On the space of ergodic measures for the horocycle flow on strata of Abelian differentials

We study the horocycle flow on the stratum of translation surfaces H (2), showing that there is a sequence of horocycle ergodic measures, supported on a sequence of periodic horocycle orbits, which weak-* converges to an SL₂ (R) -invariant but not ergodic measure. As a consequence, we show that there are points in H (2) that do not equidistribute under the horocycle flow for any measure.