGY (Theorem 7.3 in AGY06) to conclude exponential mixing for the suspension, which transfers to the flow on C via a finite-to-one semiconjugacy of full measure . However, the paper purposely does not specify the precise class of observables (it refers the reader to AGY06, Section 7, for the admissible test functions) and does not claim any uniformity of mixing constants across all affine invariant manifolds inside a stratum . The candidate solution, while following essentially the same proof outline, makes two additional claims not supported by the paper: (i) it asserts exponential correlation decay for all f,g in C^α(X) as defined by the Euclidean period-coordinate metric on the ambient component, and (ii) it claims the constants and rates depend only on S, X, α but not on the particular affine invariant manifold or measure. Neither uniform norm-to-norm comparison with period-coordinate Hölder classes nor uniformity of rates across all affine invariant submanifolds is established in the paper; the paper verifies the AGY hypotheses in a specific Finsler/symbolic framework and deliberately refrains from specifying a global class of observables or uniform constants. Thus the paper’s main claim is correct on its stated scope, but the candidate solution overstates the function class and uniformity of constants.