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

We study the horocycle flow on the stratum of abelian differentials H(2). We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but non-ergodic, measure by SL2(R). As a consequence, we show that there are points in H(2) whose horocycle flow orbits do not equidistribute towards any invariant measure