Classification of ergodic invariant measures for the Teichmüller horocycle flow

Classify all ergodic probability measures that are invariant under the Teichmüller horocycle flow (the unipotent subgroup U-action of SL_2(R)) on each stratum and component of the moduli space of unit-area holomorphic quadratic differentials Q^1. Provide a complete Ratner-type description of these measures for the horocycle flow on strata of quadratic differentials, beyond the special cases already resolved.

Background

The paper studies a measurable bijection O between the earthquake flow on the hyperbolic side and the horocycle flow on the flat (quadratic differentials) side, and develops continuity properties that allow transport of ergodic-theoretic results between these settings.

While Ratner-type classifications are known for the SL_2(R) action on strata of quadratic differentials and for certain related settings, the authors point out that the analogous problem for the Teichmüller horocycle flow (the U-action) remains unresolved in general, with complete answers known only in special cases (e.g., some eigenform loci or specific strata). This classification problem is central to understanding the invariant measures and orbit closures of the horocycle flow in Teichmüller dynamics.