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.